GeoDif de CyS con Mathematica

Dante Yván Chavil Montenegro

Differential geometry of curves studies the properties of curves and higher-dimensional curved spaces using tools from calculus and linear algebra. This study has two aspects: the classical differential geometry which started with the beginnings of calculus and the global differential geometry which is the study of the influence of the local properties on the behavior of the entire curve. The local properties involves the properties which depend only on the behavior of the curve in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Due to this, the curves considered in differential geometry will be defined by functions which can be differentiated a certain number of times. The other aspect is the so-called global differential geometry which study the influence of the local properties on the behavior of the entire curve or surface. This paper aims to give an advanced introduction to the theory of curves, and those that are curved in general.

Alfred Gray Modern Differential Geometry of Curves and Surfaces with Mathematica R Third Edition by Elsa Abbena and Simon Salamon i Preface to the Second Edition 1 Modern Differential Geometry of Curves and Surfaces is a traditional text, but it uses the symbolic manipulation program Mathematica. This important computer program, available on PCs, Macs, NeXTs, Suns, Silicon Graphics Workstations and many other computers, can be used very effectively for plotting and computing. The book presents standard material about curves and surfaces, together with accurate interesting pictures, Mathematica instructions for making the pictures and Mathematica programs for computing functions such as curvature and torsion. Although Curves and Surfaces makes use of Mathematica, the book should also be useful for those with no access to Mathematica. All calculations mentioned in the book can in theory be done by hand, but some of the longer calculations might be just as tedious as they were for differential geometers in the 19th century. Furthermore, the pictures (most of which were done with Mathematica) elucidate concepts, whether or not Mathematica is used by the reader. The main prerequisite for the book is a course in calculus, both single variable and multi-variable. In addition, some knowledge of linear algebra and a few basic concepts of point set topology are needed. These can easily be obtained from standard sources. No computer knowledge is presumed. In fact, the book provides a good introduction to Mathematica; the book is compatible with both versions 2.2 and 3.0. For those who want to use Curves and Surfaces to learn Mathematica, it is advisable to have access to Wolfram’s book Mathematica for reference. (In version 3.0 of Mathematica, Wolfram’s book is available through the help menus.) Curves and Surfaces is designed for a traditional course in differential geometry. At an American university such a course would probably be taught at the junior-senior level. When I taught a one-year course based on Curves and Surfaces at the University of Maryland, some of my students had computer experience, others had not. All of them had acquired sufficient knowledge of Mathematica after one week. I chose not to have computers in my classroom because I needed the classroom time to explain concepts. I assigned all of the problems at the end of each chapter. The students used workstations, PCs 1 This is a faithful reproduction apart from the updating of chapter references. It already incorporated the Preface to the First Edition dating from 1993. ii and Macs to do those problems that required Mathematica. They either gave me a printed version of each assignment, or they sent the assignment to me by electronic mail. Symbolic manipulation programs such as Mathematica are very useful tools for differential geometry. Computations that are very complicated to do by hand can frequently be performed with ease in Mathematica. However, they are no substitute for the theoretical aspects of differential geometry. So Curves and Surfaces presents theory and uses Mathematica programs in a complementary way. Some of the aims of the book are the following. • To show how to use Mathematica to plot many interesting curves and surfaces, more than in the standard texts. Using the techniques described in Curves and Surfaces, students can understand concepts geometrically by plotting curves and surfaces on a monitor and then printing them. The effect of changes in parameters can be strikingly portrayed. • The presentation of pictures of curves and surfaces that are informative, interesting and accurate. The book contains over 400 illustrations. • The inclusion of as many topics of the classical differential geometry and surfaces as possible. In particular, the book contains many examples to illustrate important theorems. • Alleviation of the drudgery of computing things such as the curvature and torsion of a curve in space. When the curvature and torsion become too complicated to compute, they can be graphed instead. There are more than 175 miniprograms for computing various geometric objects and plotting them. • The introduction of techniques from numerical analysis into differential geometry. Mathematica programs for numerical computation and drawing of geodesics on an arbitrary surface are given. Curves can be found numerically when their torsion and curvature are specified. • To place the material in perspective through informative historical notes. There are capsule biographies with portraits of over 75 mathematicians and scientists. • To introduce interesting topics that, in spite of their simplicity, deserve to be better known. I mention triply orthogonal systems of surfaces (Chapter 19), Björling’s formula for constructing a minimal surface containing a given plane curve as a geodesic (Chapter 22) and canal surfaces and cyclides of Dupin as Maxwell discussed them (Chapter 20). iii • To develop a dialect of Mathematica for handling functions that facilitates the construction of new curves and surfaces from old. For example, there is a simple program to generate a surface of revolution from a plane curve. • To provide explicit definitions of curves and surfaces. Over 300 Mathematica definitions of curves and surfaces can be used for further study. The approach of Curves and Surfaces is admittedly more computational than is usual for a book on the subject. For example, Brioschi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing the curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library. For example, nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Frequently, I have been asked if new mathematical results can be obtained by means of computers. Although the answer is generally no, it is certainly the case that computers can be an effective supplement to pure thought, because they allow experimentation and the graphs provide insights into complex relationships. I hope that many research mathematicians will find Curves and Surfaces useful for that purpose. Two results that I found with the aid of Mathematica are the interpretation of torsion in terms of tube twisting in Chapter 7 and the construction of a conjugate minimal surface without integration in Chapter 22. I have not seen these results in the literature, but they may not be new. The programs in the book, as well as some descriptive Mathematica notebooks, will eventually be available on the web. Sample Course Outlines There is ample time to cover the whole book in three semesters at the undergraduate level or two semesters at the graduate level. Here are suggestions for courses of shorter length. • One semester undergraduate differential geometry course: Chapters 1, 2, 7, 9 – 13, parts of 14 – 16, 27. • Two semester undergraduate differential geometry course: Chapters 1 – 3, 9 – 19, 27. • One semester graduate differential geometry course: Chapters 1, 2, 7 – 13, 15 – 19, parts of 22 – 27. • One semester course on curves and their history: Chapters 1 – 8. iv • One semester course on Mathematica and graphics Chapters 1 – 6, 7 – 11, parts of 14 – 19, 23, and their notebooks. I have tried to include more details than are usually found in mathematics books. This, plus the fact that the Mathematica programs can be used to elucidate theoretical concepts, makes the book easy to use for independent study. Curves and Surfaces is an ongoing project. In the course of writing this book, I have become aware of the vast amount of material that was well-known over a hundred years ago, but now is not as popular as it should be. So I plan to develop a web site, and to write a problem book to accompany the present text. Spanish, German, Japanese and Italian versions of Curves and Surfaces are already available. Graphics Although Mathematica graphics are very good, and can be used to create QuickTime movies, the reader may also want to consider the following additional display methods: • Acrospin is an inexpensive easy-to-use program that works on even the humblest PC. • Geomview is a program for interactive display of curves and surfaces. It works on most unix-type systems, and can be freely downloaded from http://www.geomview.org • Dynamic Visualizer is an add-on program to Mathematica that allows interactive display. Details are available from http://www.wolfram.com • AVS programs (see the commercial site http://www.avs.com) have been developed by David McNabb at the University of Maryland (http://www.umd.edu) for spectacular stereo three-dimensional images of the surfaces described in this book. A Perspective Mathematical trends come and go. R. Osserman in his article (‘The Geometry Renaissance in America: 1938–1988’ in A Century of Mathematics in America, volume 2, American Mathematical Society, Providence, 1988) makes the point that in the 1950s when he was a student at Harvard, algebra dominated mathematics, the attention given to analysis was small, and the interest in differential geometry was converging to zero. It was not always that way. In the last half of the 19th century surface theory was a very important area of mathematics, both in research and teaching. Brill, v then Schilling, made an extensive number of plaster models available to the mathematical public. Darboux’s Leçons sur la Théorie Générale des Surfaces and Bianchi’s Lezioni di Geometria Differenziale were studied intensely. I attribute the decline of differential geometry, especially in the United States, to the rise of tensor analysis. Instead of drawing pictures it became fashionable to raise and lower indices. I strongly feel that pictures need to be much more stressed in differential geometry than is presently the case. It is unfortunate that the great differential geometers of the past did not share their extraordinary intuitions with others by means of pictures. I hope that the present book contributes in some way to returning the differential geometry of curves and surfaces to its proper place in the mathematics curriculum. I wish to thank Elsa Abbena, James Anderson, Thomas Banchoff, Marcel Berger, Michel Berry, Nancy Blachman, William Bruce, Renzo Caddeo, Eugenio Calabi, Thomas Cecil, Luis A. Cordero, Al Currier, Luis C. de Andrés, Mirjana Djorić, Franco Fava, Helaman Fergason, Marisa Fernández, Frank Flaherty, Anatoly Fomenko, V.E. Fomin, David Fowler, George Francis, Ben Friedman, Thomas Friedrick, Pedro M. Gadea, Sergio Garbiero, Laura Geatti, Peter Giblin, Vladislav Goldberg, William M. Goldman, Hubert Gollek, Mary Gray, Joe Grohens, Garry Helzer, A.O. Ivanov, Gary Jensen, Alfredo Jiménez, Raj Jakkumpudi, Gary Jensen, David Johannsen, Joe Kaiping, Ben Kedem, Robert Kragler, Steve Krantz, Henning Leidecker, Stuart Levy, Mats Liljedahl, Lee Lorch, Sanchez Santiago Lopez de Medrano, Roman Maeder, Steen Markvorsen, Mikhail A. Malakhaltsev, Armando Machado, David McNabb, José J. Mencı́a, Michael Mezzino, Vicente Miquel Molina, Deanne Montgomery, Tamara Munzner, Emilio Musso, John Novak, Barrett O’Neill, Richard Palais, Mark Phillips, Lori Pickert, David Pierce, Mark Pinsky, Paola Piu, Valeri Pouchnia, Rob Pratt, Emma Previato, Andreas Iglesias Prieto, Lilia del Riego, Patrick Ryan, Giacomo Saban, George Sadler, Isabel Salavessa, Simon Salamon, Jason P. Schultz, Walter Seaman, B.N. Shapukov, V.V. Shurygin, E.P. Shustova, Sonya Šimek, Cameron Smith, Dirk Struik, Rolf Sulanke, John Sullivan, Daniel Tanrè, C. Terng, A.A. Tuzhilin, Lieven Vanhecke, Gus Vlahacos, Tom Wickam-Jones and Stephen Wolfram for valuable suggestions. Alfred Gray July 1998 vi Preface to the Third Edition Most of the material of this book can be found, in one form or another, in the Second Edition. The exceptions to this can be divided into three categories. Firstly, a number of modifications and new items had been prepared by Alfred Gray following publication of the Second Edition, and we have been able to incorporate some of these in the Third Edition. The most obvious is Chapter 21. In addition, we have liberally expanded a number of sections by means of additional text or graphics, where we felt that this was warranted. The second is Chapter 23, added by the editors to present the popular theory of quaternions. This brings together many of the techniques in the rest of the book, combining as it does the theory of space curves and surfaces. The third concerns the Mathematica code presented in the notebooks. Whilst this is closely based on that written by the author and displayed in previous editions, many programs have been enhanced and sibling ones added. This is to take account of the progressive presentation that Mathematica notebooks offer, and a desire to publish instructions to generate every figure in the book. The new edition does differ notably from the previous one in the manner in which the material is organized. All Mathematica code has been separated from the body of the text and organized into notebooks, so as to give readers interactive access to the material. There is one notebook to accompany each chapter, and it contains relevant programs in parallel with the text, section by section. An abridged version is printed at the end of the chapter, for close reference and to present a fair idea of the programs that ride in tandem with the mathematics. The distillation of computer code into notebooks also makes it easier to conceive of rewriting the programs in a different language, and a project is underway to do this for Maple. The full notebooks can be downloaded from the publisher’s site http://www.crcpress.com Their organization and layout is discussed in more detail in Notebook 0 below. They contain no output, as this can be generated at will. All the figures in the book were compiled automatically by merely evaluating the notebooks chapter by chapter, and this served to ‘validate’ the notebooks using Version 5.1 of Mathematica. It is the editors’ intention to build up an on-line database of solutions to the exercises at the end of each chapter. Those marked M are designed to be solved with the help of a suitable Mathematica program. vii The division of the material into chapters and the arrangement of later chapters has also been affected by the presence of the notebooks. We have chosen to shift the exposition of differentiable manifolds and abstract surfaces towards the end of the book. In addition, there are a few topics in the Second Edition that have been relegated to electronic form in an attempt to streamline the volume. This applies to the fundamental theorems of surfaces, and some more advanced material on minimal surfaces. (It is also a recognition of other valuable sources, such as [Oprea2] to mention one.) The Chapter Scheme overleaf provides an idea of the relationship between the various topics; touching blocks represent a group of chapters that are probably best read in numerical order, and 21, 22, 27 are independent peaks to climb. In producing the Third Edition, the editors were fortunate in having ready access to electronic versions of the author’s files. For this, they are grateful to Mike Mezzino, as well as the editors of the Spanish and Italian versions, Marisa Ferńandez and Renzo Caddeo. Above all, they are grateful to Mary Gary and Bob Stern for entrusting them with the editing task. As regards the detailed text, the editors acknowledge the work of Daniel Drucker, who diligently scanned the Second Edition for errors, and provided helpful comments on much of the Third Edition. Some of the material was used for the course ‘Geometrical Methods for Graphics’ at the University of Turin in 2004 and 2005, and we thank students of that course for improving some of the figures and associated computer programs. We are also grateful to Simon Chiossi, Sergio Console, Antonio Di Scala, Anna Fino, Gian Mario Gianella and Sergio Garbiero for proof-reading parts of the book, and to John Sullivan and others for providing photo images. The years since publication of the First Edition have seen a profileration of useful websites containing information on curves and surfaces that complements the material of this book. In particular, the editors acknowledge useful visits to the Geometry Center’s site http://www.geom.uiuc.edu, and Richard Palais’ pages http://vmm.math.uci.edu/3D-XplorMath. Finally, we are grateful for support from Wolfram Research. The editors initially worked with Alfred Gray at the University of Maryland in 1979, and on various occasions subsequently. Although his more abstract research had an enormous influence on many branches of differential geometry (a hint of which can be found in the volume [FeWo]), we later witnessed the pleasure he experienced in preparing material for both editions of this book. We hope that our more modest effort for the Third Edition will help extend this pleasure to others. Elsa Abbena and Simon Salamon December 2005 viii Chapter Scheme 22 21 16 15 20 19 18 14 13 11 5 1 25 10 24 6 7 3 17 26 12 9 2 27 4 8 23 ix 1 2 Curves in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Famous Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 3 Alternative Ways of Plotting Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4 5 New Curves from Old . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Determining a Plane Curve from its Curvature . . . . . . . . . . . . . . . . 127 6 Global Properties of Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7 Curves in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191 8 Construction of Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9 Calculus on Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 10 Surfaces in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 11 12 Nonorientable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .331 Metrics on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13 Shape and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 14 Ruled Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 15 16 Surfaces of Revolution and Constant Curvature . . . . . . . . . . . . . . . 461 A Selection of Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 17 Intrinsic Surface Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 18 Asymptotic Curves and Geodesics on Surfaces . . . . . . . . . . . . . . . 557 19 Principal Curves and Umbilic Points . . . . . . . . . . . . . . . . . . . . . . . . . . 593 20 Canal Surfaces and Cyclides of Dupin . . . . . . . . . . . . . . . . . . . . . . . . 639 21 The Theory of Surfaces of Constant Negative Curvature . . . . . . 683 22 Minimal Surfaces via Complex Variables . . . . . . . . . . . . . . . . . . . . . . 719 23 Rotation and Animation using Quaternions . . . . . . . . . . . . . . . . . . . .767 24 25 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 26 27 Abstract Surfaces and their Geodesics . . . . . . . . . . . . . . . . . . . . . . . .871 The Gauss–Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 Chapter 1 Curves in the Plane Geometry before calculus involves only the simplest of curves: straight lines, broken lines, arcs of circles, ellipses, hyperbolas and parabolas. These are the curves that form the basis of Greek geometry. Although other curves (such as the cissoid of Diocles and the spiral of Archimedes) were known in antiquity, the general theory of plane curves began to be developed only after the invention of Cartesian coordinates by Descartes1 in the early 1600s. Interesting plane curves arise in everyday life. Examples include the trajectory of a projectile (a parabola), the form of a suspension bridge (a catenary), the path of a point on the wheel of a car (a cycloid), and the orbit of a planet (an ellipse). In this chapter we get underway with the study of the local properties of curves that has developed over the last few centuries. We begin in Section 1.1 by recalling some standard operations on Euclidean space Rn , and in Section 1.2 define the notions of curves in Rn and vector fields along them. Arc length for curves in Rn is defined and discussed in Section 1.3. In Section 1.4, we define the important notion of signed curvature of a curve in the plane R2 . Section 1.5 is devoted to the problem of defining an angle function between plane curves, and this allows us to show that the signed curvature is the derivative of the turning angle. The examples discussed in Section 1.6 include the logarithmic spiral, while the arc length of the semicubical parabola is computed in Section 1.7. Our basic approach in Chapters 1 and 2 is to study curves by means of their parametric representations, whereas implicitly defined curves in R2 are discussed in Section 3.1. In this chapter, we must therefore show that any 1 René du Perron Descartes (1596–1650). French mathematician and philosopher. Descartes developed algebraic techniques to solve geometric problems, thus establishing the foundations of analytic geometry. Although most widely known as a philosopher, he also made important contributions to physiology, optics and mechanics. 1 2 CHAPTER 1. CURVES IN THE PLANE geometric invariant depending only on the point set traced out by the curve is independent of the parametrization, at least up to sign. The most important geometric quantities associated with a curve are of two types: (1) those that are totally independent of parametrization, and (2) those that do not change under a positive reparametrization, but change sign under a negative reparametrization. For example, we show in Section 1.3 that the length of a curve is independent of the parametrization chosen. The curvature of a plane curve is a more subtle invariant, because it does change sign if the curve is traversed in the opposite direction, but is otherwise independent of the parametrization. This fact we prove in Section 1.4. In Section 1.6 we give the some examples of plane curves that generalize circles, discuss their properties and graph associated curvature functions. 1.1 Euclidean Spaces Since we shall be studying curves and surfaces in a Euclidean space, we summarize some of the algebraic properties of Euclidean space in this section. Definition 1.1. Euclidean n-space Rn consists of the set of all real n-tuples Rn = (p1 , . . . , pn ) | pj is a real number for j = 1, . . . , n . We write R for R1 ; it is simply the set of all real numbers. R2 is frequently called the plane. Elements of Rn represent both points in n-dimensional space, and the position vectors of points. As a consequence, Rn is a vector space, so that the operations of addition and scalar multiplication are defined. Thus if p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ), then p + q is the element of Rn given by p + q = (p1 + q1 , . . . , pn + qn ). Similarly, for λ ∈ R the vector λp is defined by λp = (λp1 , . . . , λpn ). Furthermore, we shall denote by · the dot product (or scalar product) of Rn . It is an operation that assigns to each pair of vectors p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) the real number p·q = n X j=1 pj qj . 1.1. EUCLIDEAN SPACES 3 The norm and distance functions of Rn can then be defined by kpk = √ p · p, and distance(p, q) = kp − qk for p, q ∈ Rn . These functions have the properties p·q = (λp) · q = (p + r) · q q · p, λ(p · q) = p · (λq), kλpk = p · q + r · q, = |λ| kpk, for λ ∈ R and p, q, r ∈ Rn . Furthermore, the Cauchy–Schwarz and triangle inequalities state that for all p, q ∈ Rn we have |p · q| 6 kpk kqk and kp + qk 6 kpk + kqk. (See, for example, [MS, pages 22–24].) We shall also need the notion of the angle between nonzero vectors p, q of Rn , which is a number θ defined by: cos θ = p·q . kpk kqk The Cauchy–Schwarz inequality implies that −1 6 p·q 61 kpk kqk for nonzero p, q ∈ Rn , so that the definition of the angle makes sense. At times, it is convenient to specify the range of θ using Lemma 1.3 below. A linear map of Rn into Rm is a function A: Rn → Rm such that A(λp + µq) = λAp + µAq for λ, µ ∈ R and p, q ∈ Rn . We may regard such a map A as a m × n matrix, though to make the above definition consistent, one needs to represent elements of Rn by columns rather than rows. This is because the usual convention dictates that matrices, like functions, should act on the left. So far we have been dealing with Rn for general n. When n is 2 or 3, the vector space Rn has special structures, namely a complex structure and a vector cross product, which are useful for describing curves and surfaces respectively. In Chapters 1 – 6 we shall be studying curves in the plane, so we concentrate attention on R2 now. Algebraic properties of R3 will be considered in Section 7.1, and those of R4 in Chapter 23. For the differential geometry of curves in the plane an essential tool is the complex structure of R2 ; it is the linear map J : R2 → R2 given by J (p1 , p2 ) = (−p2 , p1 ). 4 CHAPTER 1. CURVES IN THE PLANE Geometrically, J is merely a rotation by π/2 in a counterclockwise direction. It is easy to show that the complex structure J has the properties J 2 = −1, (J p) · (J q) = p · q, (J p) · p = 0, for p, q ∈ R2 (where 1: R2 → R2 is the identity map). A point in the plane R2 can be considered a complex number via the canonical isomorphism (1.1) p = (p1 , p2 ) ↔ p1 + ip2 = Re p + i Im p, where Re p and Im p denote the real and imaginary parts of p. In a moment, we shall need descriptions of the dot product · and the complex structure J in terms of complex numbers. Recall that the complex conjugate and absolute value of a complex number p are defined by p p = Re p − i Im p and |p| = pp. The proof of the following lemma is elementary. Lemma 1.2. Identify the plane R2 with the set of complex numbers C, and let p, q ∈ R2 = C. Then (1.2) J p = i p, |p| = kpk and pq = p · q + i p · (J q). The angle between vectors in Rn does not distinguish between the order of the vectors, but there is a refined notion of angle between vectors in R2 that makes this distinction. p Jq q Θ Figure 1.1: The oriented angle 1.2. CURVES IN SPACE 5 Lemma 1.3. Let p and q be nonzero vectors in R2 . There exists a unique number θ with the properties (1.3) cos θ = p·q , kpk kqk sin θ = p · Jq , kpk kqk 0 6 θ < 2π. We call θ the oriented angle from q to p. Proof. The oriented angle θ defines p in relation to the frame (q, J q), as indicated in Figure 1.1. A formal proof proceeds as follows. Since pq/(|p||q|) is a complex number of absolute value 1, it lies on the unit circle in C; thus there exists a unique θ with 0 6 θ < 2π such that pq = eiθ . |p||q| (1.4) Then using the expressions for pq in (1.2) and (1.4), we find that (1.5) p · q + ip · (J q) = eiθ |p||q| = |p||q| cos θ + i |p||q| sin θ. When we take the real and imaginary parts in (1.5), we get (1.3). 1.2 Curves in Space In the previous section we reviewed some of the algebraic properties of Rn . We need to go one step further and study differentiation. In this chapter, and generally throughout the rest of the book, we shall use the word ‘differentiable’ to mean ‘possessing derivatives or partial derivatives of all orders’. We begin by studying Rn -valued functions of one variable. Definition 1.4. Let α: (a, b) → Rn be a function, where (a, b) is an open interval in R. We write α(t) = a1 (t), . . . , an (t) , where each aj is an ordinary real-valued function of a real variable. We say that α is differentiable provided aj is differentiable for j = 1, . . . , n. Similarly, α is piecewise-differentiable provided aj is piecewise-differentiable for j = 1, . . . , n. Definition 1.5. A parametrized curve in Rn is a piecewise-differentiable function α : (a, b) −→ Rn , where (a, b) is an open interval in R. We allow the interval to be finite, infinite or half-infinite. If I is any other subset of R, we say that α: I −→ Rn is a curve provided there is an open interval (a, b) containing I such that α can be extended as a piecewise-differentiable function from (a, b) into Rn . 6 CHAPTER 1. CURVES IN THE PLANE It is important to distinguish a curve α, which is a function, from the set of points traced out by α, which we call the trace of α. The trace of α is just the image α (a, b) , or more generally α(I). We say that a subset C of Rn is parametrized by α provided there is a subset I ⊆ R such that α : I → Rn is a curve for which α(I) = C . Definition 1.6. Let α : (a, b) → Rn be a curve with α(t) = a1 (t), . . . , an (t) . Then the velocity of α is the function α′ : (a, b) → Rn given by α′ (t) = a′1 (t), . . . , a′n (t) . The function v defined by v(t) = α′ (t) is called the speed of α. The acceleration of α is given by α′′ (t). Notice that α′ (t) is defined for those t for which a′1 (t), . . . , a′n (t) are defined. When n = 2, the acceleration can be compared with the vector J α′ (t) as shown in Figure 1.2; this will be the basis of the definition of curvature on page 14. JΑ'HtL Α''HtL Α'HtL Figure 1.2: Velocity and acceleration of a plane curve Some curves are ‘better’ than others: Definition 1.7. A curve α: (a, b) → Rn is said to be regular if it is differentiable and its velocity is everywhere defined and nonzero. If α′ (t) = 1 for a < t < b, then α is said to have unit speed. 1.2. CURVES IN SPACE 7 The simplest example of a parametrized curve in Rn is a straight line. If it contains distinct points p, q ∈ Rn , it is most naturally parametrized by (1.6) β(t) = p + t(q − p) = (1 − t)p + tq, with t ∈ R. In the plane R2 , the second simplest curve is a circle. If it has radius r and center (p1 , p2 ) ∈ R2 , it can be parametrized by (1.7) γ(t) = p1 + r cos t, p2 + r sin t , with 0 6 t < 2π. Our definition of a curve in Rn is the parametric form of a curve. In contrast, the nonparametric form of a curve would typically consist of a system of n − 1 equations F1 (p1 , . . . , pn ) = · · · = Fn−1 (p1 , . . . , pn ) = 0, where each Fi is a differentiable function Rn → R. We study the nonparametric form of curves in R2 in Section 3.1. It is usually easier to work with the parametric form of a curve, but there is one disadvantage: distinct parametrizations may trace out the same point set. Therefore, it is important to know when two curves are equivalent under a change of variables. Definition 1.8. Let α: (a, b) → Rn and β : (c, d) → Rn be differentiable curves. Then β is said to be a positive reparametrization of α provided there exists a differentiable function h: (c, d) → (a, b) such that h′ (u) > 0 for all c < u < d and β = α ◦ h. Similarly, β is called a negative reparametrization of α provided there exists a differentiable function h: (c, d) → (a, b) such that h′ (u) < 0 for all c < u < d and β = α ◦ h. We say that β is a reparametrization of α if it is either a positive or negative reparametrization of α. Next, we determine the relation between the velocity of a curve α and the velocity of a reparametrization of α. Lemma 1.9. (The chain rule for curves.) Suppose that β is a reparametrization of α. Write β = α ◦ h, where h: (c, d) → (a, b) is differentiable. Then (1.8) β′ (u) = h′ (u)α′ h(u) for c < u < d. Proof. Write α(t) = a1 (t), . . . , an (t) and β(u) = b1 (u), . . . , bn (u) . Then we have bj (u) = aj h(u) for j = 1, . . . , n. The ordinary chain rule implies that b′j (u) = a′j h(u) h′ (u) for j = 1, . . . , n, and so we get (1.8). 8 CHAPTER 1. CURVES IN THE PLANE For example, the straight line (1.6) and circle (1.7) have slightly more complicated unit-speed parametrizations: s s s kp − qk − s p+ q kp − qk kp − qk s s 7 → p1 + r cos , p2 + r sin . r r 7→ The quantities α′ , J α′ , α′′ are examples of vector ﬁelds along a curve, because they determine a vector at every point of the curve, not just at the one point illustrated. We now give the general definition. Definition 1.10. Let α: (a, b) → Rn be a curve. A vector ﬁeld along α is a function Y that assigns to each t with a < t < b a vector Y(t) at the point α(t). At this stage, we shall not distinguish between a vector at α(t) and the vector parallel to it at the origin. This means a vector field Y along a curve α is really an n-tuple of functions: Y(t) = y1 (t), . . . , yn (t) . (Later, for example in Section 9.6, we will need to distinguish between a vector at α(t) and the vector parallel to it at the origin.) Differentiability of Y means that each of the functions y1 , . . . , yn is differentiable, and in this case Y is effectively another curve and its derivative is defined in the obvious way: Y′ (t) = y1′ (t), . . . , yn′ (t) . Addition, scalar multiplication and dot product of vector fields along a curve α : (a, b) → Rn are defined in the obvious way. Furthermore, we can multiply a vector field Y along α by a function f : (a, b) → R: we define the vector field f Y by (f Y)(t) = f (t)Y(t). Finally, if n = 2 and X is a vector field along a curve with X(t) = (x(t), y(t)), then another vector field J X can be defined by J X(t) = (−y(t), x(t)). Taking the derivative is related to these operations in the obvious way: Lemma 1.11. Suppose that X and Y are differentiable vector fields along a curve α: (a, b) → Rn , and let f : (a, b) → R be differentiable. Then (i) (f Y)′ = f ′ Y + f Y′ ; (ii) (X + Y)′ = X′ + Y′ ; (iii) (X · Y)′ = X′ · Y + X · Y′ ; (iv) for n = 2, (J Y)′ = J Y′ . 1.3. LENGTH OF A CURVE 9 Proof. For example, we prove (i): (f Y)′ = (f y1 )′ , . . . , (f yn )′ = (f ′ y1 + f y1′ , . . . , f ′ yn + f yn′ ) = (y1 , . . . , yn )f ′ + f (y1′ , . . . , yn′ ) = f ′ Y + f Y′ . 1.3 The Length of a Curve One of the simplest and most important geometric quantities associated with a curve is its length. Everyone has a natural idea of what is meant by length, but this idea must be converted into an exact definition. For our purposes the simplest definition is as follows: Definition 1.12. Let α : (a, b) → Rn be a curve. Assume that α is defined on a slightly larger interval containing (a, b), so that α is defined and differentiable at a and b. Then the length of α over the interval [a, b] is given by length[α] = (1.9) Z b α′ (t) dt. a One can emphasize the role of the endpoints with the notation length[a, b][α] (compatible with Notebook 1); this is essential when we wish to compute the length of a curve restricted to an interval smaller than that on which it is defined. It is clear intuitively that length should not depend on the parametrization of the curve. We now prove this. Theorem 1.13. Let β be a reparametrization of α. Then length[α] = length[β]. Proof. We do the positive reparametrization case first. Let β = α ◦ h, where h: (c, d) → (a, b) and h′ (u) > 0 for c < u < d. Then by Lemma 1.9 we have β′ (u) = α′ h(u) h′ (u) = α′ h(u) h′ (u). Using the change of variables formula for integrals, we compute Z b Z d length[α] = α′ (t) dt = α′ h(u) h′ (u)du a c = Z d β ′ (u) du = length[β]. c In the negative reparametrization case, we have lim h(u) = b, u↓c lim h(u) = a, u↑d 10 CHAPTER 1. CURVES IN THE PLANE and β ′ (u) = α′ h(u) h′ (u) = − α′ h(u) h′ (u). Again, using the change of variables formula for integrals, Z b Z c ′ length[α] = α (t) dt = α′ h(u) h′ (u)du a d = Z d β ′ (u) du = length[β]. c The trace of a curve α can be approximated by a series of line segments connecting a sequence of points on the trace of α. Intuitively, the length of this approximation to α should tend to the length of α as the individual line segments become smaller and smaller. This is in fact true. To explain exactly what happens, let α : (c, d) → Rn be a curve and [a, b] ⊂ (c, d) a closed finite interval. For every partition (1.10) P = { a = t0 < t1 < · · · < tN = b } of [a, b], let |P | = max (tj − tj−1 ) 16j6N and ℓ(α, P ) = N X j=1 α(tj ) − α(tj−1 ) . Geometrically, ℓ(α, P ) is the length of the polygonal path in Rn whose vertices are α(tj ), j = 1, . . . , N . Then length[α] is the limit of the lengths of inscribed polygonal paths in the sense of Theorem 1.14. Let α: (c, d) → Rn be a curve, and let length[α] denote the length of the restriction of α to a closed subinterval [a, b]. Given ǫ > 0, there exists δ > 0 such that (1.11) |P | < δ length[α] − ℓ(α, P ) < ǫ. implies Proof. Given a partition (1.10), write α = (a1 , . . . , an ). For each i, the Mean (i) (i) Value Theorem implies that for 1 6 j 6 N there exists tj with tj−1 < tj < tj such that (i) ai (tj ) − ai (tj−1 ) = a′i (tj )(tj − tj−1 ). Then α(tj ) − α(tj−1 ) 2 = n X i=1 = n X i=1 2 ai (tj ) − ai (tj−1 ) (i) a′i (tj )2 (tj − tj−1 )2 = (tj − tj−1 )2 n X i=1 (i) a′i (tj )2 . 1.3. LENGTH OF A CURVE 11 Hence s n X (i) a′i (tj )2 = (tj − tj−1 )(Aj + Bj ), = (tj − tj−1 ) α(tj ) − α(tj−1 ) i=1 where Aj = s n X a′i (tj )2 ′ = α (tj ) and Bj = i=1 s n X (i) a′i (tj )2 i=1 − s n X a′i (tj )2 . i=1 Two separate applications of the triangle inequality allow us to deduce that (n) (1) |Bj | = a′1 (tj ), . . . , a′n (tj ) − α′ (tj ) 6 (n) (1) a′1 (tj ), . . . , a′1 (tj ) − α′ (tj ) = X n i=1 6 n X i=1 ′ 2 (i) a′i (tj ) − a′i (tj ) 1/2 (i) a′i (tj ) − a′i (tj ) . Let ǫ > 0. Since α is continuous on the closed interval [a, b], there exists a positive number δ1 such that ǫ |u − v| < δ1 implies a′i (u) − a′i (v) < 2n(b − a) for i = 1, . . . , n. Then |P | < δ1 implies that |Bj | < ǫ/(2(b − a)), for each j, so that N N X X ǫ ǫ (tj − tj−1 ) (tj − tj−1 )Bj 6 = . 2(b − a) 2 j=1 j=1 Hence length[α] − ℓ(α, P ) = = 6 6 length[α] − N X length[α] − N X (Aj + Bj )(tj − tj−1 ) length[α] − N X Aj (tj − tj−1 ) + Z a α(tj ) − α(tj−1 ) j=1 j=1 j=1 b α′ (t) dt − N X j=1 N X j=1 Bj (tj − tj−1 ) α′ (tj )(tj − tj−1 ) ǫ + . 2 12 CHAPTER 1. CURVES IN THE PLANE Figure 1.3: Polygonal approximation of the curve t 7→ (cos t, sin t, t/3) But N X j=1 α′ (tj )(tj − tj−1 ) is a Riemann sum approximation to the right-hand side of (1.9). Hence there exists δ2 > 0 such that |P | < δ2 implies that N X j=1 ′ α (tj )(tj − tj−1 ) − Z b α′ (t) dt < a ǫ . 2 Thus if we take δ = min(δ1 , δ2 ), we get (1.11). We need to know how the length of a curve varies when we change the limits of integration. Definition 1.15. Fix a number c with a < c < b. The arc length function sα of a curve α : (a, b) → Rn , starting at c, is defined by Z t sα (t) = length[c, t][α] = α′ (u) du, a for a 6 t 6 b. When there is no danger of confusion, we simplify sα to s. 1.3. LENGTH OF A CURVE 13 Note that if c 6 d 6 f 6 b, then length[d, f ][α] = sα (f ) − sα (d) > 0. Theorem 1.16. Let α : (a, b) → Rn be a regular curve. Then there exists a unit-speed reparametrization β of α. Proof. By the fundamental theorem of calculus, any arc length function s of α satisfies ds (t) = s′ (t) = α′ (t) . dt Since α is regular, α′ (t) is never zero; hence ds/dt is always positive. The Inverse Function Theorem implies that t 7→ s(t) has an inverse s 7→ t(s), and that 1 dt . = ds ds s(t) dt t(s) Now define β by β(s) = α t(s) . By Lemma 1.9, we have β ′ (s) = (dt/ds)α′ t(s) . Hence β′ (s) = dt ′ dt ′ dt α t(s) = α t(s) = ds ds ds s ds dt t(s) = 1. The arc length function of any unit-speed curve β : (c, d) → Rn starting at c satisfies Z t sβ (t) = β ′ (u) du = t − c. c Thus the function sβ actually measures length along β. This is the reason why unit-speed curves are said to be parametrized by arc length. We conclude this section by showing that unit-speed curves are unique up to starting point and direction. Lemma 1.17. Let α: (a, b) → Rn be a unit-speed curve, and let β : (c, d) → Rn be a reparametrization of α such that β also has unit speed. Then β(s) = α(±s + s0 ) for some constant real number s0 . Proof. By hypothesis, there exists a differentiable function h: (c, d) → (a, b) such that β = α ◦ h and h′ (u) 6= 0 for c < u < d. Then Lemma 1.9 implies that 1 = β′ (u) = α′ h(u) |h′ (u)| = |h′ (u)|. Therefore, h′ (u) = ±1 for all t. Since h′ is continuous on a connected open set, its sign is constant. Thus there is a constant s0 such that h(u) = ±s + s0 for c < u < d, s being arc length for α. 14 CHAPTER 1. CURVES IN THE PLANE 1.4 Curvature of Plane Curves Intuitively, the curvature of a curve measures the failure of a curve to be a straight line. The faster the velocity α′ turns along a curve α, the larger the curvature2 . In Chapter 7, we shall define the notion of curvature κ[α] of a curve α in Rn for arbitrary n. In the special case that n = 2 there is a refined version κ2 of κ, which we now define. We first give the definition that is the easiest to use in computations. In the next section, we show that the curvature can be interpreted as the derivative of a turning angle. Definition 1.18. Let α: (a, b) → R2 be a regular curve. The curvature κ2[α] of α is given by the formula (1.12) κ2[α](t) = α′′ (t) · J α′ (t) α′ (t) 3 . The positive function 1/ κ2[α] is called the radius of curvature of α. The notation κ2[α] is consistent with that of Notebook 1, though there is no danger in omitting α if there is only one curve under consideration. We can still speak of the curvature of a plane curve α that is only piecewisedifferentiable. At isolated points where the velocity vector α′ (t) vanishes or does not exist, the curvature κ2(t) remains undefined. Similarly, the radius of curvature 1/|κ2(t)| is undefined at those t for which either α′ (t) or α′′ (t) vanishes or is undefined. Notice that the curvature is proportional (with a positive constant) to the orthogonal projection of α′′ (t) in the direction of J α′ (t) (see Figure 1.2). If the acceleration α′′ (t) vanishes, so does κ2(t). In particular, the curvature of the straight line parametrized by (1.6) vanishes identically. It is important to realize that the curvature can assume both positive and negative values. Often κ2 is called the signed curvature in order to distinguish it from a curvature function κ which we shall define in Chapter 7. It is useful to have some pictures to understand the meaning of positive and negative curvature for plane curves. There are four cases, each of which we illustrate by a parabola in Figures 1.4 and 1.5. These show that, briefly, positive curvature means ‘bend to the left’ and negative curvature means ‘bend to the right’. 2 The notion of curvature of a plane curve first appears implicitly in the work of Apollonius of Perga (262–180 BC). Newton (1642–1727) was the first to study the curvature of plane curves explicitly, and found in particular formula (1.13). 1.4. CURVATURE OF PLANE CURVES 9 t, t2 - 1= 15 9-t, 1- t2 = Figure 1.4: Parabolas with κ2 > 0 9-t, t2 - 1= 9 t, 1 - t2 = Figure 1.5: Parabolas with κ2 < 0 Next, we derive the formula for curvature that can be found in most calculus books, and a version using complex numbers and the notation of Lemma 1.2. Lemma 1.19. If α: (a, b) → R2 is a regular curve with α(t) = (x(t), y(t)), then the curvature of α is given in the equivalent ways (1.13) κ2[α](t) = (1.14) κ2[α](t) = x′ (t)y ′′ (t) − x′′ (t)y ′ (t) 3/2 , x′2 (t) + y ′2 (t) Im α′′ (t) α′ (t) α′ (t) 3 . Proof. We have α′′ (t) = (x′′ (t), y ′′ (t)) and J α′ (t) = (−y ′ (t), x′ (t)), so that x′′ (t), y ′′ (t) · −y ′ (t), x′ (t) x′ (t)y ′′ (t) − x′′ (t)y ′ (t) κ2[α](t) = = 3/2 3/2 , x′2 (t) + y ′2 (t) x′2 (t) + y ′2 (t) proving (1.13). Equation (1.14) follows from (1.12) and (1.2) (Exercise 5). Like the function length[α] the curvature κ2[α] is a geometric quantity associated with a curve α. As such, it should be independent of the parametrization of the curve α. We now show that this is almost true. If λ is a nonzero real number, then λ/|λ| is understandably denoted by sign λ. 16 CHAPTER 1. CURVES IN THE PLANE Theorem 1.20. Let α: (a, b) → R2 be a regular curve, and let β : (c, d) → R2 be a reparametrization of α. Write β = α ◦ h, where h: (c, d) → (a, b) is differentiable. Then (1.15) κ2[β](u) = sign h′ (u) κ2[α] h(u) , wherever h′ (u) 6= 0. Thus the curvature of a plane curve is independent of the parametrization up to sign. Proof. We have β′ = (α′ ◦ h)h′ , so that J β ′ = J(α′ ◦ h) h′ , and β′′ = (α′′ ◦ h)h′2 + (α′ ◦ h)h′′ . (1.16) Thus κ2[β] = = (α′′ ◦ h)h′2 + (α′ ◦ h)h′′ · J(α′ ◦ h) h′ (α′ ◦ h)h′ h′3 |h′ |3 3 (α′′ ◦ h) · J (α′ ◦ h) (α′ ◦ h) 3 . Hence we get (1.15). The formula for κ2 simplifies dramatically for a unit-speed curve: Lemma 1.21. Let β be a unit-speed curve in the plane. Then (1.17) β ′′ = κ2[β]J β ′ . Proof. Differentiating β′ · β ′ = 1, we obtain β ′′ · β ′ = 0. Thus β ′′ must be a multiple of J β ′ . In fact, it follows easily from (1.12) that this multiple is κ2[β], and we obtain (1.17) . Finally, we give simple characterizations of straight lines and circles by means of curvature. Theorem 1.22. Let α: (a, b) → R2 be a regular curve. (i) α is part of a straight line if and only if κ2[α](t) ≡ 0. (ii) α is part of a circle of radius r > 0 if and only if κ2[α] (t) ≡ 1/r. Proof. It is easy to show that the curvature of the straight line is zero, and that the curvature of a circle of radius r is +1/r for a counterclockwise circle and −1/r for a clockwise circle (see Exercise 3). For the converse, we can assume without loss of generality that α is a unitspeed curve. Suppose κ2[α](t) ≡ 0. Lemma 1.21 implies that α′′ (t) = 0 for a < t < b. Hence there exist constant vectors p and q such that α(t) = pt + q 1.5. ANGLE FUNCTIONS 17 for a < t < b; thus α is part of a straight line. Next, assume that κ2[α](t) is identically equal to a positive constant 1/r. Without loss of generality, α has unit speed. Define a curve γ : (b, c) → R2 by γ(t) = α(t) + rJ α′ (t). Lemma 1.21 implies that γ ′ (t) = 0 for a < t < b. Hence there exists q such that γ(t) = q for a < t < b. Then α(t) − q = rJ α′ (t) = r. Hence α(t) lies on a circle of radius r centered at q. Similarly, κ2[α](t) ≡ −1/r also implies that α is part of a circle of radius r (see Exercise 4). 1.5 Angle Functions In Section 1.1 we defined the notion of oriented angle between vectors in R2 ; now we wish to define a similar notion for curves. Clearly, we can compute the oriented angle between corresponding velocity vectors. This oriented angle θ satisfies 0 6 θ < 2π, but such a restriction is not desirable for curves. The problem is that the two curves can twist and turn so that eventually the angle between them lies outside the interval [0, 2π). To arrive at the proper definition, we need the following useful lemma of O’Neill [ON1]. Lemma 1.23. Let f, g : (a, b) → R be differentiable functions with f 2 + g 2 = 1. Fix t0 with a < t0 < b and let θ0 be such that f (t0 ) = cos θ0 and g(t0 ) = sin θ0 . Then there exists a unique continuous function θ : (a, b) → R such that (1.18) θ(t0 ) = θ0 , f (t) = cos θ(t) and g(t) = sin θ(t) for a < t < b. Proof. We use complex numbers. Let h = f + ig, so hh = 1. Define Z t θ(t) = θ0 − i h′ (s)h(s) ds. (1.19) t0 Then Thus he−iθ d (he−iθ ) = e−iθ (h′ − i hθ′ ) = e−iθ (h′ − hh′ h) = 0. dt = c for some constant c. Since h(t0 ) = eiθ0 , it follows that c = h(t0 )e−iθ(t0 ) = 1. Hence h(t) = eiθ(t) , and so we get (1.18). Let θ̂ be another continuous function such that θ̂(t0 ) = θ0 , f (t) = cos θ̂(t) and g(t) = sin θ̂(t) 18 CHAPTER 1. CURVES IN THE PLANE for a < t < b. Then eiθ(t) = eiθ̂(t) for a < t < b. Since both θ and θ̂ are continuous, there is an integer n such that θ(t) − θ̂(t) = 2πn for a < t < b. But θ(t0 ) = θ̂(t0 ), so that n = 0. Hence θ and θ̂ coincide. We can now apply this lemma to deduce the existence and uniqueness of a differentiable angle function between curves in R2 . Corollary 1.24. Let α and β be regular curves in R2 defined on the same interval (a, b), and let a < t0 < b. Choose θ0 such that α′ (t0 ) · β ′ (t0 ) = cos θ0 α′ (t0 ) β ′ (t0 ) and α′ (t0 ) · J β′ (t0 ) = sin θ0 . α′ (t0 ) β ′ (t0 ) Then there exists a unique differentiable function θ : (a, b) → R such that θ(t0 ) = θ0 , α′ (t) · β′ (t) = cos θ(t) α′ (t) β ′ (t) and α′ (t) · J β ′ (t) = sin θ(t) α′ (t) β ′ (t) for a < t < b. We call θ the angle function from β to α determined by θ0 . Proof. We take f (t) = α′ (t) · β′ (t) α′ (t) β ′ (t) and g(t) = α′ (t) · J β ′ (t) α′ (t) β ′ (t) in Lemma 1.23. Α'HtL Β'HtL Figure 1.6: The angle function between curves 1.5. ANGLE FUNCTIONS 19 Intuitively, it is clear that the velocity of a highly curved plane curve changes rapidly. This idea can be made precise with the concept of turning angle. Lemma 1.25. Let α: (a, b) → R2 be a regular curve and fix t0 with a < t0 < b. Let θ0 be a number such that α′ (t0 ) = (cos θ0 , sin θ0 ). α′ (t0 ) Then there exists a unique differentiable function θ = θ[α]: (a, b) → R such that θ(t0 ) = θ0 and α′ (t) (1.20) = cos θ(t), sin θ(t) . ′ α (t) for a < t < b. We call θ[α] the turning angle determined by θ0 . Proof. Let β(t) = (t, 0); then β parametrizes a horizontal straight line and β ′ (t) = (1, 0) for all t. Write α(t) = (a1 (t), a2 (t)); then α′ (t) · β′ (t) = a′1 (t) and α′ (t) · J β ′ (t) = a′2 (t). Corollary 1.24 implies the existence of a unique that θ(t0 ) = θ0 and  α′ (t) · β ′ (t)     cos θ(t) = α′ (t) β′ (t) = (1.21)  α′ (t) · J β ′ (t)   =  sin θ(t) = α′ (t) β ′ (t) function θ : (a, b) → R such a′1 (t) , α′ (t) a′2 (t) . α′ (t) Then (1.21) is equivalent to (1.20). Α'HtL ΑHtL ΘHtL Figure 1.7: The turning angle of a plane curve 20 CHAPTER 1. CURVES IN THE PLANE Geometrically, the turning angle θ[α](t) is the angle between the horizontal and α′ (t). Next, we derive a relation (first observed by Kästner3 ) between the turning angle and the curvature of a plane curve. Lemma 1.26. The turning angle and curvature of a regular curve α in the plane are related by (1.22) θ[α]′ (t) = α′ (t) κ2[α](t). Proof. The derivative of the left-hand side of (1.20) is 1 α′′ (t) d ′ (1.23) . + α (t) dt α′ (t) α′ (t) Furthermore, by the chain rule (Lemma 1.9) it follows that the derivative of the right-hand side of (1.20) is (1.24) J α′ (t) θ ′ (t) −sin θ(t), cos θ(t) = θ′ (t) . J α′ (t) Setting (1.24) equal to (1.23) and taking the dot product with J α′ (t), we obtain α′′ (t) · J α′ (t) = α′ (t) α′ (t) θ′ (t) α′ (t) = 2 κ2[α](t). Equation (1.22) now follows.. Corollary 1.27. The turning angle and curvature of a unit-speed curve β in the plane are related by κ2[β](s) = dθ[β](s) . ds Curvature therefore measures rate of change of the turning angle with respect to arc length. 1.6 First Examples of Plane Curves The equation of a circle of radius a with center (0, 0) is x2 + y 2 = a2 . Its simplest parametrization is (1.25) 3 circle[a](t) = a(cos t, sin t), 0 6 t < 2π, Abraham Gotthelf Kästner (1719–1800). German mathematician, professor at Leipzig and Göttingen. Although Kästner’s German contemporaries ranked his mathematical and expository skills very high, Gauss found his lectures too elementary to attend. Gauss declared that Kästner was first mathematician among poets and first poet among mathematicians. 1.6. EXAMPLES OF PLANE CURVES 21 as in (1.7) but with a now representing the radius (not an initial parameter value). A more interesting parametrization is obtained by setting τ = tan(t/2), so as to give 1 − τ2 2τ , τ ∈ R, , β(τ ) = a 2 1+τ 1 + τ2 though the trace of β omits the point (−a, 0). We shall now study two classes of curves that include the circle as a special case. In both cases, we shall graph their curvature. Ellipses These are perhaps the next simplest curves after the circle. The name ‘ellipse’ (which means ‘falling short’) is due to Apollonius4 . The nonparametric form of the ellipse centered at the origin with axes of lengths a and b is x2 y2 + 2 = 1, 2 a b and the standard parametrization is (1.26) ellipse[a, b](t) = a cos t, b sin t , 0 6 t < 2π. P F- F+ Figure 1.8: An ellipse and its foci The curvature of this ellipse is κ2[ellipse[a, b]](t) = (b2 cos2 ab . t + a2 sin2 t)3/2 4 Apollonius of Perga (262–180 BC). His eight volume treatise on conic sections is the standard ancient source of information about ellipses, hyperbolas and parabolas. 22 CHAPTER 1. CURVES IN THE PLANE Its graph for a = 3, b = 4 (the ellipse in Figure 1.8) is illustrated below with an exaggerated vertical scale. The fact that there are two maxima and two minima is related to the four vertex theorem, which will be discussed in Section 6.5. 0.6 0.4 0.2 1 2 3 4 5 6 -0.2 Figure 1.9: The curvature of an ellipse An ellipse is traditionally defined as the locus of points such that the sum of the distances from two fixed points F− and F+ is constant. The points F− and F+ are called √ the foci of the ellipse. In the case of ellipse[a, b] with a > b the foci are ± a2 − b2 , 0 : Theorem 1.28. Let E be an ellipse with foci F− and F+ . Then (i) for any point P on E , the sum distance(F− , P ) + distance(F+ , P ) is constant. (ii) the tangent line to E at P makes equal angles with the line segments connecting P to F− and F+ . Proof. To establish these two properties of an ellipse, it is advantageous to use complex numbers. Clearly, the parametrization (1.26) is equivalent to (1.27) z(t) = 12 (a + b)eit + 21 (a − b)e−it (0 6 t < 2π). The vector from F∓ to P is given by z± (t) = 21 (a + b)eit + 12 (a − b)e−it ± √ a2 − b 2 . It is remarkable that this can be written as a perfect square √ 2 √ z± (t) = 21 (1.28) a + b eit/2 ± a − b e−it/2 . An easy calculation now shows that |z± (t)| = a ± p a2 − b2 cos t. Consequently, |z+ (t)| + |z− (t)| = 2a; this proves (i). 1.6. EXAMPLES OF PLANE CURVES 23 To prove (ii), we first use (1.27) to compute i i (a + b) eit − (a − b) e−it 2 2 √ √ √ i √ = a+b eit/2 + a+b e−it/2 a+b eit/2 − a+b e−it/2 . 2 It follows from (1.28) that √ √ z′ (t) z+ (t) a + b eit/2 + a + b e−it/2 √ = √ , i ′ = −i z (t) z− (t) a + b eit/2 − a + b e−it/2 z′ (t) = and (ii) follows once one expresses the three complex numbers z− (t), z′ (t), z− (t) in exponential form. Logarithmic spirals A logarithmic spiral is parametrized by (1.29) logspiral[a, b](t) = a(ebt cos t, ebt sin t). Changing the sign of b makes the sprial unwind the other way, with b = 0 corresponding to the circle. If b > 0, it is evident that the curvature decreases as t increases. We check this by graphing the function 1 √ , 1 + b2 obtained from Notebook 1, in the case corresponding to Figure 1.10. κ2[logspiral[a, b]](t) = aebt 4 2 -4 -2 2 -2 -4 Figure 1.10: The curve logspiral[1, 51 ] 4 24 CHAPTER 1. CURVES IN THE PLANE 12 10 8 6 4 2 -10 -5 5 10 Figure 1.11: The function κ2[logspiral[1, 15 ]](t) = 0.98e−0.2t Logarithmic spirals were first discussed by Descartes in 1638 in connection with a problem from dynamics. He was interested in determining all plane curves α with the property that the angle between α(t) and the tangent vector α′ (t) is constant. Such curves turn out to be precisely the logarithmic spirals, as we proceed to prove. Let α: (a, b) → R2 be a curve which does not pass through the origin. The tangent-radius angle of α is defined to be the function φ = φ[α]: (a, b) → R2 such that α′ (t) · α(t) = cos φ(t) α′ (t) α(t) and α′ (t) · J α(t) = sin φ(t) α′ (t) α(t) for a < t < b. The existence of φ[α] follows from Lemma 1.23; see Exercise 6. Lemma 1.29. Let α: (a, b) → R2 be a curve which does not pass through the origin. The following conditions are equivalent: (i) The tangent-radius angle φ[α] is constant; (ii) α is a reparametrization of a logarithmic spiral. Proof. We write α(t) = r eiθ with r = r(t) and θ = θ(t) both functions of t. Then α′ (t) = eiθ (r′ + i r θ′ ), so that p α(t) = r, α′ (t) = r′2 + r2 θ′2 and α(t) α′ (t) = r(r′ + i r θ′ ). Assume that (i) holds and let γ be the constant value of φ[α](t). Equation (1.4) implies that r′ + i r θ′ α′ (t)α(t) = √ (1.30) . eiγ = ′ α (t) α(t) r′2 + r2 θ′2 From (1.30) we get r′ cos γ = √ r′2 + r2 θ′2 and r θ′ sin γ = √ , r′2 + r2 θ′2 1.7. SEMICUBICAL PARABOLA 25 so that r′ = θ′ cot γ. r The solution of this differential equation is found to be r = a exp(θ cot γ), where a is a constant, and we obtain (1.31) α(t) = a exp (i + cot γ)θ(t) , which is equivalent to (1.29). Conversely, one can check that the tangent-radius angle of any logarithmic spiral is a constant γ. Under a reparametrization, the tangent-radius angle has the constant value ±γ. 1.7 The Semicubical Parabola and Regularity It is informative to calculate lengths and curvatures by hand in a simple case. We choose the semicubical parabola defined parametrically by sc(t) = (t2 , t3 ). (1.32) The nonparametric version of sc is obviously x3 = y 2 . The semicubical parabola has a special place in the history of curves, because it was the first algebraic curve (other than a straight line) to be found whose arc length function is algebraic. See [Lock, pages 3–11] for an extensive discussion. 1 0.5 0.5 1 -0.5 -1 Figure 1.12: The semicubical parabola 26 CHAPTER 1. CURVES IN THE PLANE We compute easily sc′ (t) = (2t, 3t2 ), so that κ2[sc](t) = J sc′ (t) = (−3t2 , 2t), (2, 6t) · (−3t2 , 2t) 3 (2t, 3t2 ) = sc′′ (t) = (2, 6t), 6 . |t|(4 + 9t2 )3/2 Also, the length of sc over the interval [0, t] is Z t Z t p 1 8 u 4 + 9 u2 du = (4 + 9t2 )3/2 − . ssc (t) = sc′ (u) du = 0 27 0 27 Let α : (a, b) → Rn be a curve, and let a < t0 < b. There are two ways that α can fail to be regular at t0 . On the one hand, regularity can fail because of the particular parametrization. For example, a horizontal straight line, when parametrized as t 7→ (t3 , 0), is not regular at 0. This kind of nonregularity can be avoided by changing the parametrization. A more fundamental kind of nonregularity occurs when it is impossible to find a regular reparametrization of a curve near a point. Definition 1.30. We say that a curve α: (a, b) → Rn is regular at t0 provided that it is possible to extend the function t 7→ α′ (t) α′ (t) to be a differentiable function at t0 . Otherwise α is said to be singular at t0 . It is easy to see that the definitions of regular and singular at t0 are independent of the parametrization of α. The notion of regular curve α: (a, b) → Rn , as defined on page 6, implies regularity at each t0 for a < t0 < b, but is stronger, because it implies that α′ (t0 ) must be nonzero. The following result may easily be deduced from the previous definition: Lemma 1.31. A curve α is regular at t0 if and only if there exists a unit-speed parametrization of α near t0 . For example, in seeking a unit-speed parametrization of sc, a computation shows that (2t, 3t2 ) sc′ (t) p = . sc′ (t) t2 (4 + 9t2 ) However, this is discontinuous at the origin because (2t, 3t2 ) (2t, 3t2 ) (−1, 0) = lim p 6= lim p = (1, 0), t↑0 t↓0 t2 (4 + 9t2 ) t2 (4 + 9t2 ) reflecting the singularity of (1.32) at t = 0. 1.8. EXERCISES 27 1.8 Exercises 1. Let a > 0. Find the arc length function and the curvature for each of the following curves, illustrated in Figure 1.13. (a) t 7→ a cos t + t sin t, sin t − t cos t ; t (b) t 7→ a cosh , t ; a (c) t 7→ a cos3 t, sin3 t ; (d) t 7→ a 2t, t2 . Figure 1.13: The curves of Exercise 1 2. Show that the velocity α′ of a differentiable curve α: (a, b) → R2 is given by α(t + δ) − α(t) α′ (t) = lim , a < t < b. δ→0 δ 3. Let β : (a, b) → R2 be a circle of radius r > 0 centered at q ∈ R2 . Using the fact that β(t) − q = r, show that κ2[β](t) = 1/r for a < t < b. 4. Suppose that β : (a, b) → R2 is a unit-speed curve whose curvature is given by κ2[β](s) = 1/r for all s, where r > 0 is a constant. Show that β is part of a circle of radius r centered at some point q ∈ R2 . 5. Verify the formula (1.14) for the curvature of a plane curve α: (a, b) → C. 6. Let α: (a, b) → R2 be a regular curve which does not pass through the origin. Fix t0 and choose φ0 such that α′ (t0 ) · α(t0 ) = cos φ0 α′ (t0 ) α(t0 ) and α′ (t0 ) · J α(t0 ) = sin φ0 . α′ (t0 ) α(t0 ) Establish the existence and uniqueness of a differentiable function φ = φ[α]: (a, b) → R such that φ(t0 ) = φ0 , α′ (t) · α(t) = cos φ(t) α′ (t) α(t) and α′ (t) · J α(t) = sin φ(t) α′ (t) α(t) 28 CHAPTER 1. CURVES IN THE PLANE for a < t < b. Geometrically, φ(t) represents the angle between the radius vector α(t) and the tangent vector α′ (t). 7. Find a unit-speed parametrization of the semicubical parabola t 7→ (t2 , t3 ), valid for t > 0. 8. Let γ a : (−∞, ∞) → R2 be the curve defined by   t, |t|a sin 1 if t 6= 0, t γ a (t) =  (0, 0) if t = 0. Figure 1.14: The curve γ 3/2 (i) Show that γ a is continuous if a > 0, but discontinuous if a 6 0. (ii) Show that γ a is differentiable if 1 < a < 2, but that the curve (γ a )′ is discontinuous. Chapter 2 Famous Plane Curves Plane curves have been a subject of much interest beginning with the Greeks. Both physical and geometric problems frequently lead to curves other than ellipses, parabolas and hyperbolas. The literature on plane curves is extensive. Diocles studied the cissoid in connection with the classic problem of doubling the cube. Newton1 gave a classification of cubic curves (see [Newton] or [BrKn]). Mathematicians from Fermat to Cayley often had curves named after them. In this chapter, we illustrate a number of historically-interesting plane curves. Cycloids are discussed in Section 2.1, lemniscates of Bernoulli in Section 2.2 and cardioids in Section 2.3. Then in Section 2.4 we derive the differential equation for a catenary, a curve that at first sight resembles the parabola. We shall present a geometrical account of the cissoid of Diocles in Section 2.5, and an analysis of the tractrix in Section 2.6. Section 2.7 is devoted to an illustration of clothoids, though the significance of these curves will become 1 Sir Isaac Newton (1642–1727).English mathematician, physicist, and astronomer. Newton’s contributions to mathematics encompass not only his fundamental work on calculus and his discovery of the binomial theorem for negative and fractional exponents, but also substantial work in algebra, number theory, classical and analytic geometry, methods of computation and approximation, and probability. His classification of cubic curves was published as an appendix to his book on optics; his work in analytic geometry included the introduction of polar coordinates. As Lucasian professor at Cambridge, Newton was required to lecture once a week, but his lectures were so abstruse that he frequently had no auditors. Twice elected as Member of Parliament for the University, Newton was appointed warden of the mint; his knighthood was awarded primarily for his services in that role. In Philosophiæ Naturalis Principia Mathematica, Newton set forth fundamental mathematical principles and then applied them to the development of a world system. This is the basis of the Newtonian physics that determined how the universe was perceived until the twentieth century work of Einstein. 39 40 CHAPTER 2. FAMOUS PLANE CURVES clearer in Chapter 5. Finally, pursuit curves are discussed in Section 2.8. There are many books on plane curves. Four excellent classical books are those of Cesàro [Ces], Gomes Teixeira2 [Gomes], Loria3 [Loria1], and Wieleitner [Wiel2]. In addition, Struik’s book [Stru2] contains much useful information, both theoretical and historical. Modern books on curves include [Arg], [BrGi], [Law], [Lock], [Sav], [Shikin], [vonSeg], [Yates] and [Zwi]. 2.1 Cycloids The general cycloid is defined by cycloid[a, b](t) = at − b sin t, a − b cos t . Taking a = b gives cycloid[a, a], which describes the locus of points traced by a point on a circle of radius a which rolls without slipping along a straight line. Figure 2.1: The cycloid t 7→ (t − sin t, 1 − cos t) In Figure 2.2, we plot the graph of the curvature κ2(t) of cycloid[1, 1](t) over the range 0 6 t 6 2π. Note that the horizontal axis represents the variable t, and not the x-coordinate t − sin t of Figure 2.1. A more faithful representation of the curvature function is obtained by expressing the parameter t as a function of x. In Figure 2.3, the curvature of a given point (x(t), y(t)) of cycloid[1, 12 ] is represented by the point (x(t), κ2(t)) on the graph vertically above or below it. 2 Francisco Gomes Teixeira (1851–1933). A leading Portuguese mathematician of the last half of the 19th century. There is a statue of him in Porto. 3 Gino Loria (1862–1954). Italian mathematician, professor at the University of Genoa. In addition to his books on curve theory, [Loria1] and [Loria2], Loria wrote several books on the history of mathematics. 2.1. CYCLOIDS 41 1 -1 1 2 3 4 5 7 6 -1 -2 -3 -4 -5 -6 -7 Figure 2.2: Curvature of cycloid[1, 1] 4 3 2 1 2 4 6 -1 -2 Figure 2.3: cycloid[1, 12 ] together with its curvature Consider now the case in which a and b are not necessarily equal. The curve cycloid[a, b] is that traced by a point on a circle of radius b when a concentric circle of radius a rolls without slipping along a straight line. The cycloid is prolate if a < b and curtate if a > b. Figure 2.4: The prolate cycloid t 7→ (t − 3 sin t, 1 − 3 cos t) 42 CHAPTER 2. FAMOUS PLANE CURVES Figure 2.5: The curtate cycloid t 7→ (2t − sin t, 2 − cos t) The final figure in this section demonstrates the fact that the tangent and normal to the cycloid always intersect the vertical diameter of the generating circle on the circle itself. A discussion of this and other properties of cycloids can be found in [Lem, Chapter 4] and [Wagon, Chapter 2]. Figure 2.6: Properties of tangent and normal Instructions for animating Figures 2.4–2.6 are given in Notebook 2. 2.2. LEMNISCATES OF BERNOULLI 43 2.2 Lemniscates of Bernoulli Each curve in the family (2.1) lemniscate[a](t) = a sin t cos t a cos t , , 1 + sin2 t 1 + sin2 t is called a lemniscate of Bernoulli4 . Like an ellipse, a lemniscate has foci F1 and F2 , but the lemniscate is the locus L of points P for which the product of distances from F1 and F2 is a certain constant f 2 . More precisely, L = (x, y) | distance (x, y), F1 distance (x, y), F2 = f 2 , where distance(F1 , F2 ) = 2f . This choice ensures that the midpoint of the segment connecting F1 with F2 lies on the curve L . Let us derive (2.1) from the focal property. Let the foci be (±f, 0) and let L be a set of points containing (0, 0) such that the product of the distances from F1 = (−f, 0) and F2 = (f, 0) is the same for all P ∈ L . Write P = (x, y). Then the condition that P lie on L is (2.2) (x − f )2 + y 2 (x + f )2 + y 2 = f 4 , or equivalently (2.3) (x2 + y 2 )2 = 2f 2 (x2 − y 2 ). √ The substitutions y = x sin t, f = a/ 2 transform (2.3) into (2.1). Figure 2.7 displays four lemniscates. Starting from the largest, each successively smaller one passes through the foci of the previous one. Figure 2.8 plots the curvature of one of them as a function of the x-coordinate. Figure 2.7: A family of lemniscates 4 Jakob Bernoulli (1654–1705). Jakob and his brother Johann were the first of a Swiss mathematical dynasty. The work of the Bernoullis was instrumental in establishing the dominance of Leibniz’s methods of exposition. Jakob Bernoulli laid basic foundations for the development of the calculus of variations, as well as working in probability, astronomy and mathematical physics. In 1694 Bernoulli studied the lemniscate named after him. 44 CHAPTER 2. FAMOUS PLANE CURVES Although a lemniscate of Bernoulli resembles the figure eight curve parametrized in the simper way by (2.4) eight[a](t) = sin t, sin t cos t , a comparison of the graphs of their respective curvatures shows the difference between the two curves: the curvature of a lemniscate has only one maximum and one minimum in the range 0 6 t < 2π, whereas (2.4) has three maxima, three minima, and two inflection points. 3 2 1 -1 -0.5 0.5 1 -1 -2 -3 Figure 2.8: Part of a lemniscate and its curvature In Section 6.1, we shall define the total signed curvature by integrating κ2 through a full turn. It is zero for both (2.1) and (2.4); for the former, this follows because the curvature graphed in Figure 2.8 is an odd function. Figure 2.9: A family of cardioids 2.3. CARDIOIDS 45 2.3 Cardioids A cardioid is the locus of points traced out by a point on a circle of radius a which rolls without slipping on another circle of the same radius a. Its parametric equation is cardioid[a](t) = 2a(1 + cos t) cos t, 2a(1 + cos t) sin t . The curvature of the cardioid can be simplified by hand to get κ2[cardioid[a]](t) = 3 . 8|a cos(t/2)| The result is illustrated below using the same method as in Figures 2.3 and 2.8. The curvature consists of two branches which meet at one of two points where the value of κ2 coincides with the cardioid’s y-coordinate. 6 4 Figure 2.10: A cardioid and its curvature 2.4 The Catenary In 1691, Jakob Bernoulli gave a solution to the problem of finding the curve assumed by a flexible inextensible cord hung freely from two fixed points; Leibniz has called such a curve a catenary (which stems from the Latin word catena, meaning chain). The solution is based on the differential equation s 2 dy d2 y 1 1+ (2.5) . = dx2 a dx 46 CHAPTER 2. FAMOUS PLANE CURVES To derive (2.5), we consider a portion pq of the cable between the lowest point p and an arbitrary point q. Three forces act on the cable: the weight of the portion pq, as well as the tensions T and U at p and q. If w is the linear density and s is the length of pq, then the weight of the portion pq is ws. U Θ q p T Figure 2.11: Definition of a catenary Let |T| and |U| denote the magnitudes of the forces T and U, and write U = |U| cos θ, |U| sin θ , with θ the angle shown in Figure 2.11. Because of equilibrium we have (2.6) |T| = |U| cos θ and w s = |U| sin θ. Let q = (x, y), where x and y are functions of s. From (2.6) we obtain dy ws = tan θ = . dx |T| (2.7) Since the length of pq is s= Z 0 xs 1+ dy dx 2 dx, the fundamental theorem of calculus tells us that s 2 ds dy (2.8) . = 1+ dx dx When we differentiate (2.7) and use (2.8), we get (2.5) with a = ω/|T |. 2.5. CISSOID OF DIOCLES 47 Although at first glance the catenary looks like a parabola, it is in fact the graph of the hyperbolic cosine. A solution of (2.5) is given by x (2.9) y = a cosh . a The next figure compares a catenary and a parabola having the same curvature at x = 0. The reader may need to refer to Notebook 2 to see which is which. 8 7 6 5 4 3 2 1 -4 -3 -2 -1 1 2 3 4 Figure 2.12: Catenary and parabola We rotate the graph of (2.9) to define t (2.10) catenary[a](t) = a cosh , t , a where without loss of generality, we assume that a > 0. A catenary is one of the few curves for which it is easy to compute the arc length function. Indeed, t |catenary[a]′ (t)| = cosh , a and it follows that a unit-speed parametrization of a catenary is given by ! r s s2 1 + 2 , arcsinh . s 7→ a a a In Chapter 15, we shall use the catenary to construct an important minimal surface called the catenoid. 2.5 The Cissoid of Diocles The cissoid of Diocles is the curve defined nonparametrically by (2.11) x3 + xy 2 − 2ay 2 = 0. 48 CHAPTER 2. FAMOUS PLANE CURVES To find a parametrization of the cissoid, we substitute y = xt in (2.11) and obtain 2at3 2at2 , y= . x= 2 1+t 1 + t2 Thus we define (2.12) cissoid[a](t) = 2at3 2at2 . , 1 + t2 1 + t2 The Greeks used the cissoid of Diocles to try to find solutions to the problems of doubling a cube and trisecting an angle. For more historical information and the definitions used by the Greeks and Newton see [BrKn, pages 9–12], [Gomes, volume 1, pages 1–25] and [Lock, pages 130–133]. Cissoid means ‘ivy-shaped’. Observe that cissoid[a]′ (0) = 0 so that cissoid is not regular at 0. In fact, a cissoid has a cusp at 0, as can be seen in Figure 2.14. The definition of the cissoid used by the Greeks and by Newton can best be explained by considering a generalization of the cissoid. Let ξ and η be two curves and A a fixed point. Draw a line ℓ through A cutting ξ and η at points Q and R respectively, and find a point P on ℓ such that the distance from A to P equals the distance from Q to R. The locus of such points P is called the cissoid of ξ and η with respect to A. Ξ Η A Q P R Figure 2.13: The cissoid of ξ and η with respect to the point A Then cissoid[a] is precisely the cissoid of a circle of radius a and one of its tangent lines with respect to the point diametrically opposite to the tangent line, as in Figure 2.14. Let us derive (2.11). Consider a circle of radius a centered at (a, 0). Let (x, y) be the coordinates of a point P on the cissoid. Then 2a = distance(A, S), so by the Pythagorean theorem we have 2.5. CISSOID OF DIOCLES 49 R P Q A S Figure 2.14: AR moves so that distance(A, P ) = distance(Q, R) (2.13) distance(Q, S)2 = distance(A, S)2 − distance(A, Q)2 2 = 4a2 − distance(A, R) − distance(Q, R) . The definition of the cissoid says that distance(Q, R) = distance(A, P ) = p x2 + y 2 . By similar triangles, distance(A, R)/(2a) = distance(A, P )/x. Therefore, (2.13) becomes 2 2a 2 2 distance(Q, S) = 4a − − 1 (x2 + y 2 ). (2.14) x On the other hand, distance(Q, S)2 = distance(R, S)2 − distance(Q, R)2 (2.15) = distance(R, S)2 − distance(A, P )2 2 2ay = − (x2 + y 2 ), x since by similar triangles, distance(R, S)/(2a) = y/x. Then (2.11) follows easily by equating the right-hand sides of (2.14) and (2.15). 50 CHAPTER 2. FAMOUS PLANE CURVES From Notebook 2, the curvature of the cissoid is given by κ2[cissoid[a]](t) = 3 , a|t|(4 + t2 )3/2 and is everywhere strictly positive. 50 40 30 20 10 -1 -0.5 0.5 1 Figure 2.15: Curvature of the cissoid 2.6 The Tractrix A tractrix is a curve α passing through the point A = (a, 0) on the horizontal axis with the property that the length of the segment of the tangent line from any point on the curve to the vertical axis is constant, as shown in Figure 2.16. Figure 2.16: Tangent segments have equal lengths 2.6. TRACTRIX 51 The German word for tractrix is the more descriptive Hundekurve. It is the path that an obstinate dog takes when his master walks along a north-south path. One way to parametrize the curve is by means of t (2.16) , tractrix[a](t) = a sin t, cos t + log tan 2 It approaches the vertical axis asymptotically as t → 0 or t → π, and has a cusp at t = π/2. To find the differential equation of a tractrix, write tractrix[a](t) = (x(t), y(t)). Then dy/dx is the slope of the curve, and the differential equation is therefore √ dy a2 − x2 (2.17) =− . dx x It can be checked (with the help of (2.19) overleaf) that the differential equation is satisfied by the components x(t) = a sin t and y(t) = a cos t + log(tan(t/2)) of (2.16), and that both sides of (2.17) equal cot t. a Hx,yL x Ha,0L Figure 2.17: Finding the differential equation The curvature of the tractrix is κ2[tractrix[1]](t) = −| tan t |. In particular, it is everywhere negative and approaches −∞ at the cusp. Figure 2.18 graphs κ2 as a function of the x-coordinate sin t, so that the curvature at a given point on the tractrix is found by referring to the point on the curvature graph vertically below. 52 CHAPTER 2. FAMOUS PLANE CURVES For future use let us record Lemma 2.1. A unit-speed parametrization of the tractrix is given by (2.18) Z sp  −s/a  −2t/a  1−e dt for 0 6 s < ∞, ,  ae 0 α(s) = Z sp    aes/a , 1 − e2t/a dt for −∞ < s 6 0. 0 Note that Z sp p p 1 − e−2t/a dt = a arctanh 1 − e−2s/a − a 1 − e−2s/a . 0 Proof. First, we compute (2.19) tractrix[a]′ (φ) = a cos φ, − sin φ + 1 . sin φ −s/a Define φ(s) = π − arcsin(e−s/a ) for s > 0. Then ; furthermore, p sin φ(s) = e −2s/a . Hence π/2 6 φ(s) < π for s > 0, so that cos φ(s) = − 1 − e (2.20) e−s/a sin φ(s) φ′ (s) = √ =− . −2s/a a cos φ(s) a 1−e 2 1 -1 -2 Figure 2.18: A tractrix and its curvature 2.7. CLOTHOIDS 53 Therefore, if we define a curve β by β(s) = tractrix[a] φ(s) , it follows from (2.19) and (2.20) that β ′ (s) = tractrix[a]′ φ(s) φ′ (s) sin φ(s) 1 − = a cos φ(s), − sin φ(s) + sin φ(s) a cos φ(s) p = − sin φ(s), − cos φ(s) = −e−s/a , 1 − e−2s/a = α′ (s). Also, β(0) = (a, 0) = α(0). Thus α and β coincide for 0 6 s < ∞, so that α is a reparametrization of a tractrix in that range. The proof that α is a reparametrization of tractrix[a] for −∞ < s 6 0 is similar. Finally, an easy calculation shows that α has unit speed. 2.7 Clothoids One of the most elegant of all plane curves is the clothoid or spiral of Cornu5 . We give a generalization of the clothoid by defining Z t n+1 n+1 Z t u u clothoid[n, a](t) = a sin du, cos du . n + 1 n +1 0 0 Clothoids are important curves used in freeway and railroad construction (see [Higg] and [Roth]). For example, a clothoid is needed to make the gradual transition from a highway, which has zero curvature, to the midpoint of a freeway exit, which has nonzero curvature. A clothoid is clearly preferable to a path consisting of straight lines and circles, for which the curvature is discontinuous. The standard clothoid is clothoid[1, a], represented by the larger one of two plotted in Figure 2.19. The quantities Z t Z t sin(πu2/2)du and cos(πu2/2)du, 0 0 6 are called Fresnel integrals; clothoid[1, a] is expressible in terms of them. Since √ Z ±∞ Z ±∞ π sin(u2 /2)du = cos(u2 /2)du = ± 2 0 0 5 Marie Alfred Cornu (1841–1902). French scientist, who studied the clothoid in connection with diffraction. The clothoid was also known to Euler and Jakob Bernoulli. See [Gomes, volume 2, page 102–107] and [Law, page 190]. 6 Augustin Jean Fresnel (1788–1827). French physicist, one of the founders of the wave theory of light. 54 CHAPTER 2. FAMOUS PLANE CURVES (as is easily checked by computer), the ends of the clothoid[1, a] curl around the √ points ± 21 a π(1, 1). The first clothoid is symmetric with respect to the origin, but the second one (smaller in Figure 2.19) is symmetric with respect to the horizontal axis. The odd clothoids have shapes similar to clothoid[1, a], while the even clothoids have shapes similar to clothoid[2, a]. 1.5 1 0.5 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -1 -1.5 Figure 2.19: clothoid[1, 1] and clothoid[2, 21 ] Although the definition of clothoid[n, a] is quite complicated, its curvature is simple: tn κ2[clothoid[n, a]](t) = − . (2.21) a In Chapter 5, we shall show how to define the clothoid as a numerical solution to a differential equation arising from (2.21). 2.8 Pursuit Curves The problem of pursuit probably originated with Leonardo da Vinci. It is to find the curve by which a vessel moves while pursuing another vessel, supposing that the speeds of the two vessels are always in the same ratio. Let us formulate this problem mathematically. Definition 2.2. Let α and β be plane curves parametrized on an interval [a, b]. We say that α is a pursuit curve of β provided that (i) the velocity vector α′ (t) points towards the point β(t) for a < t < b; that is, α′ (t) is a multiple of α(t) − β(t); 2.8. PURSUIT CURVES 55 (ii) the speeds of α and β are related by kα′ k = kkβ′ k, where k is a positive constant. We call k the speed ratio. A capture point is a point p for which p = α(t1 ) = β(t1 ) for some t1 . In Figure 2.20, α is the curve of the pursuer and β the curve of the pursued. Β Α Figure 2.20: A pursuit curve When the speed ratio k is larger than 1, the pursuer travels faster than the pursued. Although this would usually be the case in a physical situation, it is not a necessary assumption for the mathematical analysis of the problem. We derive differential equations for pursuit curves in terms of coordinates. Lemma 2.3. Write α = (x, y) and β = (f, g), and assume that α is a pursuit curve of β. Then (2.22) and (2.23) x′2 + y ′2 = k 2 (f ′2 + g ′2 ) x′ (y − g) − y ′ (x − f ) = 0. Proof. Equation (2.22) is the same as kα′ k = kkβ′ k. To prove (2.23), we observe that α(t) − β(t) = x(t) − f (t), y(t) − g(t) and α′ (t) = x′ (t), y ′ (t) . Note that the vector J α(t)−β(t) = −y(t)+g(t), x(t)−f (t) is perpendicular to α(t)−β(t). The condition that α′ (t) is a multiple of α(t)−β(t) is conveniently expressed by saying that α′ (t) is perpendicular to J α(t) − β(t) , which is equivalent to (2.23). Next, we specialize to the case when the curve of the pursued is a straight line. Assume that the curve β of the pursued is a vertical straight line passing through the point (a, 0), and that the speed ratio k is larger than 1. We want to find the curve α of the pursuer, assuming the initial conditions α(0) = (0, 0) and α′ (0) = (1, 0). 56 CHAPTER 2. FAMOUS PLANE CURVES We can parametrize β as β(t) = a, g(t) . Furthermore, the curve α of the pursuer can be parametrized as α(t) = t, y(t) . The condition (2.22) becomes 1 + y ′2 = k 2 g ′2 , (2.24) and (2.23) reduces to (y − g) − y ′ (t − a) = 0. Differentiation with respect to t yields −y ′′ (t − a) = g ′ . (2.25) From (2.24) and (2.25) we get (2.26) 1 + y ′2 = k 2 (a − t)2 y ′′2 . Let p = y ′ ; then (2.26) can be rewritten as k dp dt p . = 2 a−t 1+p This separable first-order equation has the solution 1 a−t arcsinh p = − log (2.27) , k a when we make use of the initial condition y ′ (0) = 0. Then (2.27) can be rewritten as y ′ = p = sinh arcsinh p = 12 earcsinh p − e−arcsinh p −1/k 1/k ! a−t a−t 1 = . − 2 a a Integrating, with the initial condition y(0) = 0, yields 1+ 1/k 1− 1/k ! ak a−t ak a−t ak 1 (2.28) y= 2 . − + k −1 2 k+1 a k−1 a The curve of the pursuer is then α(t) = (t, y(t)), where y is given by (2.28). Since α(t1 ) = β(t1 ) if and only if t1 = a, the capture point is ak (2.29) . p = a, 2 k −1 2.9. EXERCISES 57 The graph below depicts the case when a = 1 and k has the values 2, 3, 4, 5. As the speed ratio k becomes smaller and smaller, the capture point goes higher and higher. k=5 yHtL k=4 0.6 k=3 0.5 k=2 0.4 0.3 0.2 0.1 t 0.2 0.4 0.6 1 0.8 Figure 2.21: The case in which the pursued moves in a straight line 2.9 Exercises M 1. Graph the curvatures of the cycloids illustrated in Figures 2.4 and 2.5. Find the formula for the curvature κ2 of the general cycloid cycloid[a, b]. Then define and draw ordinary, prolate and curtate cycloids together with the defining circle such as those on page 42. M 2. A deltoid is defined by deltoid[a](t) = 2a cos t(1 + cos t) − a, 2a sin t(1 − cos t) . The curve is so named because it resembles a Greek capital delta. It is a particular case of a curve called hypocycloid (see Exercise 13 of Chapter 6). Plot as one graph the deltoids deltoid[a] for a = 1, 2, 3, 4. Graph the curvature of the first deltoid. 58 CHAPTER 2. FAMOUS PLANE CURVES M 3. The Lissajous7 or Bowditch curve8 is defined by lissajous[n, d, a, b](t) = a sin(nt + d), b sin t . Draw several of these curves and plot their curvatures. (One is shown in Figure 11.19 on page 349.) M 4. The limaçon, sometimes called Pascal’s snail, named after Étienne Pascal, father of Blaise Pascal9 , is a generalization of the cardioid. It is defined by limacon[a, b](t) = (2a cos t + b) cos t, sin t . Find the formula for the curvature of the limaçon, and plot several of them. 5. Consider a circle with center C = (0, a) and radius a. Let ℓ be the line tangent to the circle at (0, 2a). A line from the origin O = (0, 0) intersecting ℓ at a point A intersects the circle at a point Q. Let x be the first coordinate of A and y the second coordinate of Q, and put P = (x, y). As A varies along ℓ the point P traces out a curve called versiera, in Italian and misnamed in English as the witch of Agnesi10 . Verify that a parametrization of the Agnesi versiera is agnesi[a](t) = 2a tan t, 2a cos2 t . 7 Jules Antoine Lissajous (1822–1880). French physicist, who studied similar curves in 1857 in connection with his optical method for studying vibrations. 8 Nathaniel Bowditch curve (1773–1838). American mathematician and astronomer. His New American Practical Navigator, written in 1802, was highly successful. Bowditch also translated and annotated Laplace’s Mécanique Céleste. His study of pendulums in 1815 included the figures named after him. Preferring his post as president of the Essex Fire and Marine Insurance Company from 1804 to 1823, Bowditch refused chairs of mathematics at several universities. 9 Blaise Pascal (1623–1662). French mathematician, philosopher and inventor. Pascal was an early investigator in projective geometry and invented the first mechanical device for performing addition and subtraction. 10 Maria Gaetana Agnesi (1718–1799). Professor at the University of Bologna. She was the first woman to occupy a chair of mathematics. Her widely used calculus book Instituzioni Analitiche was translated into French and English. 2.9. EXERCISES 59 M 6. Define the curve tschirnhausen[n, a](t) = a sin t cos t , a (cos(t/3))n (cos(t/3))n . When n = 1, this curve is attributed to Tschirnhausen11 . Find the formula for the curvature of tschirnhausen[n, a][t] and make a simultaneous plot of the curves for 1 6 n 6 8. 7. In the special case that the speed ratio is 1, show that the equation for the pursuit curve is 2 ! a a−t a−t y(t) = , − 1 − 2 log 4 a a and that the pursuer never catches the pursued. M 8. Equation (2.28) defines the function y(t) = k(a − t)1+ 1/k ak a1/k k(a − t)1− 1/k + . − k2 − 1 2(k − 1) 2a1/k (1 + k) Plot a pursuit curve with a = 1 and k = 1.2. 11 Ehrenfried Walter Tschirnhausen (1651–1708). German mathematician, who tried to solve equations of any degree by removing all terms except the first and last. He contributed to the rediscovery of the process for making hard-paste porcelain. Sometimes the name is written von Tschirnhaus. Chapter 3 Alternative Ways of Plotting Curves So far, we have mainly been plotting curves in parametric form. In Section 3.1, we discuss implicitly defined curves, and contrast them with those parametrized in the previous chapters. A curve is defined implicitly as the ‘zero set’ or set of zeros of a differentiable function of two variables. In particular cases, this is the more natural way of defining the curve. Even so, if one is given the nonparametric form of a curve, it is sometimes easier to find a suitable parametrization before attempting to plot it. In any case, it can be important to be able to switch between the two representations of the curve. As an initial example, after lines and circles, we recall that the cissoid was first defined as the set of zeros of a cubic equation. Similar techniques can be applied to another cubic curve, the folium of Descartes. The latter is studied in Section 3.2, which provides an example of what can happen to a curve as one deforms its initial equation. Cassinian ovals, which we consider in Section 3.3, form a further class of curves that are easily plotted implicitly. This generalizes the lemniscate by requiring that the product of the distances from a point on the curve to two fixed foci be constant without regard to the distance between the foci; the resulting equation is quartic. As the second topic of this chapter, we shall use polar coordinates to describe curves in Section 3.4. We use this approach to give a simple generalization of a cardioid and limaçon, and to study related families of closed curves. We also compute lengths and curvature in polar coordinates, using the formalism of complex numbers to prove our results. The use of polar coordinates leads to an investigation of different types of spirals in Section 3.5. A number of these have especially simple polar equations. Further examples of implicit and polar plots can be found in this chapter’s exercises and notebook. 73 74 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES 3.1 Implicitly Defined Plane Curves We now consider a way of representing a plane curve that does not involve an explicit parametrization. Definition 3.1. Let F : R2 → R be any function. The set of zeros of F is F −1 (0) = p ∈ R2 | F (p) = 0 . If no restrictions are placed on F , then not much can be said about its set of zeros, so we make an additional assumption. Definition 3.2. An implicitly defined curve in R2 is the set of zeros of a differentiable function F : R2 → R. Frequently, we refer to the set of zeros as ‘the curve F (x, y) = 0’. We point out that the analogous theory for nonparametrically defined surfaces will be presented in Section 10.6. Even when F is assumed to be differentiable, the set of zeros of F may have cusps, and hence appear to be nondifferentiable. Moreover, a theorem of Whitney1 states that any closed subset of R2 is the set of zeros of some differentiable function (see [BrLa, page 56]). However, there is an important case when it is possible to find a parametrized curve whose trace is the set of zeros of F . Theorem 3.3. Let F : R2 → R be a differentiable function, and let q = (q1 , q2 ) be a point such that F (q) = 0. Assume that at least one of the partial derivatives Fx , Fy is nonzero at q. Then there is a neighborhood U of q in R2 and a parametrized curve α: (a, b) → R2 such that the trace of α is precisely p ∈ U | F (p) = 0 . Proof. Suppose, for example, that Fv (q) 6= 0. The Implicit Function Theorem states that there is a differentiable real-valued function g defined on a small neighborhood of q1 in R such that g(q1 ) = q2 and t 7→ F (t, g(t)) vanishes identically. Then we define α(t) = (t, g(t)). Definition 3.4. Let C be a subset of R2 . If F : R2 → R is a differentiable func- tion with F −1 (0) = C , we say that the equation F (x, y) = 0 is a nonparametric form or implicit form of C . If α: (a, b) → R2 is a curve whose trace is C , we say that t 7→ α(t) is a parametrization or parametric form of C . 1 Hassler Whitney (1907–1989). Influential American differential-topologist. 3.1. IMPLICITLY DEFINED PLANE CURVES 75 To start with some examples using the operations of vector calculus, let p, v ∈ R2 with v 6= 0, and define functions F [p, v], G[p, v]: R2 −→ R by (3.1) F [p, v](q) = (q − p) · v G[p, v](q) = kq − vk2 − kp − vk2 . Then F [p, v]−1 (0) is the straight line through p perpendicular to v and the curve G[p, v]−1 (0) is the circle with center v that contains p. Some curves are more naturally defined implicitly. This was the case of the cissoid, for which it is easy to pass between the two representations. For given the parametric form (2.12), we have no difficulty in spotting that t = y/x and reversing the derivation on page 47. Substituting this expression for t into the equation x = 2at/(1 + t2 ) yields the nonparametric form x= 2ay 2 x2 + y 2 of the cissoid, which is equivalent to the cubic equation (2.11). 2 1 -3 -2 -1 1 2 3 -1 -2 -3 -4 Figure 3.1: The folium of Descartes, x3 + y 3 = 3xy The trace of a curve α : (a, b) → R2 is always connected. This is a special case of the theorem from topology that states that the image under a continuous map of a connected set remains connected [Kelley]. It can easily happen however that the set of zeros of a differentiable function F : R2 → R is disconnected. 76 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES This means that the notions of parametric form and nonparametric form are essentially different. We give an example of a disconnected zero set immediately below, by perturbing a slightly different cubic equation, that used to plot Figure 3.1. 3.2 The Folium of Descartes The folium of Descartes2 is defined nonparametrically by the equation x3 + y 3 = 3xy. (3.2) In order to formalize this definition, we consider the family Fε of functions defined by Fε (x, y) = x3 + y 3 − 3 xy − ε, and their associated curves Fε = {(x, y) ∈ R2 | Fε (x, y) = 0}. Thus, Fε = Fε−1 (0) is the set of zeros of the function Fε , and F0 is the folium of Descartes. Whilst the latter is connected, Fε is disconnected for general values 1 . of ε; Figure 3.2 illustrates the case ε = − 10 2 1 -3 -2 -1 1 2 3 -1 -2 -3 -4 Figure 3.2: Perturbed folium, F−0.1 2 It was Huygens who first drew the curve correctly. See [Still1, pages 67–68] 3.2. FOLIUM OF DESCARTES 77 We can find a parametrization for F0 , by observing that F0 is close to a homogeneous polynomial. Dividing the equation F0 = 0 by x3 , we obtain 1 + t3 = 3 t, x where t = y/x (as for the cissoid, in the previous section). Solving for x, and then y = tx, yields the parametrization 3t 3t2 folium(t) = (3.3) . , 1 + t3 1 + t3 Such a rational parametrization cannot be found for the curve Fε , unless ε equals 0 or 1. This follows from the fact that Fε is a nonsingular cubic curve if ε 6= 0, 1, and from standard results on the projective geometry of cubic curves. We explain briefly how this theory applies to our example, but refer the reader to [BrKn, Kir] for more details. The equation Fε = 0 can be fully ‘homogenized’ by inserting powers of a third variable z so that each term has total degree equal to 3; the result is x3 + y 3 − 3xyz − εz 3 = 0. (3.4) This equation actually defines a curve not in R2 , but in the real projective plane RP2 , a surface that we define in Chapter 11. Linear transformations in the variables x, y, z determine projective transformations of RP3 , and can be used to find an equivalent form of the equation of the cubic. For example, applying the linear change of coordinates  1   x = 2 X + Y, (3.5) y = 21 X − Y,   z = Z − X, converts (3.4) into (3.6) (1 + ε)2 X 3 − (3ε + 34 )X 2 Z + 3εXZ 2 − εZ 3 + 3Y 2 Z = 0 (Exercise 8). Finally, we set Z = 1 in order to return to an equation in two variables, namely (3.7) Y 2 = − 31 (1 + ε)2 X 3 + (ε + 14 )X 2 − εX + 13 ε. This is a type of ‘canonical equation’ of the cubic curve Fε . If we write (3.7) as Y 2 = p(X), then p is a cubic polynomial provided ε 6= −1; if ε = −1, (3.6) is divisible by Z, and p arises from the remaining quadratic factor. The cubic curve Fε is said to be nonsingular if p has three distinct roots (two of which may be complex), and it can be checked in Notebook 3 78 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES that this is the case provided ε 6= 0, 1. If ε = 0, then the origin (0, 0) counts as a singular point in the implicit theory because there is no unique tangent line there (see Figure 3.3). The cissoid is a cubic curve with a different type of singularity. To parametrize a nonsingular cubic curve, one needs the Weierstrass ℘ function, a special case of which is defined (albeit with complex numbers) in Section 22.7. The analysis there should make it clear that there is no elementary parametrization of the curve Fε in general. 2 1 -2 -1 1 2 3 -1 -2 -3 Figure 3.3: Folium of Descartes with asymptote We conclude this section with some remarks of a more practical nature. The vector-valued function (3.3) was used to produce Figure 3.3, in which the asymptote is generated automatically by the plotting program. The latter is unable to distinguish a pair of points far apart in opposite quadrants from the many pairs of adjacent points selected on the curve. This phenomenon is one possible disadvantage of parametric plotting, though we explain in Notebook 3 how the problem can be overcome to advantage. 1.4 4 1.2 3.5 3 1 2.5 0.8 2 0.6 1.5 -30 -20 -10 0.4 1 0.2 0.5 10 20 30 -3 -2 -1 Figure 3.4: Folium curvature 1 2 3 3.3. CASSINIAN OVALS 79 The curvature of F0 can be obtained from the parametrization (3.3). To understand the resulting function t 7→ κ2[folium](t), we plot it first over the range −30 < t < 30 and then over the range −3 < t < 3. These two plots constitute Figure 3.4. The determination of the maxima and minima of the curvature can be carried out by computer, and is explained in Notebook 3. 3.3 Cassinian Ovals A Cassinian oval3 is a generalization of the lemniscate of Bernoulli, and therefore of the ellipse. It is the locus Ca,b = (x, y) | distance (x, y), F1 distance (x, y), F2 = b2 , where F1 , F2 are two fixed points with distance(F1 , F2 ) = 2a. The constants a, b are unrelated, though setting a = b = f , the curve Cf,f coincides with the lemniscate L defined on page 43. 1 0.5 -2 -1 1 2 1 2 1 2 -0.5 -1 0.75 0.5 0.25 -3 -2 -1 -0.25 -0.5 -0.75 3 1 0.5 -2 -1 -0.5 -1 Figure 3.5: Ovals with b = 2 and 200a = 199, 200, 201 respectively 3 Gian Domenico Cassini (1625–1712). Italian astronomer, who did his most important work in France. He proposed the fourth degree curves now called ovals of Cassini to describe planetary motion. 80 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES To study Cassinian ovals, we first define cassiniimplicit[a, b](x, y) = (x2 + y 2 + a2 )2 − b4 − 4a2 x2 . Generalizing the computations of Section 2.2, one can prove that Ca,b is the zero set of this function. This is the next result, whose proof we omit. Lemma 3.5. An oval of Cassini with (±a, 0) as foci is the implicitly defined curve (x2 + y 2 )2 + 2a2 (y 2 − x2 ) = b4 − a4 . The zero set of cassiniimplicit[f, f ](x, y) = (x2 + y 2 )2 + 2f 2 (y 2 − x2 ) is the lemniscate L = Cf,f , while cassiniimplicit[0, b] is of course a circle. √ The horizontal intercepts of cassiniimplicit[a, b] are (± a2 ± b2 , 0). When a2 > b2 , there are four horizontal intercepts, but when a2 < b2 there are only two horizontal intercepts, since we must exclude imaginary solutions. Similarly, cassiniimplicit[a, b] has no vertical intercepts when a2 > b2 , whereas for a2 < b2 √ the vertical intercepts are (0, ± b2 − a2 ). This is shown in Figure 3.5. Figure 3.6: A Cassinian family 3.4. PLANE CURVES IN POLAR COORDINATES 81 3.4 Plane Curves in Polar Coordinates In this section we show how to study and compute the length and curvature of a plane curve using polar coordinates. Definition 3.6. A polar parametrization is a curve γ : (a, b) → R2 of the form (3.8) γ(θ) = r[γ] θ)(cos θ, sin θ , where r[γ](θ) > 0 for a < θ < b. We call r[γ] the radius function of the curve γ, and abbreviate it to r when there is no danger of confusion. The radius function completely determines the polar parametrization, so usually a curve is described in polar coordinates simply by giving the definition of r. A polar parametrization of a plane curve is often very simple. Its description is simplified further by writing just the definition of the radius function of the curve. We shall see shortly that there are formulas for the arc length and curvature of a polar parametrization in terms of the radius function alone. Let us give a generalization of the cardioid and limaçon (see Exercise 4 of Chapter 2) by defining the radius function limaconpolar[n, a, b](θ) = 2a cos nθ + b, Observe that limaconpolar[1, a, 2a](θ) = 2a(cos θ + 1) is the radius function of a standard cardioid. More complicated examples are exhibited in Figure 3.7. An analogous polar parametrization is (3.9) pacman[n](θ) = 1 + cosn θ, and its name is justified by the example shown in Figure 3.13 and Exercise 10. Figure 3.7: The polar limaçons limaconpolar[6, 3, 1] and limaconpolar[13, 12 , 2] 82 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES Next, we derive the polar coordinate formulas for arc length and curvature. Lemma 3.7. The length and curvature of the polar parametrization (3.8) are given in terms of the radius function r = r[γ] by the formulas Z bp length[γ] = (3.10) r′ (θ)2 + r(θ)2 dθ, a (3.11) κ2[γ] = −r′′ r + 2r′2 + r2 . (r′2 + r2 )3/2 Proof. The calculations can be carried out most easily using complex numbers. Equation (3.8) can be written more succinctly as γ(θ) = r(θ)eiθ . We get Therefore (3.12) γ ′ (θ) = γ ′′ (θ) = r′ (θ) + ir(θ) eiθ r′′ (θ) + 2ir′ (θ) − r(θ) eiθ . kγ ′ (θ)k2 = r′ (θ)2 + r(θ)2 , and (3.10) follows immediately from the definition of length. Furthermore, using Lemma 1.2 we find that n o γ ′′ (θ) · Jγ ′ (θ) = Re r′′ (θ) + 2ir′ (θ) − r(θ) eiθ i(r′ (θ) + ir(θ))eiθ n o = Re r′′ (θ) + 2ir′ (θ) − r(θ) − ir′ (θ) − r(θ) = −r′′ (θ)r(θ) + 2r′ (θ)2 + r(θ)2 . Then (3.11) now follows from (3.12) and (1.12). 3.5 A Selection of Spirals A good example of a polar parametrization is the logarithmic spiral, that was studied in some detail towards the end of Chapter 1. There, we saw that its radius function is given by logspiralpolar[a, b](θ) = aebθ . The curve is characterized by Lemma 1.29 on page 24, and for this reason is sometimes called the equiangular spiral [Coxeter, 8.7]. A somewhat more naı̈ve equation for a spiral in polar coordinates is (3.13) rn = an θ, where n is a nonzero integer. Several of the resulting curves have been given particular names, and we now plot a few of them. 3.5. SELECTION OF SPIRALS 83 7.5 5 2.5 -5 -10 5 10 -2.5 -5 -7.5 -10 Figure 3.8: The spiral of Archimedes4 , r = θ 3 2 1 -3 -2 -1 1 2 3 -1 -2 -3 Figure 3.9: Fermat’s spiral r2 = θ 4 Archimedes of Syracuse (287–212 BC). Archimedes is credited with the creation of many mechanical devices such as compound pulleys, water clocks, catapults and burning mirrors. Legend has it that he was killed by a Roman soldier as he traced mathematical figures in the sand during the siege of Syracuse, at that time a Greek colony on what is now Sicily. His mathematical work included finding the area of a circle and the area under a parabola using the method of exhaustion. 84 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES The spiral (3.13) has two branches, the second obtained by allowing the radius function to be negative. The two branches join up provided the exponent n is positive. For Archimedes’ spiral, the two branhes are reflections of each other in the vertical y-axis. By contrast, for Fermat’s spiral5 in Figure 3.9, the two branches are rotations of each other by 180o . Similar behavior is seen in Figure 3.10. However, when n is negative, θ cannot be allowed to pass through zero, and this explains why the two branches fail to join up. Figure 3.10: The hyperbolic spiral r = 3/θ and lituus r2 = 1/θ Finally, we use the formula (3.11) to compute the curvature of a spiral given by (3.13), as a function of the angle θ. The result, κ2(θ) = nθ1 − a/n (1 + n + n2 θ2 ) , a(1 + n2 θ2 )3/2 is finite for all θ. In the special case n = −1 and a = 3, we get κ2(θ) = − θ6 , 3(1 + θ2 )3/2 corresponding to Figure 3.10 (left). 5 Pierre de Fermat (1601–1665). Fermat, like his contemporary Descartes, was trained as a lawyer. Fermat generalized the work of Archimedes on spirals. Like Archimedes he had the germs of the ideas of both differential and integral calculus – using the new techniques of analytic geometry – but failed to see the connections. Fermat is most remembered for stating the theorem that xn + y n = z n has no solution in integers for n > 2. 3.6. EXERCISES 85 3.6 Exercises 1. Find the nonparametric form of the following curves: (a) α(t) = (t15 , t6 ). (b) The hyperbola-like curve γ(t) = (t3 , t−4 ). (c) The strophoid defined by strophoid[a](t) = a t2 − 1 t(t2 − 1) , t2 + 1 t2 + 1 . 2. The devil’s curve is defined nonparametrically as the zeros of the function devilimplicit[a, b](x, y) = y 2 (y 2 − b2 ) − x2 (x2 − a2 ). Find its intercepts, and compare the result with Figure 3.11. 3 2 1 -4 -2 2 4 -1 -2 -3 Figure 3.11: The devil’s curve M 3. Kepler’s folium6 is the curve defined nonparametrically as the set of zeros of the function keplerimplicit[a, b](x, y) = (x − b)2 + y 2 x(x − b) + y 2 − 4a(x − b)y 2 . Plot Kepler’s folium and some of its perturbations. 6 Johannes Kepler (1571–1630). German astronomer, who lived in Prague. By empirical observations Kepler showed that a planet moves around the sun in an elliptical orbit having the sun at one of its two foci, and that a line joining the planet to the sun sweeps out equal areas in equal times as the planet moves along its orbit. 86 CHAPTER 3. ALTERNATIVE WAYS OF PLOTTING CURVES M 4. Plot the following curves with interesting singularities ([Walk, page 56]): (a) x3 − x2 + y 2 = 0, (b) x4 + x2 y 2 − 2x2 y − xy 2 + y 2 = 0, (c) 2x4 − 3x2 y + y 2 − 2y 3 − y 4 = 0, (d) x3 − y 2 = 0, (e) (x2 + y 2 )2 + 3x2 y − y 3 = 0, (f) (x2 + y 2 )3 − 4x2 y 2 = 0. M 5. The parametric equations for the trisectrix of Maclaurin7 are trisectrix[a](t) = a(4 cos2 t − 3), a(1 − 4 cos2 t) tan t . Plot it, using its simpler polar equation θ 3 polartrisectrix[a](θ) = a sec . 6. A nephroid is the kidney-shaped curve epicycloid[2b, b] = b 3 cos t − cos 3t, 2 sin t − sin 2t , that results from setting a = 2b in Exercise 10 on page 144. Show that its equation in polar coordinates is r = 2b θ sin 2 32 θ + cos 2 32 32 . 7. The general semicubical parabola is defined implicitly by y 2 = ax3 + bx2 + cx + d Find appropriate values of a, b, c, d so as to duplicate Figure 3.12. 7 Colin Maclaurin (1698–1746). Scottish mathematician. His Geometrica organica, sive discriptio linearum curvarum universalis dealt with general properties of conics and higher plane curves. In addition to Maclaurin’s own results, this book contained the proofs of many important theorems that Newton had given without proofs. Maclaurin is also known for his work on power series and the defense of his book Theory of Fluxions against the religious attacks of Bishop George Berkeley. 3.6. EXERCISES 87 Figure 3.12: Semicubical curves y 2 = a x3 + b x2 + c x + d M 8. Verify that the transformation (3.5) does indeed convert the equation (3.4) on page 77 into (3.6) and (3.7). 9. Show that, up to sign, the signed curvature of the curve g(x, y) = 0 is κ2(x, y) = gxx gy2 − 2gxy gx gy + gyy gx2 . 3/2 gx2 + gy2 M 10. Draw a selection of the curves (3.9) for n between 1 and 1000, paying attention to the difference in behavior when n is even or odd. Figure 3.13: pacman[999] Show that pacman[n] has finite curvature at θ = 0 but infinite curvature at θ = π. Chapter 4 New Curves from Old Any plane curve gives rise to other plane curves through a variety of general constructions. Each such construction can be thought of as a function which assigns one curve to another, and we shall discover some new curves in this manner. Four classic examples of constructing one plane curve from another are studied in the present chapter: evolutes in Sections 4.1 and 4.2, involutes in Section 4.3, parallel curves in Section 4.5 and pedal curves in Section 4.6. Along the way, we show how to construct normal and tangent lines to a curve, and osculating circles to curves. We explain in Section 4.4 that the circle through three points on a plane curve tends to the osculating circle as the three points become closer and coincide. For the same reason, the evolute of a plane curve can be visualized by plotting a sufficient number of normal lines to the curve, as illustrated by the well-known design in Figure 4.12 on page 111. 4.1 Evolutes A point p ∈ R2 is called a center of curvature at q of a curve α: (a, b) → R2 , provided that there is a circle γ with center p which is tangent to α at q such that the curvatures of the curves α and γ, suitably oriented, are the same at q. We shall see that this implies that there is a line ℓ from p to α which meets α perpendicularly at q, and the distance from p to q is the radius of curvature of α at q, as defined on page 14. An example is shown in Figure 4.2. The centers of curvature form a new plane curve, called the evolute of α, whose precise definition is as follows. Definition 4.1. The evolute of a regular plane curve α is the curve given by (4.1) evolute[α](t) = α(t) + 99 J α′ (t) 1 . κ2[α](t) α′ (t) 100 CHAPTER 4. NEW CURVES FROM OLD 1 0.5 -1.5 -1 -0.5 0.5 1 1.5 -0.5 -1 Figure 4.1: The ellipse and its evolute It turns out that a circle with center evolute[α](t) and radius 1/|κ2[α](t)| will be tangent to the plane curve α at α(t). This is the circle, called the osculating circle, that best approximates α near α(t); it is studied in Section 4.4. Using Formula (1.12), page 14, we see that the formula for the evolute can be written more succinctly as (4.2) evolute[α] = α + kα′ k2 J α′ . · J (α′ ) α′′ An easy consequence of (4.1) and (1.15) is the following important fact. Lemma 4.2. The definition of evolute of a curve α is independent of parametrization, so that evolute[α ◦ h] = evolute[α] ◦ h, for any differentiable function h: (c, d) → (a, b). The evolute of a circle consists of a single point. The evolute of any plane curve γ can be described physically. Imagine light rays starting at all points of the trace of γ and propagating down the normals of γ. In the case of a circle, these rays focus perfectly at the center, so for γ the focusing occurs along the centers of best fitting circles, that is, along the evolute of γ. A more interesting example is the evolute of the ellipse x2 /a2 + y 2 /b2 = 1, parametrized on page 21. This evolute is the curve γ defined by 2 (a − b2 ) cos3 t (a2 − b2 ) sin3 t γ(t) = (4.3) , , a b 4.1. EVOLUTES 101 and is an astroid1 (see Exercise 1 of Chapter 1). Of course, setting a = b confirms that the evolute of a circle is simply its center-point. Figure 4.1 represents an ellipse and its evolute simultaneously, though the focussing property is best appreciated by viewing Figure 4.12. p Α Γ q Figure 4.2: A center of curvature on a cubic curve The notions of tangent line and normal line to a curve are clear intuitively; here is the mathematical definition. Definition 4.3. The tangent line and normal line to a curve α: (a, b) → R2 at α(t) are the straight lines passing through α(t) with velocity vectors equal to α′ (t) and J α′ (t), respectively. Next, we obtain a characterization of the evolute of a curve in terms of tangent lines and normal lines, and also determine the singular points of the evolute. Theorem 4.4. Let β: (a, b) → R2 be a unit-speed curve. Then (i) the evolute of β is the unique curve of the form γ = β + f J β ′ for some function f for which the tangent line to γ at each point γ(s) coincides with the normal line to β at β(s). (ii) Suppose that κ2[β] is nowhere zero. The singular points of the evolute of β occur at those values of s for which κ2[β]′ (s) = 0. Proof. When we differentiate (4.1) and use Lemma 1.21, page 16, we obtain (4.4) evolute[β]′ = − κ2[β]′ ′ 2 J β . κ2[β] 1 Note that an astroid is a four-cusped curve, but that an asteroid is a small planet. This difference persists in English, French, Spanish and Portugese, but curiously in Italian the word for both notions is asteroide. 102 CHAPTER 4. NEW CURVES FROM OLD Hence the tangent line to evolute[β] at evolute[β](s) coincides with the normal line to β at β(s). Conversely, suppose γ = β + f J β′ . Again, using Lemma 1.21, we compute γ ′ = 1 − f κ2[β] β ′ + f ′ J β ′ . If the tangent line to γ at each point γ(s) coincides with the normal line to β at β(s), then f = 1/κ2[β], and so γ is the evolute of β. This proves (i); then (ii) is a consequence of (4.4). For the parametrization ellipse[a, b], it is verified in Notebook 4 that κ2′ (t) = 3ab(b2 − a2 ) sin 2t . 2(a2 sin2 t + b2 cos2 t)5/2 It therefore follows that the evolute of an ellipse is singular when t assumes one of the four values 0, π/2, π, 3π/2. This is confirmed by both differentiating (4.3) and inspecting the plot in Figure 4.1. 4.2 Iterated Evolutes An obvious problem is to see what happens when one applies the evolute construction repeatedly. Computations are carried out in Notebook 4 to find the evolute of an evolute of a curve, and to iterate the procedure. As an example of this phenomenon, Figure 4.3 plots the first three evolutes of the cissoid, parametrized on page 48. In spite of the fact that the cissoid has a cusp, its first evolute (passing through (−2, 0)) and the third evolute (passing through (0, 0)) do not. Another example of a curve with a cusp whose evolute has no cusp is the tractrix, parametrized on page 51. It turns out that the evolute of a tractrix is a catenary. By direct computation, the evolute of tractrix[a] is the curve t 7→ a 1 t . , log tan sin t 2 To see that this curve is actually a catenary, let τ = tan(t/2) and u = log τ . Then 2 τ2 + 1 = . eu + e−u = τ sin t Thus the evolute of the tractrix can be reparametrized as u 7→ a(cosh u, u), which is a multiple of catenary[1] as defined on page 47. 4.2. ITERATED EVOLUTES 103 4 2 -4 -2 2 4 -2 -4 Figure 4.3: Iterated evolutes of cissoid[1] For Figure 4.4, the evolute of the tractrix was computed and plotted automatically; it is the smooth curve that passes through the cusp. A catenary was then added to the picture off-center, but close enough to emphasize that it is the same curve as the evolute. Figure 4.4: tractrix[1] and its catenary evolute 104 CHAPTER 4. NEW CURVES FROM OLD 4.3 Involutes The involute is a geometrically important operation that is inverse to the map α 7→ evolute[α] that associates to a curve its evolute. In fact, the evolute is related to the involute in the same way that differentiation is related to indefinite integration. Just like the latter, the operation of taking the involute depends on an arbitrary constant. Furthermore, we shall prove (Theorem 4.9) that the evolute of the involute of a curve γ is again γ; this corresponds to the fact that the derivative of the indefinite integral of a function f is again f . We first give the definition of the involute of a unit-speed curve. Definition 4.5. Let β: (a, b) → R2 be a unit-speed curve, and let a < c < b. The involute of β starting at β(c) is the curve given by (4.5) ′ involute[β, c](s) = β(s) + (c − s)β (s). Whereas the evolute of a plane curve β is a linear combination of β and J β ′ , an involute of β is a linear combination of β and β ′ . Note that although we use s as the arc length parameter of β, it is not necessarily an arc length parameter for the involute of β. The formula for the involute of an arbitrary-speed curve needs the arc length function defined on page 12. Lemma 4.6. Let α: (a, b) → R2 be a regular arbitrary-speed curve. Then the involute of α starting at c (where a < c < b) is given by (4.6) α′ (t) , α′ (t) involute[α, c](t) = α(t) + sα (c) − sα (t) where t 7→ sα (t) denotes the arc length of α measured from an arbitary point. The involute of a curve can be described geometrically. Theorem 4.7. An involute of a regular plane curve β is formed by unwinding a taut string which has been wrapped around β. This result is illustrated in Figure 4.5, in which the string has been ‘cut’ at the point β(c) on the curve, and gradually unwound from that point. Proof. Without loss of generality, we may suppose that β has unit-speed. Then involute[β, c](s) − β(s) = (c − s)β ′ (s), so that (4.7) involute[β, c](s) − β(s) = |s − c|. Here |s − c| is the distance from β(s) to β(c) measured along the curve β, while the left-hand side of (4.7) is the distance from involute[β, c](s) to β(s) measured along the tangent line to β emanating from β(s). 4.3. INVOLUTES 105 ΒHsL ΒHcL Figure 4.5: Definition of an involute The most famous involute is that of a circle. With reference to (1.25), involute[circle[a], b](t) = a cos t + (−b + t) sin t, (b − t) cos t + sin t . An example is visible in Figure 4.7 on page 107, in which the operation of taking the involute has been iterated. (This is carried out analytically in Exercise 6). The involute of the figure eight (2.4) requires a complicated integral, which is computed numerically in Notebook 4 to obtain Figure 4.6 and a selection of normal lines. Figure 4.6: Involute of a figure eight 106 CHAPTER 4. NEW CURVES FROM OLD Next, we find a useful relation between the curvature of a curve and that of its involute. Lemma 4.8. Let β: (a, b) → R2 be a unit-speed curve, and let γ be the involute of β starting at c, where a < c < b. Then the curvature of γ is given by sign κ2[β](s) (4.8) . κ2[γ](s) = |s − c| Proof. First, we use (4.5) and Lemma 1.21 to compute (4.9) γ ′ (s) = (c − s)β′′ (s) = (c − s)κ2[β](s) J β ′ (s), and (4.10) γ ′′ (s) = −κ2[β](s) J β ′ (s) + (c − s)κ2[β]′ (s) J β ′ (s) +(c − s)κ2[β](s) J β ′′ (s) = −κ2[β](s) + (c − s)κ2[β]′ (s) J β ′ (s) 2 −(c − s) κ2[β](s) β′ (s). From (4.9) and (4.10), we get 3 γ ′′ (s) · J γ ′ (s) = (c − s)2 κ2[β](s) . (4.11) Now (4.8) follows from (4.9), (4.11) and the definition of κ2[γ]. Lemma 4.8 implies that the absolute value of the curvature of the involute of a curve is always decreasing as a function of s in the range s > c. This can be seen clearly in Figures 4.6 and 4.7. Theorem 4.9. Let β: (a, b) → R2 be a unit-speed curve and let γ be the involute of β starting at c, where a < c < b. Then the evolute of γ is β. Proof. By definition the evolute of γ is the curve ζ given by (4.12) ζ(s) = γ(s) + 1 J γ ′ (s) . κ2[γ](s) γ ′ (s) When we substitute (4.5), (4.8) and (4.9) into (4.12), we get ζ(s) = β(s) + (c − s)β′ (s) + = β(s). Thus β and ζ coincide. |c − s| (c − s)κ2[β](s)J 2 β′ (s) sign κ2[β](s) (c − s)κ2[β](s)J β ′ (s) 4.4. OSCULATING CIRCLES 107 The previous result is consistent with thinking of ‘evolution’ as differentiation, and ‘involution’ as integration, as explained at the start of this section. That being the case, one would expect a sequel to Theorem 4.9 to assert that the involute of the evolute of a curve β is the same as β ‘up to a constant’. The appropriate notion is contained in Definition 4.13 below: it can be shown that the involute of the evolute of β is actually a parallel curve to β. Examples are given in Notebook 4. 30 20 10 -15 -10 -5 5 10 Figure 4.7: A circle and three successive involutes 4.4 Osculating Circles to Plane Curves Just as the tangent is the best line that approximates a curve at one of its points p, the osculating circle is the best circle that approximates the curve at p. Definition 4.10. Let α be a regular plane curve defined on an interval (a, b), and let a < t < b be such that κ2[α](t) = 6 0. Then the osculating circle to α at α(t) is the circle of radius 1/|κ2[α](t)| and center α(t) + 1 J α′ (t) . κ2[α](t) α′ (t) 108 CHAPTER 4. NEW CURVES FROM OLD The dictionary definition of ‘osculating’ is kissing. In fact, the osculating circle at a point p on a curve approximates the curve much more closely than the tangent line. Not only do α and its osculating circle at α(t) have the same tangent line and normal line, but also the same curvature. It is easy to see from equation (4.1) that Lemma 4.11. The centers of the osculating circles to a curve form the evolute to the curve. The osculating circles to a logarithmic spiral are a good example of close approximation to the curve. Figure 4.8 represents these circles without the spiral itself. Figure 4.8: Osculating circles to logspiral[1, −1.5] Next, we show that an osculating circle to a plane curve is the limit of circles passing through three points of the curve as the points tend to the point of contact of the osculating circle. Theorem 4.12. Let α be a plane curve defined on an interval (a, b), and let a < t1 < t2 < t3 < b. Denote by C (t1 , t2 , t3 ) the circle passing through the points α(t1 ), α(t2 ), α(t3 ) provided these points are distinct and do not lie on the same straight line. Assume that κ2[α](t0 ) 6= 0. Then the osculating circle to α at α(t0 ) is the circle C = lim C (t1 , t2 , t3 ). t1 →t0 t2 →t0 t3 →t0 4.4. OSCULATING CIRCLES 109 Proof. Denote by p(t1 , t2 , t3 ) the center of C (t1 , t2 , t3 ), and define f : (a, b) → R by f (t) = α(t) − p(t1 , t2 , t3 ) 2 . Then (4.13) ( f ′ (t) = 2α′ (t) · α(t) − p(t1 , t2 , t3 ) , f ′′ (t) = 2α′′ (t) · α(t) − p(t1 , t2 , t3 ) + 2 α′ (t) 2 . Since f is differentiable and f (t1 ) = f (t2 ) = f (t3 ), there exist u1 and u2 with t1 < u1 < t2 < u2 < t3 such that (4.14) f ′ (u1 ) = f ′ (u2 ) = 0. Similarly, there exists v with u1 < v < u2 such that f ′′ (v) = 0. (4.15) (Equations (4.14) and (4.15) follow from Rolle’s theorem2 . See for example, [Buck, page 90].) Clearly, as t1 , t2 , t3 tend to t0 , so do u1 , u2 , v. Equations (4.13)–(4.15) imply that (4.16) where   α′ (t0 ) · (α(t0 ) − p) = 0,  α′′ (t ) · (α(t ) − p) = − α′ (t ) 2 , 0 0 0 p= lim t1 →t0 t2 →t0 t3 →t0 p(t1 , t2 , t3 ). It follows from (4.16) and the definition of κ2 that α(t0 ) − p = −1 J α′ (t0 ) . κ2[α](t0 ) α′ (t0 ) Thus, by definition, C is the osculating circle to α at α(t). Figure 4.9 plots various circles, each passing through three points on the parabola 4y = x2 . As the three points converge to the vertex of the parabola, the circle through the three points converges to the osculating circle at the vertex. Since the curvature of the parabola at the vertex is 1/2, the osculating circle at the vertex has radius exactly 2. 2 Michel Rolle (1652–1719). French mathematician, who resisted the infinitesimal techniques of calculus. 110 CHAPTER 4. NEW CURVES FROM OLD 15 12.5 10 7.5 5 2.5 -7.5 -5 -2.5 2.5 5 7.5 Figure 4.9: Circles converging to an osculating circle of a parabola 4.5 Parallel Curves It is appropriate to begin this section by illustrating the concept of tangent and normal lines. The tangent line to a plane curve at a point p on the curve is the best linear approximation to the curve at p, and the normal line is the tangent line rotated by π/2. Figure 4.10: Tangent lines to ellipse[ 32 , 1] Having drawn some short tangent lines to an ellipse (parametrized on page 21), as if to give it fur, we draw normal lines to the same ellipse in Figure 4.11. Longer normal lines may intersect one another, as we see from Figure 4.12, in which we can clearly distinguish the evolute of the ellipse. 4.5. PARALLEL CURVES 111 Figure 4.11: Normal lines to ellipse[ 32 , 1] Figure 4.12: Intersecting normals to the same ellipse We shall now construct a curve γ at a fixed distance r > 0 from a given curve α, where r is not too large. Let α and γ be defined on an interval (a, b); then we require kγ(t) − α(t)k = r and γ(t) − α(t) · α′ (t) = 0, for a < t < b. This leads us to the next definition. Definition 4.13. A parallel curve to a regular plane curve α at a distance r is the plane curve given by (4.17) parcurve[α, r](t) = α(t) + r J α′ (t) . α′ (t) 112 CHAPTER 4. NEW CURVES FROM OLD Actually, we can now allow r in (4.17) to be either positive or negative, in order to obtain parallel curves on either side of α, without changing t. The definition of parallel curve does not depend on the choice of positive reparametrization. In fact, it is not hard to prove Lemma 4.14. Let α: (a, b) → R2 be a plane curve, and let h: (c, d) → (a, b) be differentiable. Then parcurve[α ◦ h, r](u) = parcurve[α, r sign h′ ] h(u) . Figure 4.13: Four parallel curves to ellipse[2, 1] Figure 4.13 illustrates some parallel curves to an ellipse. It shows that if |r| is too large, a parallel curve may intersect itself and |r| will not necessarily represent distance to the original curve. We can estimate when this first happens, and at the same time compute the curvature of the parallel curve. Lemma 4.15. Let α be a regular plane curve. Then the curve parcurve[α, r] is regular at those t for which 1 − r κ2[α](t) 6= 0. Furthermore, its curvature is given by κ2[α](t) κ2[parcurve[α, r]](t) = . 1 − r κ2[α](t) Proof. By Theorem 1.20, page 16, and Lemma 4.14 we can assume that α has unit speed. Write β(t) = parcurve[α, r](t). Then β = α + r J α′ so that by Lemma 1.21 we have β ′ = α′ + r J α′′ = α′ + r J 2 κ2[α]α′ = 1 − r κ2[α] α′ . The regularity statement follows. Also, we compute β ′′ = 1 − rκ2[α] κ2[α]J α′ − r κ2[α]′ α′ . 4.6. PEDAL CURVES 113 Hence 2 1 − r κ2[α] κ2[α] κ2[α] β ′′ · J β ′ , = κ2[β] = ′ 3 = 3 kβ k 1 − r κ2[α] 1 − rκ2[α] as stated. 4.6 Pedal Curves Let α be a curve in the plane, and let p ∈ R2 . The locus of base points β(t) of a perpendicular line let down from p to the tangent line to α at α(t) is called the pedal curve to α with respect to p. It follows that β(t) − p is the projection of α(t) − p in the J α′ (t) direction, as shown in Figure 4.15. This enables us to give a more formal definition: Definition 4.16. The pedal curve of a regular curve α: (a, b) → R2 with respect to a point p ∈ R2 is defined by pedal[α, p](t) = p + α(t) − p · J α′ (t) α′ (t) 2 J α′ (t). ΒHtL ΑHtL p Figure 4.14: The definition of a pedal curve The proof of the following lemma follows the model of Theorem 1.20; see Exercise 13. Lemma 4.17. The definition of the pedal curve of α is independent of the parametrization of α, so that pedal[α ◦ h, p] = pedal[α, p] ◦ h. 114 CHAPTER 4. NEW CURVES FROM OLD Other examples are: (i) The pedal curve of a parabola with respect to its vertex is a cissoid; see Exercise 12. (ii) The pedal curve of a circle with respect to any point p other than the center of the circle is a limaçon. If p lies on the circumference of the circle, the limaçon reduces to a cardioid. The pedal curve of a circle with respect to its center is the circle itself; see Exercise 17. (iii) The pedal curve of cardioid[a] (see page 45) with respect to its cusp point is called Cayley’s sextic3 . Its equation is 3t t t . cayleysextic[a](t) = 4a (2 cos t − 1) cos4 , sin cos3 2 2 2 Figure 4.16 plots the cardioid and Cayley’s sextic simultaneously. Figure 4.15: Cardioid and its pedal curve (iv) The pedal curve of 3 t + 21 , α(t) = t, 3 illustrated in Figure 4.14, has a singularity, discussed in [BGM, page 5]. 3 Arthur Cayley (1821–1895). One of the leading English mathematicians of the 19th century; his complete works fill many volumes. Particularly known for his work on matrices, elliptic functions and nonassociative algebras. His first 14 professional years were spent as a lawyer; during that time he published over 250 papers. In 1863 Cayley was appointed Sadleirian professor of mathematics at Cambridge with a greatly reduced salary. 4.7. EXERCISES 115 1.5 1 0.5 -2 -1 1 2 -0.5 -1 -1.5 Figure 4.16: The cubic y = 1 3 3x + 1 2 and its pedal curve For more information about pedal curves see [Law, pages 46–49], [Lock, pages 153–160] and [Zwi, pages 150–158]. We shall return to pedal curves, and parametrize them with an angle, as part of the study of ovals in Section 6.8. 4.7 Exercises M 1. Plot evolutes of a cycloid, a cardioid and a logarithmic spiral. Show that the evolute of a cardioid is another cardioid, and that the evolute of a logarithmic spiral is another logarithmic spiral. 2. Determine conditions under which the evolute of a cycloid is another cycloid. 3. Prove Lemma 4.2. M 4. Plot normal lines to a cardioid, making the lines sufficiently long so that they intersect. M 5. Plot as one graph four parallel curves to a lemniscate. Do the same for a cardioid and a deltoid (see page 57). 6. Show that the curve defined by γ(t) = aeit n X (−it)k k=0 k! is the nth involute starting at (a, 0) of a circle of radius a. 116 CHAPTER 4. NEW CURVES FROM OLD 7. A strophoid of a curve α with respect to a point p ∈ R2 is a curve γ such that α(t) − γ(t) = α(t) − p and γ(t) = sα(t), for some s. Show that α has two strophoids and find the equations for them. (A special case was defined in Exercise 1 on page 85.) 8. Show that the involute of a catenary is a tractrix. M 9. A point of inflection of a plane curve α is defined to be a point α(t0 ) for which κ2[α](t0 ) = 0; a strong inflection point is a point α(t0 ) for which there exists ε > 0 such that κ2[α](t) is negative for t0 − ε < t < t0 and positive for t0 < t < t0 + ε, or vice versa. (a) Show that if α(t0 ) is a strong inflection point, and if t 7→ κ2[α](t) is continuous at t0 , then α(t0 ) is an inflection point. (b) For the curve t 7→ (t3 , t5 ) show that (0, 0) is a strong inflection point which is not an inflection point. Plot the curve. 10. Let α(t0 ) be a strong inflection point of a curve α: (a, b) → R2 . Show that any involute of α must have discontinuous curvature at t0 . This accounts for the cusps on the involutes of a figure eight and of a cubic parabola. M 11. Find and draw the pedal curve of (i) an ellipse with respect to its center, and (ii) a catenary with respect to the origin. 12. Find the parametric form of the pedal curve of the parabola t 7→ (2at, t2 ) with respect to the point (0, b), where a2 6= b. Show that the nonparametric form of the pedal curve is (x2 + y 2 )y + (a2 − b)x2 − 2b y 2 + b2 y = 0. 13. Prove Lemma 4.17. 14. An ordinary pendulum swings back and forth in a circular arc. The oscillations are not isochronous. To phrase it differently, the time it takes for the (circular) pendulum to go from its staring point to its lowest point will be almost, but not quite independent, of the height from which the pendulum is released. In 1680 Huygens4 observed two important facts: 4 Christiaan Huygens (1629–1695). A leading Dutch scientist of the 17th century. As a mathematician he was a major precursor of Leibniz and Newton. His astronomical contributions include the discovery of the rings of Saturn. In 1656, Huygens patented the first pendulum clock. 4.7. EXERCISES 117 (a) a pendulum that swings back and forth in an inverted cycloidial arc is isochronous, and (b) the involute of a cycloid is a cycloid. Combined, these facts show that the involute of an inverted cycloidial arc can be used to constrain a pendulum so that it moves in a cycloidial arc. Prove (b). For (a), see the end of Notebook 4. Figure 4.17: A cycloidal pendulum in action 15. Show that the formula for the pedal curve of α with respect to the origin can be written as αα′ − αα′ . pedal[α, 0] = 2α′ See [Zwi, page 150]. 16. The contrapedal of a plane curve α is defined analogously to the pedal of α. It is the locus of bases of perpendicular lines let down from a point p to a variable normal line to α. Prove that the exact formula is α(t) − p · α′ (t) ′ contrapedal[α, p](t) = p + α (t). 2 α′ (t) M 17. Show that the pedal curve of a circle with respect to a point on the circle is a cardioid. Plot pedal curves of a circle with respect to points inside the circle, and then with respect to points outside the circle. Chapter 5 Determining a Plane Curve from its Curvature In this chapter we confront the following question: to what extent does curvature determine a plane curve? This question has two parts: when do two curves have the same curvature, and when can a curve be determined from its curvature? To answer the first part, we begin by asking: which transformations of the plane preserve curvature? We have seen (in Theorem 1.20 on page 16) that the curvature of a plane curve is independent of the parametrization, at least up to sign. Therefore, without loss of generality, we can assume in this chapter that all curves have unit speed. It is intuitively clear that the image of a plane curve α under a rotation or translation has the same curvature as the original curve α. Rotations and translations are examples of Euclidean motions of R2 , meaning those maps of R2 onto itself which do not distort distances. We discuss Euclidean motions in general in Section 5.1, and introduce the important concept of a group. We specialize Euclidean motions to the plane in Section 5.2, while the case of orthogonal transformations and rotations of R3 will be taken up in Chapter 23. The invariance of curvature under Euclidean motions is established in Section 5.3, in which we also prove the Fundamental Theorem of Plane Curves. This important theorem states that two unit-speed curves in R2 that have the same curvature differ only by a Euclidean motion. The second part of our question now becomes: can a unit-speed curve in R2 be determined up to a Euclidean motion from its curvature? In Section 5.3, we give an explicit system of differential equations for determining a plane curve from its curvature. In simple cases this system can be solved explicitly, but in the general case only a numerical solution is possible. Such solutions are illustrated in Section 5.4. So the answer to the second part of the question is effectively ‘yes’. 127 128 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE 5.1 Euclidean Motions In order to discuss the invariance of the curvature of plane curves under rotations and translations, we need to make precise what we mean by ‘invariance’. This requires discussing various kinds of maps of R2 to itself. We begin by defining transformations of Rn , before restricting to n = 2. First, we define several important types of maps that are linear, a concept introduced in Chapter 1. The linearity property A(ap + b q) = aAp + b Aq, for a, b ∈ R, implies that each of these maps takes the origin into the origin. Definition 5.1. Let A: Rn → Rn be a nonsingular linear map. (i) We say that A is orientation-preserving if det A is positive, or orientationreversing if det A is negative. (ii) A is called an orthogonal transformation if Ap · Aq = p · q for all p, q ∈ Rn . (iii) A rotation of Rn is an orientation-preserving orthogonal transformation. (iv) A reﬂection of Rn is an orthogonal transformation of the form reflq , where (5.1) reflq (p) = p − 2(p · q) q kqk2 for all p ∈ Rn and some fixed q 6= 0. Lemma 5.2. Let A: Rn → Rn be an orthogonal transformation. Then det A = ±1. Proof. This well-known result follows from the fact that, if we represent A by a matrix relative to an orthonormal basis, and p, q by column vectors, then the orthogonality property is equivalent to asserting that (5.2) (Ap)T (Aq) = pT q, where AT denotes the transpose of A. It follows that A is orthogonal if and only if ATA is the identity matrix, and 1 = det(ATA) = det(AT ) det A = (det A)2 . 5.1. EUCLIDEAN MOTIONS 129 Figure 5.1: Reflection in 3 dimensions We can characterize reflections geometrically. Lemma 5.3. Let reflq : Rn → Rn be a reflection, q 6= 0. Then det(reflq ) = −1. Furthermore, reflq restricts to the identity map on the orthogonal complement Π of q (the hyperplane of Rn of dimension n − 1 of the vectors orthogonal to q) and reverses directions parallel to q. Conversely, any hyperplane gives rise to a reflection reflq which restricts to the identity map on Π and reverses directions on the orthogonal complement of Π . Proof. Let Π be the orthogonal complement of q, that is Π = {p ∈ Rn | p · q = 0}. Clearly, dim(Π ) = n − 1, and it is easy to check that reflq (p) = p for all p ∈ Π . Since reflq (q) = −q, it follows that reflq reverses directions on the orthogonal complement of Π . To show that det(reflq ) = −1, choose a basis {f1 , . . . , fn−1 } of Π . It follows that {q, f1 , . . . , fn−1 } is a basis of Rn . Since the matrix of reflq with respect to this basis is   −1 0 . . . 0    0 1 ... 0   . .. . . .  ,  . . ..   . . 0 0 ... 1 we have det(reflq ) = −1. Conversely, given a hyperplane Π , let q be a nonzero vector perpendicular to Π . Then the reflection reflq defined by (5.1) has the required properties. 130 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE Definition 5.4. Let q ∈ Rn . (i) An afﬁne transformation of Rn is a map F : Rn → Rn of the form F (p) = Ap + q for all p ∈ Rn , where A is a linear transformation of Rn . We shall call A the linear part of F . An affine transformation F is orientation-preserving if det A is positive, or orientation-reversing if det A is negative. (ii) A translation is an affine map as above with A the identity, that is a mapping tranq : Rn → Rn of the form tranq (p) = p + q for all p ∈ Rn . (iii) A Euclidean motion of Rn is an affine transformation whose linear part is an orthogonal transformation. (iv) An isometry of Rn is a map F : Rn → Rn that preserves distance, so that kF (p1 ) − F (p2 )k = kp1 − p2 k for all p1 , p2 ∈ Rn . It is easy to see that an isometry must be an injective map; we shall see below that it is necessarily bijective. Given an affine transformation, we can study its effect on any geometrical object in Rn . Figure 5.2 illustrates a logarithmic spiral in R2 and its image under an affine transformation F , a process formalized in Definition 5.11 below. Notice that F is orientation-reversing, as the inward spiralling is changed from clockwise to counterclockwise. 4 3 2 1 -2 2 4 6 -1 -2 Figure 5.2: The effect of an affine transformation 5.1. EUCLIDEAN MOTIONS 131 To put these notions in their proper context, let us recall the following abstract algebra. Definition 5.5. A group consists of a nonempty set G and a multiplication ◦ : G × G −→ G with the following properties. (i) There exists an identity element 1 ∈ G , that is, an element 1 such that a◦1 =1◦a = a for all a ∈ G . (ii) Multiplication is associative; that is, (a ◦ b) ◦ c = a ◦ (b ◦ c). for all a, b, c ∈ G . (iii) Every element a ∈ G has an inverse; that is, for each a ∈ G there exists an element a−1 ∈ G (that can be proved to be unique) such that a ◦ a−1 = a−1 ◦ a = 1. Moreover, a subgroup of a group G is a nonempty subset S of G closed under multiplication and inverses, that is a◦b∈S and a−1 ∈ S , for all a, b ∈ S . It is easy to check that a subgroup of a group is itself a group. See any book on abstract algebra for more details on groups, for example [Scott] or [NiSh]. Many sets of maps form groups. Recall that if F : Rm −→ Rn and G: Rn −→ Rp are maps, their composition G ◦ F : Rm → Rp is defined by (G ◦ F )(p) = G F (p) for p ∈ Rm . Denote by GL(n) the set of all nonsingular linear maps of Rn into itself. It is easy to verify that composition makes GL(n) into a group. Similarly, we may speak of the following groups: • the group Affine(Rn ) of affine maps of Rn , 132 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE • the group Orthogonal(Rn ) of orthogonal transformations of Rn , • the group Translation(Rn ) of translations of Rn , • the group Isometry(Rn ) of isometries of Rn , • the group Euclidean(Rn ) of Euclidean motions of Rn . The verification that each is a group is easy. Moreover: Theorem 5.6. A map F : Rn → Rn is an isometry of Rn if and only if it is a composition of a translation and an orthogonal transformation of Rn . As a consequence, the group Euclidean(Rn ) coincides with the group Isometry(Rn ). Proof. It is easy to check that any orthogonal transformation or translation of Rn preserves the distance function. Hence the composition of an orthogonal transformation and a translation also preserves the distance function. This shows that Euclidean(Rn ) ⊆ Isometry(Rn ). Conversely, let F : Rn → Rn be an isometry, and define G: Rn → Rn by G(p) = F (p) − F (0). Then G is an isometry and G(0) = 0. To show that G preserves the scalar product of Rn , we calculate −2 G(p) · G(q) = kG(p) − G(q)k2 − kG(p) − G(0)k2 − kG(q) − G(0)k2 = kp − qk2 − kp − 0k2 − kq − 0k2 = −2 p · q. To show that G is linear, let {e1 , . . . , en } be an orthonormal basis of Rn . By definition, each ei has unit length and is perpendicular to every other ej . Then {G(e1 ), . . . , G(en )} is also an orthonormal basis of Rn . Thus, for any p ∈ Rn we have n n X X (p · ej )G(ej ) G(p) · G(ej ) G(ej ) = G(p) = j=1 j=1 (see Exercise 1). Because each mapping p 7→ p · ej is linear, so is G. Hence G is an orthogonal transformation of Rn , so that F is the composition of the orthogonal transformation G and the translation tranF (0) . We have proved that Isometry(Rn ) ⊆ Euclidean(Rn ). Since we have already proved the reverse inclusion, it follows that Isometry(Rn ) = Euclidean(Rn ). 5.2. ISOMETRIES OF THE PLANE 133 Figure 5.3: Bouquet of clothoids 5.2 Isometries of the Plane We now specialize to R2 . First, we give a characterization of transformations of R2 in terms of the linear map J , defined on page 3, represented by the matrix ! 0 −1 . 1 0 Lemma 5.7. Let A be an orthogonal transformation of R2 . Then (5.3) A J = ε J A, (5.4) det A = ε, where ε= ( +1 if A is orientation-preserving, −1 if A is orientation-reversing. Furthermore, det A = −1 if and only if A is a reflection. Proof. Let p ∈ R2 be a nonzero vector. Then JAp · Ap = 0; since A is orthogonal, AJ p · Ap = J p · p = 0. Thus both AJ p and J Ap are perpendicular to Ap, so that for some λ ∈ R we have AJ p = λJ Ap. Since both A and J are orthogonal transformations, it is 134 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE easy to see that |λ| = 1. One can now obtain both (5.3) and (5.4) by computing the matrix of A with respect to the basis {p, J p} of R2 . Lemma 5.3 implies that the determinant of any reflection of R2 is −1. Conversely, an orthogonal transformation A: R2 → R2 with det A = −1 must have 1 and −1 as its eigenvalues. Let q ∈ R2 be such that A(q) = −q and kqk = 1. Then A = reflq . The following theorem is due to Chasles1 . Theorem 5.8. Every isometry of R2 is the composition of translations, reflections and rotations. Proof. Theorem 5.6 implies that an isometry F of R2 is a Euclidean motion. There is an orthogonal transformation A: R2 → R2 and q ∈ R2 such that F (p) = A(p) + q. Otherwise said, F = tranq ◦ A. From Lemma 5.7 we know that A is a rotation if det A = 1 or a reflection if det A = −1. A standard procedure for simplifying the equation ax2 + 2bxy + cy 2 + dx + ey + f = 0 of a conic into ‘normal form’ corresponds to first rotating the vector (x, y) so as to eliminate the term in xy (in the new coordinates) and then translating it to eliminate the terms in x and y. But it is sometimes convenient to eliminate the linear terms first, so as determine the center, and then rotate. These two contrasting methods are illustrated in the Figure 5.4, starting with the ellipse in (say) the first quadrant. By expressing an arbitrary vector p ∈ R2 as a linear combination of two nonzero vectors q, J q, and using (5.3), we may deduce Lemma 5.9. Let A be a rotation of R2 . Then the numbers Ap · p kpk2 and AJp · p kpk2 are each independent of the nonzero vector p. 1 Michel Chasles (1793–1880). French geometer and mathematical historian. He worked on algebraic and projective geometry. 5.2. ISOMETRIES OF THE PLANE 135 4 2 -2 2 4 6 8 10 -2 -4 -6 -8 Figure 5.4: Translations and rotations of a conic Definition 5.10. Let A be a rotation of R2 . The angle of rotation of A is any number θ such that cos θ = Ap · p kpk2 and sin θ = AJp · p kpk2 for all nonzero p ∈ R2 . Hence using the identification (1.1) of C with R2 , we can write Ap = (cos θ)p + (sin θ)Jp = eiθ p for all p ∈ R2 . It is clear intuitively that any mapping of R2 into itself maps a curve into another curve. Here is the exact definition: Definition 5.11. Let F : R2 → R2 be a map, and let α: (a, b) → R2 be a curve. The image of α under F is the curve F ◦ α. If for example F is an affine transformation with linear part represented by a matrix A, then the image of α(t) is found by premultiplying the column vector α(t) by A and then adding the translation vector. This fact is the basis of the practical implementation of rotations in Notebooks 5 and 23. It was used to produce Figure 5.3, which shows a clothoid and its images under two rotations. 136 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE 5.3 Intrinsic Equations for Plane Curves Since a Euclidean motion F of the plane preserves distance, it cannot stretch or otherwise distort a plane curve. We now show that F preserves the curvature up to sign. Theorem 5.12. The absolute value of the curvature and the derivative of arc length of a curve are invariant under Euclidean motions of R2 . The signed curvature κ2 is preserved by an orientation-preserving Euclidean motion of R2 and changes sign under an orientation-reversing Euclidean motion. Proof. Let α: (a, b) → R2 be a curve, and let F : R2 → R2 be a Euclidean motion. Let A denote the linear part of F , so that for all p ∈ R2 we have F (p) = Ap + F (0). Define a curve γ : (a, b) → R2 by γ = F ◦ α; then for a < t < b we have γ(t) = Aα(t) + F (0). Hence γ ′ (t) = Aα′ (t) and γ ′′ (t) = Aα′′ (t). Let sα and sγ denote the arc length functions with respect to α and γ. Since A is an orthogonal transformation, we have s′γ (t) = γ ′ (t) = Aα′ (t) = s′α (t). We compute the curvature of γ using (5.3): κ2[γ](t) = γ ′′ (t) · J γ ′ (t) γ ′ (t) 3 = Aα′′ (t) · εA J α′ (t) = =ε Aα′ (t) 3 α′′ (t) · J α′ (t) α′ (t) 3 = ε κ2[α](t). Next, we prove the converse of Theorem 5.12. Theorem 5.13. (Fundamental Theorem of Plane Curves, Uniqueness) Let α and γ be unit-speed regular curves in R2 defined on the same interval (a, b), and having the same signed curvature. Then there is an orientation-preserving Euclidean motion F of R2 mapping α into γ. Proof. Fix s0 ∈ (a, b). Clearly, there exists a translation of R2 taking α(s0 ) into γ(s0 ). Moreover, we can find a rotation of R2 that maps α′ (s0 ) into γ ′ (s0 ). Thus there exists an orientation-preserving Euclidean motion F of R2 such that F α(s0 ) = γ(s0 ) and F α′ (s0 ) = γ ′ (s0 ). To show that F ◦ α coincides with γ, we define a real-valued function f by f (s) = (F ◦ α)′ (s) − γ ′ (s) 2 5.3. INTRINSIC EQUATIONS 137 for a < s < b. The derivative of f is easily computed to be f ′ (s) = 2 (F ◦ α)′′ (s) − γ ′′ (s) · (F ◦ α)′ (s) − γ ′ (s) (5.5) = 2 (F ◦ α)′′ (s) · (F ◦ α)′ (s) + γ ′′ (s) · γ ′ (s) −(F ◦ α)′′ (s) · γ ′ (s) − (F ◦ α)′ (s) · γ ′′ (s) . Since both F ◦ α and γ have unit speed, it follows that (F ◦ α)′′ (s) · (F ◦ α)′ (s) = γ ′′ (s) · γ ′ (s) = 0. Hence (5.5) reduces to (5.6) f ′ (s) = −2 (F ◦ α)′′ (s) · γ ′ (s) + (F ◦ α)′ (s) · γ ′′ (s) . Let A denote the linear part of the motion F . Since (F ◦ α)′ (s) = Aα′ (s) and (F ◦ α)′′ (s) = Aα′′ (s), we can rewrite (5.6) as (5.7) f ′ (s) = −2 Aα′′ (s) · γ ′ (s) + Aα′ (s) · γ ′′ (s) . Now we use the assumption that κ2[α] = κ2[γ] and (5.7) to get f ′ (s) = −2 κ2[α](s)A J α′ (s) · γ ′ (s) + Aα′ (s) · κ2[γ](s)J γ ′ (s) = −2κ2[α](s) J Aα′ (s) · γ ′ (s) + Aα′ (s) · J γ ′ (s) = 0. Since f (s0 ) = 0, we conclude that f (s) = 0 for all s. Hence (F ◦ α)′ (s) = γ ′ (s) for all s, and so there exists q ∈ R2 such that (F ◦ α)(s) = γ(s) − q for all s. In fact, q = 0 because F α(s0 ) = γ(s0 ). Thus the Euclidean motion F maps α into γ. Next, we turn to the problem of explicitly determining a plane curve from its curvature. Theory from Chapter 1 enables us to prove Theorem 5.14. (Fundamental Theorem of Plane Curves, Existence) A unitspeed curve β : (a, b) → R2 whose curvature is a given piecewise-continuous function k : (a, b) → R is parametrized by  Z Z   β(s) =  cos θ(s)ds + c, sin θ(s)ds + d ,  (5.8) Z     θ(s) = k(s)ds + θ0 , where c, d, θ0 are constants of integration. Proof. We define β and θ by (5.8); it follows that   β ′ (s) = cos θ(s), sin θ(s) , (5.9)  θ′ (s) = k(s). Thus β has unit speed, so Corollary 1.27 on page 20 tells us that the curvature of β is k. 138 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE One classical way to describe a plane curve α is by means of a natural equation, which expresses the curvature κ2[α] (at least implicitly) in terms of the arc length s of α. Although such an equation may not be the most useful equation for calculational purposes, it shows clearly how curvature changes with arc length and is obviously invariant under translations and rotations. To find the natural equation of a curve t 7→ α(t), one first computes the curvature and arc length as functions of t and then one tries to eliminate t. Conversely, the ordinary equation can be found (at least in principle) by solving the natural equation and using Theorem 5.14. The following table lists some examples, while Figure 5.5 plots variants of its last two entries. Curve Natural equation straight line κ2 = 0 circle of radius a catenary[a] clothoid[n, a] logspiral[a, b] involute[circle[a]] 1 a a κ2 = − 2 a + s2 sn κ2 = − n+1 a 1 κ2 = bs 1 κ2 = √ 2as κ2 = 1.5 0.6 1 0.4 0.2 0.5 -0.4 -0.2 -0.4-0.2 0.2 0.4 0.6 0.8 0.2 -0.2 -0.5 √ Figure 5.5: Curves with κ2(s) = 5/(s + 1) and 5/ s + 1 0.4 5.3. INTRINSIC EQUATIONS 139 Natural equations arose as a reaction against the use of Cartesian and polar coordinates, which, despite their utility, were considered arbitrary. In 1736, Euler2 proposed the use of the arc length s and the radius of curvature 1/|κ2| (see [Euler1]). Natural equations were also studied by Lacroix3 and Hill4 , and excellent discussions of them are given in [Melz, volume 2, pages 33–41] and [Ces]. The book [Ces] of Cesàro5 uses natural equations as its starting point; it contains many plots (quite remarkable for a book published at the end of the 19th century) of plane curves given by means of natural equations. For example, let us find the natural equation of the catenary defined by (5.10) catenary[a](t) = t a cosh , t . a A computation tells us that the curvature and arc length are given by (5.11) κ2(t) = − sech(t/a)2 a and t s(t) = a sinh . a Then (5.11) implies that (5.12) (s2 + a2 )κ2 = −a. Although (5.10) and (5.12) both describe a catenary, (5.12) is considered more natural because it does not depend on the choice of parametrization of the catenary. 2 Leonhard Euler (1707–1783). Swiss mathematician. Euler was a geometer in the broad sense in which the term was used during his time. Not only did he contribute greatly to the evolution and systematization of analysis – in particular to the founding of the calculus of variation and the theories of differential equations, functions of complex variables, and special functions – but he also laid the foundations of number theory as a rigorous discipline. Moreover, he concerned himself with applications of mathematics to fields as diverse as lotteries, hydraulic systems, shipbuilding and navigation, actuarial science, demography, fluid mechanics, astronomy, and ballistics. 3 Sylvestre Francois Lacroix (1765–1843). French writer of mathematical texts who was a student of Monge. 4 Thomas Hill (1818–1891). American mathematician who became president of Harvard University. 5 Ernesto Cesàro (1859–1906). Italian mathematician, born in Naples, professor in Palermo and Rome. Although his most important contribution was his monograph Lezioni di Geometria Intrinseca, he is also remembered for his work on divergent series. 140 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE 5.4 Examples of Curves with Assigned Curvature A curve β parametrized by arc length s whose curvature is a given function κ2 = k(s) can be found by solving the system (5.9). Notebook 5 finds numerical solutions of the resulting ordinary differential equations. In this way, we may readily draw plane curves with assigned curvature. 4 3 2 1 -1.5 -1 -0.5 0.5 1 1.5 Figure 5.6: A curve with κ2(s) = s sin s Quite simple curvature functions can be used to produce very interesting curves. We have already seen in Section 2.7 that the curve clothoid[1, 1] has curvature given by κ2(s) = −s, where s denotes arc length. The clothoids in Figure 5.3 were in fact generated by using this program. 3 2 1 -4 -2 2 4 -1 Figure 5.7: Curves with κ2(s) = 1.6 sin s and κ2(s) = 2.4 sin s 5.4. CURVES WITH ASSIGNED CURVATURE 141 Figure 5.6 displays the case κ2(s) = s sin s. But setting κ2 to be a mere multiple of sin s presents an intriguing situation. Figure 5.7 plots simultaneously two curves whose curvature has the form κ2(s) = c sin s with c constant; the one in the middle with c = 2.4 appears to be a closed curve with this property. Notebook 5 includes an animation of all the intermediate values of c. Finally, we consider several functions that oscillate more wildly, namely those of the form s 7→ sf (s), where f is one of the Bessel6 functions Ji . As well as solving a natural differential equation, these functions occur naturally as coefficients of the Laurent series of the function exp t(z − z −1 ) . This and other useful expressions for them can be found in books on complex analysis, such as [Ahlf]. Below we display both the function f and the resulting curve, omitting the axis scales for visual clarity. Figure 5.8: Curves with κ2(s) = sJn (s) with n = 1, 2, 3, 4 6 Friedrich Wilhelm Bessel (1784–1846). German astronomer and friend of Gauss. In 1800, Bessel was appointed director of the observatory at Königsberg, where he remained for the rest of his life. Bessel functions were introduced in 1817 in connection with a problem of Kepler of determining the motion of three bodies moving under mutual gravitation. 142 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE 5.5 Exercises 1. Let {e1 , . . . , en } be an orthonormal basis of Rn . Show that any vector v ∈ Rn can be written as v = (v · e1 )e1 + · · · + (v · en )en . 2. A homothety of Rn is a map F : Rn → Rn for which there exists a constant λ > 0 such that kF (p1 ) − F (p2 )k = λkp1 − p2 k for all p1 , p2 ∈ Rn . Show that a homothety is a bijective affine transformation. 3. Show that a Euclidean motion F : R2 → R2 preserves evolutes, involutes, parallel curves and pedal curves. More precisely, if α: (a, b) → R2 is a curve, show that the following formulas hold for a < t < b: F ◦ evolute[α] (t), involute[F ◦ α, c](t) = F ◦ involute[α, c] (t), . pedal[F (p), F ◦ α](t) = F ◦ pedal[p, α] (t), evolute[F ◦ α](t) = and  F ◦ parcurve[α][s] (t)      if F is orientation-preserving, parcurve[F ◦ α][s](t) =   F ◦ parcurve[α][−s] (t)    if F is orientation-reversing. 4. An algebraic curve is an implicitly-defined curve of the form P (x, y) = 0, where P is a polynomial in x and y. The order of an algebraic curve is the order of the polynomial P . Show that an affine transformation preserves the order of an algebraic curve. 5. Find the natural equations of the first three involutes of a circle. 6. Nielsen’s spiral7 is defined as nielsenspiral[a](t) = a Ci(t), Si(t) , 7 Niels Nielsen (1865–1931). Danish mathematician. 5.5. EXERCISES 143 where the sine and cosine integrals are defined by Si(t) = Z 0 t sin u du, u Ci(t) = − Z t ∞ cos u du u for t > 0. It is illustrated in Figure 5.9. (a) Explain why the spiral is asymptotic to the x-axis as t → −∞, and find its limit as t → ∞. (b) Show that the function Ci(t) can be defined in the equivalent way Ci(t) = Z t 0 (cos u − 1) du + log t + γ, u again for t > 0, where γ is Euler’s constant 0.5772 . . . 2 1.5 1 0.5 -2.5 -2 -1.5 -1 -0.5 0.5 1 -0.5 Figure 5.9: Nielson’s spiral plotted with −2.5 < t < 15 7. Show that the natural equation of the spiral in the previous exercise is κ2(s) = es/a . a M 8. Plot curves whose curvature is ns sin s for n = ±2, ±2.5, ±3. M 9. Plot a curve whose curvature is s 7→ Γ(s + 12 ), where −5 < s < 5, and the Gamma function is defined by Z ∞ Γ(z) = tz−1 e−t dt. 0 144 CHAPTER 5. DETERMINING A CURVE FROM ITS CURVATURE M 10. An epicycloid is the curve that is traced out by a point p on the circumference of a circle (of radius b) rolling outside another circle (of radius a). Similarly, a hypocycloid is the curve that is traced out by a point p on the circumference of a circle (of radius b) rolling inside another circle (of radius a). 4 2 -4 -2 2 4 -2 -4 Figure 5.10: An epicycloid with 3 cusps These curves are parametrized by (a + b)t epicycloid[a, b](t) = (a + b) cos t − b cos , b (a + b)t , (a + b) sin t − b sin b (a − b)t , hypocycloid[a, b](t) = (a − b) cos t + b cos b (a − b)t (a − b) sin t − b sin , b and illustrated in Figures 5.10 and 5.11. Show that the natural equation of an epicycloid is 2 a + 2b a2 s2 + (5.13) = 16 b2 (a + b)2 , κ2 and find that of the hypocycloid. M 11. Let αn : (−∞, ∞) → R2 be a curve whose curvature is n sin2 s, where n is an integer and s denotes arc length. Show that αn is a closed curve if and only if n is not divisible by 4. Plot αn for 1 6 n 6 12. 5.5. EXERCISES 145 Figure 5.11: A hypocycloid with 7 cusps 12. Show that the natural equation of a tractrix is 1 . κ2 = − √ 2s/a a e −1 M 13. The Airy8 functions, denoted AiryAi and AiryBi, are linearly independent solutions of the differential equation y ′′ (t) − ty(t) = 0. Plot curves whose signed curvature is respectively s 7→ AiryAi(s) and s 7→ AiryBi(s). 8 Sir George Biddel Airy (1801–1892) Royal astronomer of England. Chapter 6 Global Properties of Plane Curves The geometry of plane curves that we have been studying in the previous chapters has been local in nature. For example, the curvature of a plane curve describes the bending of that curve, point by point. In this chapter, we consider global properties that are concerned with the curve as a whole. In Section 6.1, we define the total signed curvature of a plane curve α, by integrating its signed curvature κ2. The total signed curvature is an overall measure of curvature, directly related to the turning angle that was defined in Chapter 1. For a regular closed curve, the total signed curvature gives rise to a turning number, which is an integer. These concepts are well illustrated with reference to epitrochoids and hypotrochoids, curves formed by rolling wheels, described in Section 6.2. An example is given of relevant curvature functions. Section 6.3 is devoted to the rotation index of a closed curve, which is defined topologically as the degree of an associated continuous mapping. We then show that it coincides with the turning number. The concept of homotopy for maps from a circle is introduced, and used to show that the turning number of a simple closed curve has absolute value 1. Convex plane curves are considered in Section 6.4, where it is shown that a closed plane curve is convex if and only if its signed curvature does not change sign. We prove the four vertex theorem for such curves in Section 6.5, and illustrate it with the sine oval, parametrized using an iterated sine function. Section 6.6 is concerned with more general ovals, closed curves for which κ2 not only does not change sign, but is either strictly positive or negative. We also establish basic facts about curves of constant width, and then give two classes of examples in Section 6.7. The formula for an oval in terms of its support function is derived in Section 6.8, using the envelope of a family of straight lines. A number of examples are investigated in the text and subsequent exercises. 153 154 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES 6.1 Total Signed Curvature The signed curvature κ2 of a plane curve α was defined on page 14, and measures the bending of the curve at each of its points. A measure of the total bending of α is given by an integral involving κ2. Definition 6.1. The total signed curvature of a curve α : [a, b] → R2 is TSC[α] = Z b κ2[α](t) α′ (t) dt, a where κ2[α] denotes the signed curvature of α. Although the definition depends upon the endpoints a, b, we assume that the mapping α is in fact defined on an open interval I containing [a, b]. First, let us check that the total signed curvature is a geometric concept. Lemma 6.2. The total signed curvature of a plane curve remains unchanged under a positive reparametrization, but changes sign under a negative one. Proof. Let γ = α ◦ h where γ : (c, d) → R2 and α : (a, b) → R2 are curves. We do the case when h′ (u) > 0 for all u. Using Theorem 1.20 on page 16, and the formula from calculus for the change of variables in an integral, we compute Z d Z b κ2[α] h(u) α′ h(u) h′ (u)du κ2[α](t) α′ (t) dt = TSC[α] = c a = Z d κ2[γ](u) γ ′ (u) du c = TSC[γ]. The proof that TSC[γ] = −TSC[α] when γ is a negative reparametrization of α is similar. There is a simple relation linking the total signed curvature and the turning angle of a curve defined in Section 1.5. Lemma 6.3. The total signed curvature can be expressed in terms of the turning angle θ[α] of α by (6.1) TSC[α] = θ[α](b) − θ[α](a). Proof. Equation (6.1) results when (1.22), page 20, is integrated. Since θ[α] represents the direction of a unit tangent vector to the curve, TSC[α] measures the rotation of this vector. This is illustrated by the curve in Figure 6.2 for which θ[α] varies between −π/4 and 5π/4. 6.1. TOTAL SIGNED CURVATURE 155 The total signed curvature of a closed curve is especially important. First, we define carefully the notion of closed curve. Definition 6.4. A regular curve α : (a, b) → Rn is closed provided there is a constant c > 0 such that (6.2) α(t + c) = α(t) for all t. The least such number c is called the period of α. Equation 6.2 expresses closure in the topological sense, as it ensures that α determines a continuous mapping from a circle into Rn , though we shall generally only consider closed curves that are regular. Exercise 1 provides a particular example of a nonregular curve satisfying (6.2). Figure 6.1: Hypotrochoids with turning numbers 2,4,6,8 Clearly, we can use (6.2) to define α(t) for all t; that is, the domain of definition of a closed curve can be extended from (a, b) to R. Just as in the case of a circle, the trace C of a regular closed curve α is covered over and over again by α. Intuitively, when we speak of the length of a closed curve, we mean the length of the trace C . Therefore, in the case of a closed curve we can use either R, or a closed interval [a, b], for the domain of definition of the curve, where b − a is the period. Furthermore, we modify the definition of length given on page 9 as follows: Definition 6.5. Let α: R → Rn be a regular closed curve with period c. By the length of α we mean the length of the restriction of α to [0, c], namely, Z c α′ (t) dt. 0 Next, we find the relation between the period and length of a closed curve. Lemma 6.6. Let α : R → Rn be a closed curve with period c, and let β : R → Rn be a unit-speed reparametrization of α. Then β is also closed; the period of β is L, where L is the length of α. 156 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Proof. Let s denote the arc length function of α starting at 0; we can assume (by Lemma 1.17, page 13) that α(t) = β s(t) . We compute s(t + c) = Z Z c+t α′ (u) du = 0 c α′ (u) du + 0 = L+ Z c+t α′ (u) du c Z c+t α′ (u) du c = L+ Z t α′ (u) du = L + s(t). 0 Thus β s(t) + L = β s(t + c) = α(t + c) = α(t) = β s(t) , for all s(t); it follows that β is closed. Furthermore, since c is the least positive number such that α(t + c) = α(c) for all t, it must be the case that L is the least positive number such that β(s + L) = β(s) for all β. We return to plane curves, that is, to the case n = 2. Definition 6.7. The turning number of a closed curve α: R → R2 is Turn[α] = 1 2π Z c κ2[α](t) α′ (t) dt, 0 where c denotes the period of α. Thus the turning number of a closed curve α is just the total signed curvature of α divided by 2π, so that TSC[α] = 2π Turn[α]. For example, let α : [0, 2nπ] −→ R2 be the function t 7→ a(cos t, sin t), so that α covers a circle n times. It is easy to compute 1 and α′ (t) = a. κ2(α) = a Hence TSC[α] = 2nπ and Turn[α] = n are independent of a. More generally, if α : R → R2 is a regular closed curve with trace C , then the mapping α′ (t) t 7→ (6.3) α′ (t) 6.2. TROCHOIDS 157 gives rise to a mapping Φ from C to the unit circle S 1 (1) of R2 . Note that Φ(p) is just the end point of the unit tangent vector to α at p. It is intuitively clear that when a point p goes around C once, its image Φ(p) goes around S 1 (1) an integral number of times. The turning number of a regular closed curve should therefore be an integer; this will be proved rigorously in Section 6.3. Α'HaL Α'HbL Figure 6.2: Turning of the tangent direction If α denotes the figure eight in Figure 6.2, then Turn[α] = 0 because the ‘clock hand’ never reaches 6 pm. By contrast, each curve in Figure 6.1 has turning number equal to one less than the number of loops; a more explicit representation of a different example is displayed in Figure 6.4, after we have discussed the trochoids in more detail. 6.2 Trochoid Curves The epitrochoid and hypotrochoid are good curves to illustrate turning number. They are defined by (a + b)t , epitrochoid[a, b, h](t) = (a + b) cos t − h cos b (a + b)t (a + b) sin t − h sin b hypotrochoid[a, b, h](t) = (a − b)t (a − b) cos t + h cos , b (a − b)t (a − b) sin t − h sin b . One can describe epitrochoid[a, b, h] as the parametrized curve that is traced out by a point p fixed relative to a circle of radius b rolling outside a fixed circle of 158 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES radius a (see Figure 6.18 at the end of the chapter). Here h denotes the distance from p to the center of the rolling circle. In the case that h = b, the loops degenerate into points, and the resulting curve is a epicycloid (parametrized in Exercise 10 of Chapter 5). Similarly, hypotrochoid[a, b, h] is the curve traced out by a point p fixed relative to one circle (of radius b) rolling inside another circle (of radius a). The curves in Figure 6.1 are hypotrochoid[2k−1, 1, k] for k = 2, 3, 4, 5, respectively. A hypocycloid is a hypotrochoid for which the loops degenerate to points, which again occurs for h = b. Abstractly speaking, the parametriztion of the hypotrochoid is obtained from that of the epitrochoid by simultaneously changing the signs of b and h (Exercise 6). Both epitrochoid[a, b, h] and hypotrochoid[a, b, h] are precisely contained in a circle of radius a + b + h. Each has a/b loops as t ranges over the interval [0, 2π] provided a/b is an integer. More generally, if a, b are integers, it can be verified experimentally that the number of loops is a/gcd(a, b), where gcd(a, b) denotes the greatest common denominator (or highest common factor) of a, b. Whilst the curve will eventually close up provided a/b is rational, it never closes if a/b is irrational; this can be exploited to draw pictures like Figure 6.3. √ Figure 6.3: t 7→ epitrochoid[7, 2, 8](t) with 0 6 t 6 16π As an example to introduce the next section, consider an epitrochoid with 5 inner loops, given by epitrochoid[5, 1, 3](t) = 6 cos t − 3 cos 6t, 6 sin t − 3 sin 6t . It is clear from Figure 6.4 that the curvature has constant sign. But it varies considerably, attaining its greatest (absolute) value during the inner loops. On 6.2. TROCHOIDS 159 the other hand, its total signed curvature (plotted rising steadily in Figure 6.5) is surprisingly linear, since the fluctuations of t 7→ κ2[α](t)kα′ (t)k (the base curve in Figure 6.5) are relatively small. Figure 6.4: α = epitrochoid[5, 1, 3] 35 30 25 20 15 10 5 1 2 3 4 5 6 Figure 6.5: TSC[α] and its integrand Using the turning angle interpretation from Lemma 6.3, it is already clear from Figure 6.4 that α has turning number 6. Indeed, the mini-arrows were positioned by solving an equation asserting that the tangent vector points in the direction of the positive x axis, and they show that the tangent vector undergoes 6 full turns in traversing the curve once. We shall now prove that the turning number of a regular closed curve is always an integer. 160 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES 6.3 The Rotation Index of a Closed Curve In Section 6.1, we gave the definition of turning number in terms of the total signed curvature. In this section, we determine the turning number more directly in terms of the mapping (6.3). First, an auxiliary definition. Definition 6.8. Let φ: S 1 (1) → S 1 (1) be a continuous function, where S 1 (1) e R → R be a denotes a circle of radius 1 and center the origin in R2 . Let φ: continuous function such that e e (6.4) φ(cos t, sin t) = cos φ(t), sin φ(t) e e = 2πn. for all t. The degree of φ is the integer n such that φ(2π) − φ(0) The choice of φe is not unique, since we can certainly add integer multiples of 2π to its value without affecting (6.4). But we have e Lemma 6.9. The definition of degree is independent of the choice of φ. Proof. Let φb : R → R be another continuous function satisfying the same cone Then we have ditions as φ. b − φ(t) e = 2π n(t) φ(t) where n(t) is an integer. Since n(t) is continuous, it must be constant. Thus e e = φ(2π) b b φ(2π) − φ(0) − φ(0). We next use the notion of degree of a map to define the rotation index of a curve. Let α : R → R2 be a regular closed curve. Rescale the parameter so that its period is 2π and α determines a mapping with domain S 1 (1). Definition 6.10. The rotation index of α is the degree of the corresponding mapping Φ[α]: S 1 (1) → S 1 (1) defined by (6.5) Φ[α](t) = α′ (t) . kα′ (t)k Notice that the rotation index of a curve is defined topologically. By contrast, the turning number is defined analytically, as an integral. It is now easy for us to prove that these integers are the same. Theorem 6.11. The rotation index of a regular closed curve α coincides with the turning number of α. Proof. Without loss of generality, we can assume that α has unit speed. Then by (5.9) on page 137, Φ[α](s) = α′ (s) = cos θ(s), sin θ(s) , 6.3. ROTATION INDEX 161 where θ = θ[α] denotes the turning angle of α. Therefore, we can choose g = θ, and appeal to Lemma 6.3 and Corollary 1.27: Φ[α] Z 2π dθ(s) 1 1 θ(2π) − θ(0) = ds degree Φ[α] = 2π 2π 0 ds Z 2π 1 = κ2[α](s)ds 2π 0 = Turn[α]. Those closed curves which do not cross themselves form an important subclass. Definition 6.12. A regular closed curve α : (a, b) → Rn with period c is simple provided α(t1 ) = α(t2 ) if and only if t1 − t2 = c. For example, an ellipse is a simple closed curve, but the curves in Figure 6.1 are not. It is clear intuitively that the rotation index of a simple closed curve is ±1. To prove this rigorously, we need the important topological notion of homotopy. Let α, γ : [0, L] −→ X be two continuous mappings such that (6.6) α(0) = α(L) and γ(0) = γ(L) Here, X can be any topological space, although in our setting it suffices to take X to equal R2 , so that α, γ may be thought of as curves. (Actually, in the proof below, the traces of the two curves will both be unit circles.) What is more, the condition (6.6) ensures that the domain of these curves can also be regarded as a circle (recall (6.2)). Definition 6.13. The mappings α and γ are homotopic (as maps from a circle) if there exists a continuous mapping F : [0, 1] × [0, L] −→ X, such that (i) F (0, t) = α(t) for each t, (ii) F (1, t) = γ(t) for each t, (iii) F (u, 0) = F (u, L) for each u. The map F is called a homotopy between α and γ. 162 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Applying Lemma 6.8, we may define the degree of the mapping t → F (u, t) for each fixed u. But this map varies continuously with u, so its degree must in fact be constant, as in Lemma 6.9. In conclusion, homotopic curves have the same degree. This is used to prove the following theorem, due to H. Hopf1 [Hopf1]. Theorem 6.14. The turning number of a simple closed plane curve α is ±1. Proof. Fix a point p on the trace C of α with the property that C lies entirely to one side of the tangent line at p. This is always possible: choose a line that does not meet C and then translate it until it becomes tangent to C . The idea of the proof is then to construct a homotopy between the unit tangent map Φ[α] of (6.5), whose degree we know to be Turn[α], and a mapping of degree ±1 defined by secants passing through p. B=H0,LL C=HL,LL A=H0,0L Figure 6.6: A homotopy square Denote by L the length of α and consider the triangular region T = { (t1 , t2 ) | 0 6 t1 6 t2 6 L } shown in Figure 6.6. Let β be a reparametrization of α with β(0) = p. The secant map Σ: T → S 1 (1) is defined by 1 Heinz Hopf (1894–1971). Professor at the Eidgenössische Technische Hochschule in Zürich. The greater part of his work was in algebraic topology, motivated by an exceptional geometric intuition. In 1931, Hopf studied homotopy classes of maps from the sphere S 3 to the sphere S 2 and defined what is now known as the Hopf invariant. 6.3. ROTATION INDEX Σ(t1 , t2 ) =                      163 β ′ (t) β ′ (t) − β ′ (0) β ′ (0) β(t2 ) − β(t1 ) β(t2 ) − β(t1 ) for t1 = t2 = t, for t1 = 0 and t2 = L otherwise. Since β is regular and simple, Σ is continuous. Let A = (0, 0), B = (0, L) and C = (L, L) be the vertices of T , as in Figure 6.6. Because the restriction of Σ to the side AC is β ′ /kβ′ k, the degree of this restriction is the turning number of β. Thus, by construction, β′ /kβ′ k is homotopic to the restriction of Σ to the path consisting of the sides AB and BC joined together. We must show that the degree of the latter map is ±1. q p r Figure 6.7: Secants and tangents Assume that β is oriented with respect to R2 , so that the angle from β ′ (0) to −β′ (0) is π. The restriction of Σ to AB is represented by the family of unit vectors parallel to those (partially) shown emanating from p in Figure 6.7, and covers one half of S 1 (1), by the judicious choice of p. Similarly, the restriction of Σ to BC covers the other half of S 1 (1). Hence the degree of Σ restricted to AB and BC is +1. Reversing the orientation, we obtain −1 for the degree. This completes the proof. Corollary 6.15. If β is a simple closed unit-speed curve of period L, then the map s 7→ β′ (s) maps the interval [0, L] onto all of the unit circle S 1 (1). There is actually a far-reaching generalization of Theorem 6.14: 164 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Theorem 6.16. (Whitney-Graustein) Two curves that have the same turning number are homotopic. For a proof of this theorem see [BeGo, page 325]. 6.4 Convex Plane Curves Any straight line ℓ divides R2 into two half-planes H1 and H2 such that H1 ∪ H2 = R 2 and H1 ∩ H2 = ℓ. We say that a curve C lies on one side of ℓ provided either C is completely contained in H1 or C is completely contained in H2 . Definition 6.17. A plane curve is convex if it lies on one side of each of its tangent lines. Since the half-planes are closed, a straight line is certainly convex. We shall however be more concerned with closed curves in this section. Obviously, any ellipse is a convex curve, though we shall encounter many other examples. For a characterization of convex curves in terms of curvature, one needs the notion of a monotone function. Definition 6.18. Let f : (a, b) → R be a function, not necessarily continuous. We say that f is monotone increasing provided that s 6 t implies f (s) 6 f (t), and monotone decreasing provided that s 6 t implies f (s) > f (t). If f is either monotone decreasing or monotone increasing, we say that f is monotone. It is easy to find examples of noncontinuous monotone functions, and also of continuous monotone functions that are not differentiable. In the differentiable case we have the following well-known result: Lemma 6.19. A function f : (a, b) → R is monotone if and only if the deriva- tive f ′ does not change sign on (a, b). More precisely, f ′ > 0 implies monotone increasing and f ′ 6 0 implies monotone decreasing. A glance at any simple closed convex curve C convinces us that the signed curvature of C does not change sign. We now prove this rigorously. Theorem 6.20. A simple closed regular plane curve C is convex if and only if its curvature κ2 has constant sign; that is, κ2 is either always nonpositive or always nonnegative. 6.4. CONVEX PLANE CURVES 165 Proof. Parametrize C by a unit-speed curve β whose turning angle is θ = θ[β] (see Section 1.5). Since θ′ = κ2[β], we must show that θ is monotone if and only if β is convex, and then use Lemma 6.19. Suppose that θ is monotone, but that β is not convex. Then there exists a point p on β for which β lies on both sides of the tangent line ℓ to β at p. Since β is closed, there are points q1 and q2 on opposite sides of ℓ that are farthest from ℓ. q1 p q2 Figure 6.8: Parallel tangent lines on a nonconvex curve The tangent lines ℓ1 at q1 and ℓ2 at q2 must be parallel to ℓ; see Figure 6.8. If this were not the case, we could construct a line ℓ̃ through q1 (or q2 ) parallel to ℓ. Since ℓ̃ would pass through q1 (or q2 ) but would not be tangent to β, there would be points on β on both sides of ℓ̃. There would then be points on β more distant from ℓ than q1 (or q2 ). Two of the three points p, q1 , q2 must have tangents pointing in the same direction. In other words, if p = β(s0 ), q1 = β(s1 ) and q2 = β(s2 ), then there exist si and sj with si < sj such that β′ (si ) = β ′ (sj ) and θ(sj ) = θ(si ) + 2nπ, for some integer n. Since θ is monotone, Theorem 6.14 implies that n = 0, 1 or −1. If n = 0, then θ(si ) = θ(sj ), and the monotonicity of θ implies that θ is constant on the interval [si , sj ]. If n = ±1, then θ is constant on the intervals [0, si ] and [sj , L]. In either case, one of the segments of β between β(si ) and β(sj ) is a straight line. Hence the tangent lines at β(si ) and β(sj ) coincide. But ℓ, ℓ1 and ℓ2 are distinct. Thus we reach a contradiction, and so β must be convex. To prove the converse, assume that β is convex, but that the turning angle θ is not monotone. Then we can find s0 , s1 and s2 such that s1 < s0 < s2 with 166 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES θ(s1 ) = θ(s2 ) 6= θ(s0 ). Corollary 6.15 says that s 7→ β′ (s) maps the interval [0, L] onto all of the unit circle S 1 (1); hence there is s3 such that β ′ (s3 ) = −β′ (s1 ). If the tangent lines at β(s1 ), β(s2 ) and β(s3 ) are distinct, they are parallel, and one lies between the other two. This cannot be the case, since β is convex. Thus two of the tangent lines coincide, and there are points p and q of β lying on the same tangent line. We show that the curve β is a straight line connecting p and q. Let ℓ(p, q) denote the straight line segment from p to q. Suppose that some point r of ℓ(p, q) is not on β. Let ℓb be the straight line perpendicular to ℓ(p, q) at r. Since β is convex, ℓb is nowhere tangent to β. Thus ℓb intersects β in at least two points r1 and r2 . If r1 denotes the point closer to r, then the tangent line to β at r1 has r2 on one side and one of p, q on the other, contradicting the assumption that β is convex. Hence r cannot exist, and so the straight line segment ℓ(p, q) is contained in the trace of β. Thus p and q are β(s1 ) and β(s2 ), so that the restriction of β to the interval [s1 , s2 ] is a straight line. Therefore, θ is constant on [s1 , s2 ], and the assumption that θ is not monotone leads to a contradiction. It follows that κ2[β] has constant sign. The proof of Theorem 6.20 contains that of the following result. Corollary 6.21. Let α be a regular simple closed curve with turning angle θ[α]. If θ[α](t1 ) = θ[α](t2 ) with t1 < t2 , then the restriction of α to the interval [t1 , t2 ] is a straight line. 6.5 The Four Vertex Theorem In this section, we prove a celebrated global theorem about plane curves. To understand the result, let us first consider an example. The sine oval curve is defined by sinoval[n, a](t) = a cos t, a sin(n) (t) , where sin(n) (t) denotes the application sin(sin · · · (sin t)) {z } | n of the sine function n times. (This iterated function is easily computable, as we shall see in Notebook 6.) Clearly, sinoval[1, a] is a circle of radius a. Next, sinoval[2, a] is the curve t 7→ a cos t, a sin(sin t) , and so forth. As n increases, the top and bottom of sinoval[n, a] are pushed together more and more. A typical plot is shown in Figure 6.9. 6.5. FOUR VERTEX THEOREM 167 Figure 6.9: sinoval[3, 1] The curvature of the sine oval has four maxima (all absolute) and four minima (two absolute and two local), as shown in the curvature graph in Figure 6.10. 1.8 1.6 1.4 1.2 1 2 3 4 5 6 0.8 0.6 0.4 Figure 6.10: Curvature of sinoval[3, 1] A simpler example is an ellipse. The curvature of an ellipse has exactly 2 maxima and 2 minima (see Exercise 9). Further examples lead naturally to the conjecture that the signed curvature of any simple closed convex curve has at least two maxima and two minima. To prove this conjecture, we first make the following definition. Definition 6.22. A vertex of a regular plane curve is a point where the signed curvature has a relative maximum or minimum. On any closed curve, the continuous function κ2 must attain a maximum and a minimum, so there are at least two vertices, and they come in pairs. To find the vertices of a simple closed convex curve, we must determine those points where the derivative of the curvature vanishes. It follows from differentiating equation (1.15), page 16, that the definition of vertex is independent of the choice of regular parametrization. We need an elementary lemma: 168 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Lemma 6.23. Let ℓ be a line in the plane. Then there exist constant vectors a, c ∈ R2 with c 6= 0 such that z ∈ ℓ if and only if (z − a) · c = 0. Proof. If we parametrize ℓ by α(t) = p + tq, we can take a = p and c = Jq. Finally, we are ready to prove the conjecture. Theorem 6.24. (Four Vertex Theorem) A simple closed convex curve α has at least four vertices. Proof. The derivative κ2′ vanishes at each vertex of α. If κ2 is constant on any segment of α, then every point on the segment is a vertex, and we are done. We can therefore assume that α contains neither circular arcs nor straight line segments, and that α has at least two distinct vertices p, q ∈ R2 ; without loss of generality, α(0) = p. We now show that the assumption that p and q are the only vertices leads to a contradiction. Because vertices come in pairs, this will complete the proof of the theorem. Let ℓ be the straight line joining p and q; then ℓ divides α into two segments. Since we have assumed that there are exactly two vertices, it must be the case that κ2′ is positive on one segment of α and negative on the other. Lemma 6.23 says that there are constant vectors a and c 6= 0 such that z ∈ ℓ if and only if (z − a) · c = 0. Because α is convex, (z − a) · c is positive on one segment of α and negative on the other. It can be checked case by case that κ2′ (s) α(s) − a · c does not change sign on α. Hence there must be an s0 for which κ2′ (s0 ) α(s0 ) − a · c 6= 0, and so the integral of κ2′ (s) α(s0 ) − a · c from 0 to L is nonzero, where L is the length of C . Integrating by parts, we obtain Z L 0 6= κ2′ (s) α(s) − a · c ds 0 Z L L (6.7) = κ2(s) α(s) − a · c − κ2(s) α′ (s) · c ds 0 0 Z L ′ = − κ2(s)α (s) · c ds. 0 Lemma 1.21, page 16, implies that Jα′′ (s) = −κ2(s)α′ (s), and so the last integral of (6.7) can be written as Z L L α′′ (s) · c ds = α′ (s) · c . 0 ′ 0 ′ But α (L) = α (0), so we reach a contradiction. It follows that the assumption that C had only two vertices is false. It turns out that any simple closed curve, convex or not, has at least four vertices; this is the result of Mukhopadhyaya [Muk]. On the other hand, it is easy to find nonsimple closed curves with only two vertices; see Exercise 5. 6.6. CURVES OF CONSTANT WIDTH 169 6.6 Curves of Constant Width Why is a manhole cover (at least in the United States) round? Probably a square manhole cover would be easier to manufacture. But a square manhole cover, when rotated, could slip through the manhole; however, a circular manhole cover can never slip underground. The reason is that a circular manhole cover has constant width, but the width of a square manhole cover varies between a √ and a 2, where a is the length of a side. Are there other curves of constant width? This question was answered affirmatively by Euler [Euler4] over two hundred years ago. A city governed by a mathematician might want to use manhole covers in the shape of a Reuleaux triangle or the involute of a deltoid, both to be discussed in Sections 6.7. In this section we concentrate on the basic theory of curves of constant width. The literature on this subject is large: see, for example, [Bar], [Bieb, pages 27-29], [Dark], [Euler4], [Fischer, chapter 4], [HC-V, page 216], [MiPa, pages 66-71], [RaTo1, pages 137-150], [Strub, volume 1, pages 120-124] and [Stru2, pages 47-51]. Definition 6.25. An oval is a simple closed plane curve for which the signed curvature κ2 is always strictly positive or always strictly negative. By Theorem 6.20, an oval is convex. The converse is false; for example, the curve x4 + y 4 = 1 has points with vanishing curvature (see Notebook 6). If α is an oval whose signed curvature is always negative, consider instead t 7→ α(−t); thus the signed curvature of an oval can be assumed to be positive. Let β : R → R2 be a unit-speed oval and let β(s) be a point on β. Corollary 6.15 implies that there is a point β(so ) on β for which β ′ (so ) = −β ′ (s), and the reasoning in the proof of Theorem 6.20 implies that β(so ) is unique. b : R → R2 be the curve such that We call β(so ) the point opposite to β(s). Let β o b β(s) = β(s ) for all s. Let us write T(s) = β′ (s). Since T(s) and JT(s) are linearly independent, we can write (6.8) b β(s) − β(s) = λ(s)T(s) + µ(s)JT(s), where λ, µ: R → R are piecewise-differentiable functions. Note that even though b may not have unit-speed. We put T(s) b ′ (s)/kβ b ′ (s)k. b β has unit-speed, β =β Definition 6.26. The spread and width of a unit-speed oval β : R → R2 are the functions λ and µ. We say that an oval has constant width if µ is constant. Figure 6.11 shows that |λ(s)| (respectively |µ(s)|) is the distance p between the b normal (respectively, tangent) lines at β(s) and β(s). Moreover, λ(s)2 + µ(s)2 is the distance between the two opposite points. 170 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES p q Figure 6.11: Spread and width Next, we prove a fundamental theorem about constant width ovals. Theorem 6.27. (Barbier2) An oval of constant width w has length π w. Proof. We parametrize the oval by a unit-speed curve β : R → R2 . From (6.8) and Lemma 1.21, it follows that b ′ = 1 + dλ − µκ2 T + λκ2 + dµ JT. (6.9) β ds ds b we have On the other hand, if sb denotes the arc length function of β so that b′ β = −T b′ β and d sb b′ , = β ds b ′ = − d sbT. β ds From (6.9) and (6.10) we obtain d sb dλ dµ 1+ (6.11) JT = 0. + − µκ2 T + λκ2 + ds ds ds (6.10) Let ϑ = θ = θ[β] denote the turning angle, so that κ2 = dϑ . ds 2 Joseph Émile Barbier (1839–1889). French mathematician who wrote many excellent papers on differential geometry, number theory and probability. 6.6. CURVES OF CONSTANT WIDTH 171 We can use this to rewrite (6.11) as dϑ dµ dϑ d(s + sb) dλ T+ λ JT = 0, + −µ + ds ds ds ds ds from which we conclude that (6.12) dϑ d(s + sb) dλ + −µ =0 ds ds ds and λ dϑ dµ + = 0. ds ds Now suppose that µ has the constant value w. Since the curvature of β is always positive, the second equation of (6.12) tells us that λ = 0, and so the first equation of (6.12) reduces to d(s + sb) dϑ −w = 0. ds ds (6.13) Fix a point p on the oval, and let s0 and s1 be such that β(s0 ) = p and b , where p b is the point on the oval opposite to p. If L denotes the β(s1 ) = p length of the oval, we have from (6.13) that Z s1 Z s1 Z π d(s + sb) dϑ L= w ds = w dϑ = w π. ds = ds ds s0 s0 0 An elegant generalization of Barbier’s theorem, giving formulas for the width and spread of a general oval, has been proved by Mellish3 . Theorem 6.28. (Mellish) The width µ of an oval parametrized by a unit-speed curve β, as a function of ϑ, is a solution of the differential equation d2 µ + µ = f (ϑ), dϑ2 (6.14) where f (ϑ) = 1 1 + . κ2(ϑ) κ2(ϑ + π) Moreover, if we set U (c) = Z c f (t) cos t dt, V (c) = 0 then Z c f (t) sin t dt, 0 µ(ϑ) = U (ϑ) − 21 U (π) sin ϑ − V (ϑ) − 21 V (π) cos ϑ. Proof. We can rewrite (6.12) in terms of differentials: 3 Arthur ds + d sb + dλ − µdϑ = 0 and λdϑ + dµ = 0. Preston Mellish (1905–1930). Canadian mathematician. 172 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Clearly, this implies that (6.15) But db s dλ ds + + −µ=0 dϑ dϑ dϑ 1 ds = dϑ κ2(ϑ) and and λ+ dµ = 0. dϑ db s 1 = , dϑ κ2(ϑ + π) so that (6.15) becomes (6.16) f (ϑ) + dλ −µ=0 dϑ and λ+ dµ = 0. dϑ Elimination of λ in (6.16) yields (6.14). The general solution of (6.14) is ! ! Z ϑ Z ϑ µ(ϑ) = sin ϑ f (t) sin t dt + C2 , f (t) cos t dt + C1 − cos ϑ 0 0 and the arbitrary constants C1 and C2 can be determined by observing that µ, λ, f are all periodic functions of ϑ with period π. There is a similar formula for the spread λ(ϑ) of the oval (see Exercise 13). 6.7 Reuleaux Polygons and Involutes If we relax the condition that our closed curves be regular, simple examples with constant width can be constructed from regular polygons with odd numbers of sides. Let P[n, a] be a regular polygon with 2n + 1 sides, where a denotes the length of any side. Corresponding to each vertex p, there is a side of P[n, a] that is most distant from p. Let p1 and p2 be its vertices, and let p[ 1 p2 be the arc of the circle with center p connecting p1 and p2 . Definition 6.29. The Reuleaux4 polygon is the curve R[n, a] made up of the circular arcs p[ 1 p2 formed when p ranges over the vertices of P[n, a]. The Reuleaux polygon R[n, a] consists of 2n + 1 arcs of a circle of radius a, each subtending an angle of π/(2n+1); thus the length of R[n, a] equals πa. If we parametrize R[n, a] by arc length s, the associated mapping Φ: S 1 (1) → S 1 (1) (recall (6.3)) is undefined at the points s = kπa/(2n + 1) with k = 0, 1, . . . , 2n. Ignoring this finite number of points, the images of Φ and −Φ exactly cover the circle, and no point has an opposite in the sense of the previous section! Despite this defect, we may still assert that the width of the convex curve R[n, a] is constant and equal to a. For however we orient the ‘manhole’, it will always fit snugly into a pipe of diameter a. 4 Franz Reuleaux, (1829–1905). German professor of machine design. 6.7. REULEAUX POLYGONS AND INVOLUTES 173 Figure 6.12: The Reuleaux triangle R[1, 1] and a family of lines The unit Reuleaux triangle is shown in Figure 6.12, together with straight lines joining points of the curve an arc length π/2 apart. The curve R[1, 1] is the model for a cross section of the rotor in the Wankel5 engine. Curves with constant width can be effectively constructed as the involutes of a suitable curve. We shall illustrate this in terms of the deltoid, a special hypocycloid first defined in Exercise 2 of Chapter 2 on page 57. Let D be a deltoid with vertices a, b, c, together with a flexible cord attached c Keep the cord attached to a point p on bc, c and let both to the curved side bc. ends of the cord unwind. The end of the cord that was originally attached to b traces out a curve, part of D’s involute, and similarly for the end of the cord attached to c. Let b move to q and c to r, and denote by pq and pr the line segments from p to q and from p to r. By definition of the involute, c length pq = length pb c and p cc denote arcs of D. where pb 5 Felix and cc, length pr = length p Heinrch Wankel (1902–1988). German engineer. The Wankel engine differs greatly from conventional engines. It retains the familiar intake, compression, power, and exhaust cycle but uses a rotor in the shape of a Reuleaux triangle, instead of a piston, cylinder, and mechanical valves. The Wankel engine has 40 percent fewer parts and roughly one third the bulk and weight of a comparable reciprocating engine. Within the Wankel, three chambers are formed by the sides of the rotor and the wall of the housing. The shape, size, and position of these chambers are constantly altered by the rotor’s clockwise rotation. The engine is unique in that the power impulse is spread over approximately 270o degrees of crank shaft rotation, as compared to 180o degrees for the conventional reciprocating two-stroke engine. 174 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES Since the line segments pq and pr are both tangent to the deltoid at p, they are part of the straight line segment qr connecting q to r. Consequently, c length qr = length along the deltoid of bc. Therefore, qr has the same length, no matter the position of p. It follows that the involute of D has constant width. This construction was originally carried out by Euler [Euler4] in more generality. Figure 6.13: A deltoid and its involute The involute of a deltoid is given by a t t t 7→ 8 cos + 2 cos t − cos 2t, −8 sin + 2 sin t + sin 2t . 3 2 2 Unlike any Reuleaux polygon, this is a regular curve of constant width. It is plotted in Figure 6.13. 6.8 The Support Function of an Oval Given a simple closed curve C , choose a point o inside C . Let m denote the tangent line to C at a point r on C . Let ℓ denote the line through o meeting m perpendicularly at a point p. As r traces out C , so p traces out out the associated pedal curve defined in Section 4.6. Take o to be the origin of coordinates, and let ψ be the angle between ℓ and the x-axis. Set p = length op; the diagram is Figure 6.17 on page 178. 6.8. SUPPORT FUNCTION 175 The point r is given in polar coordinates as reiθ , where r = length or, and θ is the angle between or and the x-axis. Then p = r cos(ψ − θ), so that the line m is given by (6.17) p = r cos(ψ − θ) = x cos ψ + y sin ψ. Since x, y can themselves be expressed in terms of ψ, we ultimately obtain p as a function of ψ. Conversely, let p(ψ) be a given function of ψ; for each value of ψ, (6.17) defines a straight line, and so we obtain a family of lines. This family is defined by (6.18) F (x, y, ψ) = 0, where F (x, y, ψ) = p(ψ) − x cos ψ − y sin ψ. The discourse that follows applies in greater generality to a family of straight lines (or indeed, curves). When a given ψ is replaced by ψ + δ, we obtain a new line implicitly defined by (6.19) F (x, y, ψ + δ) = 0. The set of points that belong to both curves satisfies (6.20) F (x, y, ψ + δ) − F (x, y, ψ) = 0. δ When we take the limit as δ tends to zero in (6.20), we obtain (6.21) ∂F (x, y, ψ) = 0. ∂ψ One calls the curve implicitly defined by eliminating ψ from (6.18) and (6.21) the envelope of the family of lines. In imprecise but descriptive language, we say that the envelope consists of those points which belong to each pair of infinitely near curves in the family (6.18). A similar argument shows that the evolute of a plane curve is the envelope of its normals, exhibited in Figure 4.12 on page 111. In the case at hand, (6.18) and (6.21) become   x cos ψ + y sin ψ = p(ψ), (6.22)  −x sin ψ + y cos ψ = p′ (ψ). Their solution (6.22) is   x = p(ψ) cos ψ − p′ (ψ) sin ψ, (6.23)  y = p(ψ) sin ψ + p′ (ψ) cos ψ, and the parameter ψ can be used to parametrize C . 176 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES 3 2 1 -3 -2 -1 1 2 3 -1 -2 -3 Figure 6.14: The family of straight lines for p(ψ) = csc(1 + c sin ψ) Definition 6.30. Let C be a plane curve and o a point. The support function of C with respect to o is the function p defined by (6.17). The pedal parametrization of C with respect to o is (6.24) oval[p](ψ) = p(ψ) cos ψ − p′ (ψ) sin ψ, p(ψ) sin ψ + p′ (ψ) cos ψ . This formula defines a curve for any function p. However, it will only be closed if p is periodic; this fails in the example illustrated by Figure 6.16. The curve defined by (6.24) will be an oval if and only if it is a simple closed curve and κ2 is never zero. This is certainly the case in Figure 6.15, the envelope determined by the lines in Figure 6.14. 2 1.5 1 0.5 1 2 3 Figure 6.15: An egg and its curvature 4 5 6 6.8. SUPPORT FUNCTION 177 Lemma 6.31. Let p: R → R be a differentiable function. The curvature of oval[p] is given by κ2(ψ) = (6.25) 1 . p(ψ) + p′′ (ψ) As a consequence, oval[p] is an oval if and only if it is a simple closed curve and p(ψ) + p′′ (ψ) is never zero. Proof. Equation (6.25) is an easy calculation from (6.24) and the definition of κ2. It can also be checked by computer; see Exercise 10. Figure 6.16: The nonclosed curve with p(ψ) = sin(2ψ)/ψ There is a formula relating the width and support functions of an oval. Lemma 6.32. Let p be the support function of an oval C with respect to some point o inside C , and let µ denote the width function of C . Then (6.26) µ(ψ) = p(ψ) + p(ψ + π) for 0 6 ψ 6 2π. Proof. Let p be a point on C , and let p1 = po denote the point on C opposite to p. By definition the ray op from o to p meets the tangent line to C at p b . Hence the rays op and op1 are part of a perpendicularly, and similarly for p line segment ℓ that meets each of the two tangent lines perpendicularly. The b . Since the length of length of ℓ is the width of the oval measured at p or at p op is p(ψ) and the length of op1 is p(ψ + π), we obtain (6.26). 178 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES r Ψ-Θ Θ o Figure 6.17: Parametrizing an oval by p(ψ) 6.9 Exercises 1. Consider the piriform defined by piriform[a, b](t) = a(1 + sin t), b cos t(1 + sin t) . Find a point where this curve fails to be regular, and plot piriform[1, 1]. 2. Finish the proof of Lemma 6.2. M 3. Show that the figure eight curve t 7→ (sin t, sin t cos t), (from page 44) and the lemniscate a cos t a sin t cos t t 7→ , , 1 + sin2 t 1 + sin2 t 0 6 t 6 2π 0 6 t 6 2π (from page 43), are both closed curves with total signed curvature and turning number equal to zero. Check the results by computer. 6.9. EXERCISES 179 M 4. Show that the limaçon t 7→ (2a cos t + b)(cos t, sin t), 0 6 t 6 2π (from page 58) has turning number equal to 2. Why is the turning number of a limaçon different from that of a figure eight curve or a lemniscate? 5. Verify that limacon[1, 1] is a nonsimple closed curve that has exactly two vertices, and plot its curvature. 6. Check that epitrochoid[a, b, h](t) = hypotrochoid[a, −b, −h](t), and explain this with reference to the definitions and Figure 6.18. Figure 6.18: Definition of the epitrochoid M 7. Plot hypotrochoid[18, 2, 6] and compute its turning number. M 8. Find an explicit formula for the signed curvature of a general epitrochoid. 9. Verify that a noncircular ellipse has exactly four vertices and plot its curvature. M 10. Prove (6.25) by computer. 11. Show that a parallel curve P to a closed curve C of constant width also has constant width, provided C is interior to P. 180 CHAPTER 6. GLOBAL PROPERTIES OF PLANE CURVES 12. Referring to Figure 6.17, suppose that o = (0, 0) is the center of an ellipse E parametrized as (a cos t, b sin t). Find a relationship between tan t and tan ψ, and deduce that the pedal parametrization of E can be obtained frrom the equation q p= (a2 − b2 ) cos2 t + b2 13. Complete the determination of C1 , C2 in the proof of Theorem 6.28. In the same notation, verify that the spread of an oval is given by λ(ϑ) = − U (ϑ) − 12 U (π) cos ϑ − V (ϑ) − 12 V (π) sin ϑ. 14. The straight lines in Figure 6.13 join opposite points on R[1, 1], where ‘opposite’ now means ‘half the total arc length apart’. Investigate the deltoid-shaped curve formed as the envelope of this family. Chapter 7 Curves in Space In previous chapters, we have seen that the curvature κ2[α] of a plane curve α measures the failure of α to be a straight line. In the present chapter, we define a similar curvature κ[α] for a curve in Rn ; it measures the failure of the curve to be a straight line in space. The function κ[α] reduces to the absolute value of κ2[α] when n = 2. For curves in R3 , we can also measure the failure of the curve to lie in a plane by means of another function called the torsion, and denoted τ [α]. We shall need the Gibbs1 vector cross product to study curves in R3 , just as we needed a complex structure to study curves in R2 . We recall the definition and some of the properties of the vector cross product on R3 in Section 7.1. For completeness, and its use in Chapter 22, we take the opportunity to record analogous definitions for the complex vector spaces Cn and C3 . Curvature and torsion are defined in Section 7.2. We shall also define three orthogonal unit vector fields {T, N, B} along a space curve, that constitute the Frenet frame field of the curve. In Chapters 1 – 5, we made frequent use of the frame field {T, JT} to study the geometry of a plane curve. The Frenet frame {T, N, B} plays the same role for space curves, but the algebra is somewhat different. The twisting and turning of the Frenet frame field can be measured by curvature and torsion. In fact, the Frenet formulas use curvature and torsion to express the derivatives of the three vector fields of the Frenet frame field in terms 1 Josiah Willard Gibbs (1839–1903). American physicist. When he tried to make use of Hamilton’s quaternions (see Chapter 23), he found it more useful to split quaternion multiplication into a scalar part and a vector part, and to regard them as separate multiplications. Although this caused great consternation among some of Hamilton’s followers, the formalism of Gibbs eventually prevailed; engineering and physics books using it started to appear in the early 1900s. Gibbs also played a large role in reviving Grassman’s vector calculus. 191 192 CHAPTER 7. CURVES IN SPACE of the vector fields themselves. We establish the Frenet formulas for unit-speed space curves in Section 7.2, and for arbitrary-speed space curves in Section 7.4. The simplest space curves are discussed in the intervening Section 7.3. A number of other space curves, including Viviani’s curve, the intersection of a sphere and a cylinder, are introduced and studied in Section 7.5. Construction of the Frenet frame of a space curve leads one naturally to consider tubes about such curves. They are introduced in Section 7.6, and constitute our first examples of surfaces in R3 . The torus is a special case, and we quickly generalize its definition to the case of elliptical cross-sections, though the detailed study of such surfaces does not begin until Chapter 10. Knots form one of the most interesting classes of space curves. In the final section of this chapter we study torus knots, that is, knots which lie on the surface of a torus. Visualization of torus knots is considerably enhanced by making them thicker, and in practice considering the associated tube. We observe that such a tube can twist on itself, and explain how this phenomenon is influenced by the curvature and torsion of the torus knot. 7.1 The Vector Cross Product Let us recall the notion of vector cross product on R3 . First, let us agree to use the notation i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). For a ∈ R3 we can write either a = (a1 , a2 , a3 ) or a = a1 i + a2 j + a3 k. Deﬁnition 7.1. Let a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) be vectors in R3 . Then the vector cross product of a and b is formally given by   i j k    a × b = det   a1 a2 a3  . b1 b2 b3 More explicitly,  a × b = det  a2 a3 b2 b3    i − det  a1 a3 b1 b3    j + det  The vector cross product enjoys the following properties: a × c = −c × a, (a + b) × c = a × c + b × c, a1 a2 b1 b2   k. 7.1. VECTOR CROSS PRODUCT 193 (λa) × c = λ(a × c) = a × (λc), (7.1) (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c), (7.2) ka × bk2 = kak2 kbk2 − (a · b)2 , (7.3) a × (b × c) = (a · c)b − (a · b)c, for λ ∈ R and a, b, c, d ∈ R3 . Frequently, (7.2), which is a special case of (7.1), is referred to as Lagrange’s identity2 . For a, b, c ∈ R3 we define the vector triple product [a b c] by     a1 a2 a3 a     [a b c] = det b  = det b1 b2 b3 , c c1 c2 c3 where a = (a1 , a2 , a3 ), and so forth. The vector triple product is related to the dot product and the cross product by the formulas (7.4) [a b c] = a · (b × c) = b · (c × a) = c · (a × b), reflecting properties of the determinant. In particular, a × b is perpendicular to both a and b, and its direction is uniquely determined by the ‘right-hand rule’. Finally, note the following consequence of (7.2): (7.5) ka × bk = kak kbk sin θ, where θ is the angle θ with 0 6 θ 6 π between a and b. As a trivial consequence a × a = 0 for any a ∈ R3 . Equation (7.5) can be interpreted geometrically; it says that ka × bk equals the area of the parallelogram spanned by a and b. Complex Vector Algebra For the sequel, it will be helpful to point out the differences between real and complex vector algebra. This subsection can be omitted on a first reading. Deﬁnition 7.2. Complex Euclidean n-space Cn consists of the set of all complex n-tuples: Cn = 2 (p1 , . . . , pn ) | pj is a complex number for j = 1, . . . , n . Joseph Louis Lagrange (1736–1813). Born in Turin, Italy, Lagrange succeeded Euler as director of mathematics of the Berlin Academy of Science in 1766. In 1787 he left Berlin to become a member of the Paris Academy of Science, where he remained for the rest of his life. He made important contributions to mechanics, the calculus of variations and differential equations, in addition to differential geometry. During the 1790s he worked on the metric system and advocated a decimal base. He died in the same month as Beethoven, March 1813 194 CHAPTER 7. CURVES IN SPACE The elements of Cn are called complex vectors, and one makes Cn into a complex vector space in the same way that we made Rn into a real vector space in Section 1.1. Thus if p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) are complex vectors, we define p + q to be the element of Cn given by p + q = (p1 + q1 , . . . , pn + qn ). Similarly, for λ ∈ C the vector λp is defined by λp = (λp1 , . . . , λpn ). The complex dot product of Cn is given by the same formula as its real counterpart: (7.6) p·q = n X pj qj . j=1 In addition to these operations, Cn also has a conjugation p 7→ p, defined by p = (p1 , . . . , pn ). Conjugation has the following properties: p + q = p + q, λp = λ p, p · q = p · q, p · p > 0. The complex dot product should not be confused with the so-called Hermitian product of Cn , in which one of the two vectors in (7.6) is replaced by its conjugate. The latter is present in the definition of the norm kpk = p p p · p = |p1 |2 + · · · + |pn |2 of a vector in Cn , to ensure that this number is nonnegative. Whilst p · p = 0 does not imply that p = 0, the latter does follow if kpk = 0. Definition 7.1 extends to complex 3-tuples. Let a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) be vectors in C3 . Written out in coordinates, the vector cross product of a and b is given by a × b = a 2 b 3 − a 3 b 2 , a3 b 1 − a 1 b 3 , a1 b 2 − a 2 b 1 . It is easy to check that a × b = a × b, and that all of the identities for the real vector cross product given on page 193, with one exception, hold also for the complex vector cross product. The exception is the Lagrange identity (7.2), whose modification we include in the following lemma: 7.2. CURVATURE AND TORSION 195 Lemma 7.3. For a, b ∈ C3 we have ka × bk2 = kak2 kbk2 − |a · b|2 , (7.7) ka × ak = kak2 (7.8) if a · a = 0, a × a = 2i Im(a2 a3 ), Im(a3 a1 ), Im(a1 a2 ) , (7.9) where a = (a1 , a2 , a3 ). Proof. This is straightforward; for example, (7.8) follows directly because ka × ak2 = a × a · a × a = = −(a × a) · (a × a) −(a · a)(a · a) + (a · a)2 = kak4 . 7.2 Curvature and Torsion of Unit-Speed Curves We first define the curvature of a unit-speed curve in Rn . The definitions are more straightforward in this case, though arbitrary-speed curves in R3 will be considered in Section 7.4. Deﬁnition 7.4. Let β : (c, d) → Rn be a unit-speed curve. Write κ[β](s) = kβ′′ (s)k. Then the function κ[β]: (c, d) → R is called the curvature of β. We abbreviate κ[β] to κ when there is no danger of confusion. Intuitively, curvature measures the failure of a curve to be a straight line. More precisely, a straight line in Rn is characterized by the fact that its curvature vanishes, as we show in Lemma 7.5. Let β : (c, d) → Rn be a unit-speed curve. The following conditions are equivalent: (i) κ ≡ 0; (ii) β ′′ ≡ 0; (iii) β is a straight line segment. Proof. It is clear from the definition that (i) and (ii) are equivalent. To show that (ii) and (iii) are equivalent, suppose that β ′′ ≡ 0. Integration of this n-tuple of differential equations yields (7.10) β(s) = us + v, where u, v ∈ Rn are constant vectors with kuk = 1. Then (7.10) is a unit-speed parametrization of a straight line in Rn . Conversely, any straight line in Rn has a parametrization of the form (7.10), which implies that β ′′ ≡ 0. 196 CHAPTER 7. CURVES IN SPACE In contrast to the signed curvature κ2 defined in Chapter 1, the curvature κ is manifestly nonnegative. For a curve in the plane, the two quantities are however related in the obvious way: Lemma 7.6. Let β : (a, b) → R2 be a unit-speed curve. Then κ[β] = κ2[β] . Proof. From Lemma 1.21 we have κ[β] = kβ′′ k = κ2[β]Jβ ′ = κ2[β] kβ′ k = κ2[β] . Deﬁnition 7.7. Let β : (c, d) → Rn be a unit-speed curve. Then T = β′ is called the unit tangent vector ﬁeld of β. We need a general fact about a vector field along a curve. (The definition of vector field along a curve was given in Section 1.2.) Lemma 7.8. If F is a vector field of unit length along a curve α: (a, b) → Rn , then F′ · F = 0. Proof. Differentiating the equation F(t) · F(t) = 1 gives 2 F(t) · F′ (t) = 0. Taking F = T in Lemma 7.8, we see that T′ · T = 0. We now restrict our attention to unit-speed curves in R3 whose curvature is strictly positive. This implies that T′ = β′′ never vanishes. Now we can define the torsion, as well as the vector fields N and B. Deﬁnition 7.9. Let β : (c, d) → R3 be a unit-speed curve, and suppose that κ(s) > 0 for c < s < d. The vector field N= 1 ′ T κ is called the principal normal vector ﬁeld and B = T × N is called the binormal vector ﬁeld. The triple {T, N, B} is called the Frenet3 frame ﬁeld on β. 3 Jean Frédéric Frenet (1816–1900). French mathematician. Professor at Toulouse and Lyon. 7.2. CURVATURE AND TORSION 197 The Frenet frame at a given point of a space curve is illustrated in Figure 7.1. Figure 7.1: Frenet frame on the helix We are now ready to establish the Frenet formulas; they form one of the basic tools for the differential geometry of space curves. Theorem 7.10. Let β : (c, d) → R3 be a unit-speed curve with κ(s) > 0 for c < s < d. Then: (i) kTk = kNk = kBk = 1 and T · N = N · B = B · T = 0. (ii) Any vector field F along β can be expanded as (7.11) F = (F · T)T + (F · N)N + (F · B)B. (iii) The Frenet formulas hold:  ′   T = (7.12) N′ = −κT   ′ B = κN, +τ B, −τ N. Here, the function τ = τ [β] is called the torsion of the curve β. 198 CHAPTER 7. CURVES IN SPACE Proof. By definition kTk = 1; furthermore, kNk = kβ′′ k 1 ′ T = = 1, κ |κ| and by Lemma 7.8 we have T · N = 0. Therefore, the Lagrange identity (7.2) implies that kBk2 = kT × Nk2 = kTk2 kNk2 − (T · N)2 = kTk2 kNk2 = 1. Finally, (7.4) implies that B · T = B · N = 0. Since T, N and B are mutually orthogonal, they form a basis for the vector fields along β. Hence there exist functions λ, µ, ν such that (7.13) F = λT + µN + νB. When we take the dot product of both sides of (7.13) with T and use (i), we find that λ = F · T. Similarly, µ = F · N and ν = F · B, proving (ii). For (iii), we first observe that the first equation of (7.12) holds by definition of N. To prove the third equation of (7.12), we differentiate B · T = 0, obtaining B′ · T + B · T′ = 0; then B′ · T = −B · T′ = −B · κ N = 0. Lemma 7.8 implies B′ · B = 0. Since B′ is perpendicular to T and B, it must be a multiple of N by part (ii). This means that we can define the torsion τ by the equation B′ = −τ N. We have established the first and third of the Frenet formulas. To prove the second, we use the orthonormal expansion of N′ in terms of T, N, B given by (ii), namely, (7.14) N′ = (N′ · T)T + (N′ · N)N + (N′ · B)B. The coefficients in (7.14) are easy to find. First, differentiating N · T = 0 we get N′ · T = −N · T′ = −N · κ N = −κ. That N′ · N = 0 follows from Lemma 7.8. Finally, N′ · B = −N · B′ = −N · (−τ N) = τ . Hence (7.14) reduces to the second Frenet formula. 7.2. CURVATURE AND TORSION 199 The first book containing a systematic treatment of space curves is Clairaut’s Recherches sur les courbes à double courbure 4 . After that the term ‘courbe à double courbure’ became a technical term for a space curve. The theory of space curves became much simpler after Frenet discovered the formulas named after him in 1847. Serret5 found the formulas independently in 1851, and for this reason they are sometimes called the Frenet-Serret formulas (see [Frenet] and [Serret1]). In spite of their simplicity and usefulness, many years passed before they gained wide acceptance. The Frenet formulas were actually discovered for the first time in 1831 by Senff6 and his teacher Bartels7 . Needless to say, the isolation of Senff and Bartels in Dorpat, then a part of Russia, prevented their work from becoming widely known. (See [Reich] for details.) It is impossible to define N for a straight line parametrized as a unit-speed curve β, since T is a constant vector and κ vanishes. However, one is at liberty to take N and B to be arbitrary constant vector fields along β, and to define the torsion of β to be zero. Then the formulas (7.12) remain valid (see Exercise 5 for details). The following lemma shows that the torsion measures the failure of a curve to lie in a plane. Lemma 7.11. Let β : (c, d) → R3 be a unit-speed curve with κ(s) > 0 for c < s < d. The following conditions are equivalent: (i) β is a plane curve; (ii) τ ≡ 0. When (i) and (ii) hold, B is perpendicular to the plane containing β. 4 Alexis Claude Clairaut (1713–1765). French mathematician and astronomer, who at the age of 18 was elected to the French Academy of Sciences for his work on curve theory. In 1736–1737 he took part in an expedition to Lapland led by Maupertuis, the purpose of which was to verify Newton’s theoretical proof that the earth is an oblate spheroid. Clairaut’s precise calculations led to a near perfect prediction of the arrival of Halley’s comet in 1759. 5 Joseph Alfred Serret (1819–1885). French mathematician. Serret with other Paris mathematicians greatly advanced differential calculus in the period 1840–1865. Serret also worked in number theory and mechanics. Another Serret (Paul Joseph (1827–1898)) wrote a book Théorie nouvelle géométrique et mécanique des lignes à double courbure emphasizing space curves. 6 Karl Eduard Senff (1810–1849). German professor at the University of Dorpat (now Tartu) in Estonia. 7 Johann Martin Bartels (1769–1836). Another German professor at the University of Dorpat. Bartels was the first teacher of Gauss (in Brunswick); he and Gauss kept in contact over the years. 200 CHAPTER 7. CURVES IN SPACE Proof. The condition that a curve β lie in a plane Π can be expressed analytically as (β(s) − p) · q ≡ 0, where q is a nonzero vector perpendicular to Π . Differentiation yields β ′ (s) · q ≡ 0 ≡ β′′ (s) · q. Thus both T and N are perpendicular to q. Since B is also perpendicular to T and N, it follows that B(s) = ±q/kqk for all s. Therefore B′ ≡ 0, and from the definition of torsion, it follows that τ ≡ 0. Conversely, suppose that τ ≡ 0; then (7.12) implies that B′ ≡ −τ N ≡ 0. Thus s 7→ B(s) is a constant curve; that is, there exists a unit vector u ∈ R3 such that B ≡ u. Choose t0 with c < t0 < d and consider the real-valued function f : (c, d) → R3 given by f (s) = β(s) − β(t0 ) · u. Then f (t0 ) = 0 and f ′ (s) ≡ β ′ (s) · u ≡ T · B ≡ 0, so that f is identically zero. Thus β(s)− β(t0 ) · u ≡ 0, and it follows that β lies in the plane perpendicular to u that passes through β(t0 ). 7.3 The Helix and Twisted Cubic The circular helix is a curve in R3 that resembles a spring. Its usual parametrization is (7.15) helix[a, b](t) = a cos t, a sin t, b t , where a > 0 is the radius and b is the slope or incline of the helix. Observe that the projection of R3 onto the xy-plane maps the helix onto the circle (a cos t, a sin t) of radius a. Figure 7.2 displays the helix t 7→ (cos t, sin t, 15 t). The helix is one of the few curves for which a unit-speed parametrization is easy to find. In fact, a unit-speed parametrization of the helix is given by s s bs β(s) = a cos √ , a sin √ , √ . a2 + b 2 a2 + b 2 a2 + b 2 We use Theorem 7.10 to compute κ, τ and T, N, B for β. Firstly, 1 s s T(s) = β′ (s) = √ −a sin √ , a cos √ , b ; a2 + b 2 a2 + b 2 a2 + b 2 thus kT(s)k = 1 and β is indeed a unit-speed curve. Moreover, s 1 s √ √ −a cos T′ (s) = 2 , −a sin , 0 . a + b2 a2 + b 2 a2 + b 2 7.3. HELIX AND TWISTED CUBIC 201 -1 -0.5 0 0.5 1 2 1 0 Figure 7.2: Part of the trace of helix[1, 0.2] Since a > 0, (7.16) and (7.17) N(s) = κ(s) = kT′ (s)k = T′ (s) = kT′ (s)k a2 a , + b2 s s , − sin √ , 0 . − cos √ a2 + b 2 a2 + b 2 From the formula B = T × N we get 1 s s B(s) = √ b sin √ , −b cos √ , a . a2 + b 2 a2 + b 2 a2 + b 2 Finally, to compute the torsion we note that 1 s s √ √ B′ (s) = 2 b cos , b sin , 0 . a + b2 a2 + b 2 a2 + b 2 When we compare this to (7.17), and use the Frenet formula B′ = −τ N, we see that b . τ (s) = 2 a + b2 In conclusion, both the curvature and torsion of a helix are constant, and there is no need to graph them! 202 CHAPTER 7. CURVES IN SPACE Although we know that every regular curve has a unit-speed parametrization, in practice it is very difficult to find it. For example, consider the twisted cubic defined by twicubic(t) = (t, t2 , t3 ). (7.18) One can readily check that this curve does not lie in a plane. For any such plane would have to be parallel to all three vectors γ ′ (0) = (1, 0, 0), γ ′′ (0) = (0, 2, 0), γ ′′′ (0) = (0, 0, 6), where γ = twicubic. But this is clearly impossible. Figure 7.3: twicubic and the plane z = 0 Since γ ′ (0) is a unit vector, it coincides with T at t = 0. On the other hand, γ ′ (t) is not a unit vector unless t = 0, so γ ′′ (0) is not parallel to N. Nonetheless, γ ′′ (0) is a linear combination of T and N, and both pairs {γ ′ (0), γ ′′ (0)} and {T, N} (when applied to the origin) generate the xy-plane, shown in Figure 7.3. The arc length function of the curve (7.18) is s(t) = Z 0 t Z tp kγ (u)k du = 1 + 4u2 + 9u4 du. ′ 0 The inverse of s(t), which is needed to find the unit-speed parametrization, is an elliptic function that too complicated to be of much use. We shall show how to compute the curvature and torsion of (7.18) more directly in the next section. 7.4. ARBITRARY-SPEED CURVES 7.4 Arbitrary-Speed Curves in R 203 3 For efficient computation of the curvature and torsion of an arbitrary-speed curve, we need formulas that bypass finding a unit-speed parametrization. Although we shall define the curvature and torsion of an arbitrary-speed curve in terms of the curvature and torsion of its unit-speed parametrization, ultimately we shall find formulas for these quantities that avoid finding a unit-speed parametrization explicitly. The theoretical definition is e : (c, d) → R3 Deﬁnition 7.12. Let α: (a, b) → R3 be a regular curve, and let α e s(t) , where s(t) is be a unit-speed reparametrization of α. Write α(t) = α e and τe the curvature and torsion of α e, the arc length function. Denote by κ e N, e B} e be the Frenet frame field of α. e Then we define respectively. Also, let {T, e s(t) , κ(t) = κ τ (t) = τe s(t) , e s(t) , N(t) = N e s(t) , B(t) = B e s(t) . T(t) = T Thus, the curvature, torsion and Frenet frame field of an arbitrary-speed curve α are reparametrizations of those of a unit-speed parametrization of α. Next, we generalize the Frenet formulas (7.12) to arbitrary-speed curves. Theorem 7.13. Let α : (a, b) → R3 be a regular curve with speed v = kα′ k = s′ . Then the following generalizations of the Frenet formulas hold:  ′  v κ N,  T =   (7.19) N′ = −v κ T +v τ B,     B′ = −v τ N. Proof. By the chain rule we have  ′ ′ e ′ s(t) = v(t)T e ′ s(t) ,   T (t) = s (t)T   e ′ s(t) = v(t)N e ′ s(t) , (7.20) N′ (t) = s′ (t)N     B′ (t) = s′ (t)B e ′ s(t) = v(t)B e ′ s(t) . e , it follows that Thus from the Frenet formulas for α e s(t) = v(t)κ(t)N(t). T′ (t) = v(t)e κ s(t) N The other two formulas of (7.20) are proved similarly. 204 CHAPTER 7. CURVES IN SPACE Lemma 7.14. The velocity α′ and acceleration α′′ of a regular curve α are given by (7.21) α′ = vT, (7.22) α′′ = dv T + v 2 κN, dt where v denotes the speed of α. e s(t) , where α e is a unit-speed parametrization of α. Proof. Write α(t) = α By the chain rule we have e s(t) = v(t)T(t), e ′ s(t) s′ (t) = v(t)T α′ (t) = α proving (7.21). Next, we take the derivative of (7.21) and use the first equation of (7.19) to get α′′ = v ′ T + vT′ = v ′ T + v 2 κ N, and in stating the result we have chosen to write v ′ = dv/dt. Now we can derive useful formulas for the curvature and torsion for an arbitrary-speed curve. These formulas avoid finding a unit-speed reparametrization. Theorem 7.15. Let α: (a, b) → R3 be a regular curve with nonzero curvature. Then α′ , kα′ k (7.23) T = (7.24) N = B × T, (7.25) B = (7.26) κ = kα′ × α′′ k , kα′ k3 (7.27) τ = [α′ α′′ α′′′ ] . kα′ × α′′ k2 α′ × α′′ , kα′ × α′′ k Proof. Clearly, (7.23) is equivalent to (7.21), and (7.24) is an algebraic consequence of the definition of vector cross product. Furthermore, it follows from (7.21) and (7.22) that dv ′ ′′ 2 α × α = vT × (7.28) T + κv N dt dv = v T × T + κv 3 T × N = κ v 3 B. dt 7.4. ARBITRARY-SPEED CURVES 205 Taking norms in (7.28), we get (7.29) kα′ × α′′ k = kκv 3 Bk = κ v 3 . Then (7.29) implies (7.26). Furthermore, (7.28) and (7.29) imply that B= α′ × α′′ α′ × α′′ = , 3 v κ kα′ × α′′ k and so (7.25) is proved. To prove (7.27) we need a formula for α′′′ analogous to (7.21) and (7.22). Actually, all we need is the component of α′′′ in the B direction, because we want to take the dot product of α′′′ with α′ × α′′ . So we compute ′ dv α′′′ = T + κ v2 N = κ v 2 N′ + · · · (7.30) dt = κ τ v3 B + · · · where the dots represent irrelevant terms. It follows from (7.28) and (7.30) that (7.31) (α′ × α′′ ) · α′′′ = κ2 τ v 6 . Now (7.27) follows from (7.31) and (7.29). To conclude this section, we compute the curvature and torsion of the curve γ = twicubic defined by (7.18). Instead of finding a unit-speed reparametrization of γ, we use Theorem 7.15. The computations are easy enough to do by hand: γ ′ (t) = (1, 2t, 3t2 ), γ ′′ (t) = (0, 2, 6t), γ ′ (t) × γ ′′ (t) = (6t2 , −6t, 2), γ ′′′ (t) = (0, 0, 6), so that kγ ′ (t)k = (1 + 4t2 + 9t4 )1/2 , kγ ′ (t) × γ ′′ (t)k = (4 + 36t2 + 36t4 )1/2 . Therefore, the curvature and torsion of the twisted cubic are given by κ(t) = (4 + 36t2 + 36t4 )1/2 (1 + 4t2 + 9t4 )3/2 τ (t) = 3 . 1 + 9t2 + 9t4 Their respective maximum values are 2 and 3, and the graphs are plotted in Figure 7.4. 206 CHAPTER 7. CURVES IN SPACE 3 2.5 2 1.5 1 0.5 -1 -0.5 0.5 1 Figure 7.4: Curvature and torsion of a cubic The behavior of the Frenet frame of the twisted cubic for large values of |t| is discussed on page 794 in Chapter 23. 7.5 More Constructions of Space Curves Just as the helix sits over the circle, there are helical analogs of many other plane curves. Suppose that we are given a plane curve t 7→ α(t) = (x(t), y(t)). A general formula for the helix with slope c over α is (7.32) helical[α, c](t) = x(t), y(t), ct As an example, we may construct the helical curve over the logarithmic spiral. Using the paraemtrization on page 23, we obtain helical[logspiral[a, b], c](t) = aebt cos t, aebt sin t, ct . A similar but more complicated example formed from the clothoid is shown in Figure 7.8, at the end of this section on page 209. The formula (7.32) can of course be generalized by defining the third coordinate or ‘height’ z to equal an arbitrary function of the parameter t. One can use this idea to graph the signed curvature of a plane curve. Suppose that t 7→ α(t) = (x(t), y(t)) is a plane curve with signed curvature function κ2(t), and define β(t) = x(t), y(t), κ2(t) . Figure 7.5 displays β when α is one of the epitrochoids defined on page 157. 7.5. MORE SPACE CURVES 207 4 5 2 2.5 0 0 -2.5 -5 Figure 7.5: The curvature of epitrochoid[3, 1, 32 ] Let a > 0. Consider the sphere (7.33) x2 + y 2 + z 2 = 4a2 of radius 2a and center the origin, and the cylinder (7.34) (x − a)2 + y 2 = a2 of radius a containing the z axis and passing through the point (2a, 0, 0). This point lies on Viviani’s curve, which is by definition the intersection of (7.33) and (7.34). Figure 7.6 shows one quarter of Viviani’s curve; the remaining quarters are generated by reflection in the planes shown. Using the identity cos t = 1−2 sin2 (t/2), one may verify that this intersection is parametrized by t viviani[a](t) = a(1 + cos t), a sin t, 2a sin (7.35) , −2π 6 t 6 2π, 2 Viviani8 studied the curve in 1692. See [Stru2, pages 10–11] and [Gomes, volume 2, pages 311–320]. Figure 7.6 displays part of the curve in a way that emphasizes that it lies on a sphere. 8 Vincenzo Viviani (1622–1703). Student and biographer of Galileo. In 1660, together with Borelli, Viviani measured the velocity of sound by timing the difference between the flash and the sound of a cannon. In 1692, Viviani proposed the following problem which aroused much interest. How is it possible that a hemisphere has 4 equal windows of such a size that the remaining surface can be exactly squared? The answer involved the Viviani curve. 208 CHAPTER 7. CURVES IN SPACE Figure 7.6: The trace of t 7→ viviani[1](t) for 0 6 t 6 π Computation of the curvature and torsion of Viviani’s curve in Notebook 7 yields the results √ 13 + 3 cos t κ(t) = , a(3 + cos t)3/2 τ (t) = 6 cos(t/2) . a(13 + 3 cos t) These functions are graphed simultaneously in Figure 7.7 for a = 1, 0 6 t 6 2π. The torsion vanishes when t = π, and the most ‘planar’ parts of Viviani’s curve occur at the sphere’s poles. Notebook 7 includes an animation of the Frenet frame on Viviani’s curve; see also Figure 24.8 on page 794. 1 0.8 0.6 0.4 0.2 1 2 3 4 5 6 -0.2 -0.4 Figure 7.7: The curvature (above) and torsion (below) of viviani[1] 7.6. TUBES AND TORI 209 Figure 7.8: Helical curve over clothoid[1, 1] 7.6 Tubes and Tori By definition, the tube of radius r around a set C is the set of points at a distance r from C . In particular, let γ : (a, b) → R3 be a regular curve whose curvature does not vanish. Since the normal N and binormal B are perpendicular to γ, the circle θ 7→ − cos θ N(t) + sin θ B(t) is perpendicular to γ at γ(t). As this circle moves along γ it traces out a surface about γ, which will be the tube about γ, provided r is not too large. We can parametrize this surface by (7.36) tubecurve[γ, r](t, θ) = γ(t) + r −cos θ N(t) + sin θ B(t) , where a 6 t 6 b and 0 6 θ 6 2π. Figure 7.9 displays the tube of radius 1/2 around the trace of helix[2, 12 ]. A tube about a curve γ in R3 has the following interesting property: the volume depends only on the length of γ and radius of the tube. In particular, the volume of the tube does not depend on the curvature or torsion of γ. Thus, for example, tubes of the same radius about a circle and a helix of the same length will have the same volume. For the proofs of these facts and the study of tubes in higher dimensions, see [Gray]. 210 CHAPTER 7. CURVES IN SPACE Figure 7.9: Tube around a helix Figure 7.10 shows tubes around two ellipses. Actually, the ‘horizontal’ tube is a circular torus, formed by rotating a circle around another circle, but both tubes have circular cross sections. More generally, an elliptical torus of revolution is parametrized by the function (7.37) torus[a, b, c](u, v) = (a + b cos v) cos u, (a + b cos v) sin u, c sin v , where u represents the angle of rotation about the z axis. Setting u = 0 gives the ellipse (a + b cos v, 0, c sin v) in the xz-plane centered at (a, 0, 0), waiting to be rotated. Suppose for the time being that a, b, c > 0. In accordance with Figure 7.11, the ratio b/c determines how ‘flat’ (b > c) or how ‘slim’ (b < c) the torus is. On the other hand, the bigger a is, the bigger the ‘hole’ in the middle. Figure 7.10: Tubes around linked ellipses 7.7. TORUS KNOTS 211 Figure 7.11: Tori formed by revolving ellipses Elliptical tori will be investigated further in Section 10.4. Canal surfaces are generalizations of tubes that will be studied in Chapter 20. 7.7 Torus Knots Curves that wind around a torus are frequently knotted. Let us define torusknot[a, b, c][p, q](t) = torus[a, b, c](pt, qt) = a + b cos(qt) cos(pt), a + b cos(qt) sin(pt), c sin(qt) (refer to (7.37)). It follows that torusknot[a, b, c][p, q](t) lies on an elliptical torus. For this reason, we have called the curve a torus knot; it may or may not be truly knotted, depending on p and q. In fact, torusknot[a, b, c][1, q] is unknotted, whereas torusknot[8, 3, 5][2, 3] is the trefoil knot. Of the many books on the interesting subject of knot theory, we mention [BuZi], [Kauf1],[Kauf2] and [Rolf]. Figure 7.12: torusknot[8, 3, 5][2, 5] 212 CHAPTER 7. CURVES IN SPACE For the remainder of this section, we shall study the curve α = torusknot[8, 3, 5][2, 5] displayed in Figure 7.12. This representation is not as informative as we would like, because at an apparent crossing it is not clear which part of the curve is in front and which part is behind. To see what really is going on, let us draw a tube about this curve using the construction of the previous section. The crossings are now clear; furthermore, we can describe the winding a little more precisely by saying that α spirals around the torus 2 times in the longitudinal sense and 5 times in the meridianal sense. (This is the terminology of [Costa2].) More generally, torusknot[a, b, c][p, q] will spiral around an elliptical torus p times in the longitudinal sense and q times in the meridianal sense. Figure 7.13 raises a new issue. The tube itself seems to twist violently in several places. (This is even clearer when the graphics is rendered with fewer sample points in Notebook 7.) A careful look at the plot of the tube shows that this increased twisting occurs in five different places, in spite of the fact that other parts of the tube partially obscure two of these. Moreover, nearby to where the violent twisting occurs, the actual knot curve is straighter. This evidence leads us to suspect that there are five points on the original torus knot where simultaneously the curvature κ is small and the absolute value of the torsion τ is large. Figure 7.13: Tube around a torus knot The graphs of the curvature and torsion of α can be used to test this empirical evidence; they confirm that indeed there are five values of t at which the curvature of α(t) is small when the absolute value of the torsion is large. Furthermore, the graph seems to indicate these points are π/5, 3π/5, π, 7π/5 and 9π/5. In fact, this is the case; the numerical values at π/5 are κ[α](π/5) = 0.076, τ [α](π/5) = −0.414. 7.8. EXERCISES 213 At t = 2π/5 the curvature is a little larger and the absolute value of the torsion is quite small: κ[α](2π/5) = 0.107, τ [α](2π/5) = −0.002. See Figure 7.14. The Frenet formulas (7.19), Page 203, explain why small curvature and large absolute torsion produce so much twisting in the tube. The construction of the tube involves N and B, but not T. When the derivatives of N and B are large, the twisting in the tube will also be large. But a glance at the Frenet formulas shows that small curvature and large absolute torsion create exactly this effect. 0.3 0.2 0.1 0.2 0.4 0.6 0.8 1 -0.1 -0.2 -0.3 -0.4 Figure 7.14: Curvature (positive) and torsion of torusknot[8, 3, 5][2, 5] 7.8 Exercises 1. Establish the following identities for a, b, c, d ∈ R3 : (a) (a × b) × (c × d) = [a c d]b − [b c d]a. (b) (d × (a × b)) · (a × c) = [a b c](a · d). (c) a × (b × c) + c × (a × b) + b × (c × a) = 0. (d) (a × b) · (b × c) × (c × a) = [a b c]2 . M 2. Graph helical curves over a cycloid, a cardioid and a figure eight curve, and plot the curvature and torsion of each. 3. Let α be an arbitrary-speed curve in Rn . Define κn[α] = kα′ kα′′ − kα′ k′ α′ kα′ k3 Show that κn[α] is the curvature of α for n = 3. . 214 CHAPTER 7. CURVES IN SPACE M 4. Plot the following space curves and graph their curvature and torsion: (1 + s)3/2 (1 − s)3/2 s , ,√ . (a) s 7→ 3 3 2 p (b) t 7→ (f (t) cos t, f (t) sin t, a cos 10t where f (t) = 1 − a2 cos2 10t, and a = 0.3. (c) t 7→ 0.5 (1 + cos 10t) cos t, (1 + cos 10t) sin t, 1 + cos 10t . (d) t 7→ (a cosh t, a sinh t, b t). √ (e) t 7→ tet , e−t , 2 t . (f) t 7→ a t − sin t, 1 − cos t, 4 cos(t/2) . 5. Let β : (a, b) → R3 be a straight line parametrized as β(s) = se1 + q, where e1 , q ∈ R3 with e1 a unit vector. Let e2 , e3 ∈ R3 be such that {e1 , e2 , e3 } is an orthonormal basis of R3 . Define constant vector fields by T(s) = e1 , N(s) = e2 and B(s) = e3 for a < s < b. Show that the Frenet formulas (7.12) hold provided we take κ(s) = τ (s) = 0 for a < s < b. 6. Suppose that a < b and choose ǫ to be 1 or −1. Show that the mapping p bicylinder[a, b, ǫ](t) = a cos t, a sin t, ǫ b2 − a2 sin2 t parametrizes one component of the intersection of a cylinder of radius a and an appropriately-positioned cylinder of radius b. M 7. A 3-dimensional version of the astroid is defined by ast3d[n, a, b](t) = (a cosn t, b sinn t, cos 2t). Compute the curvature and torsion of ast3d[n, 1, 1] and plot ast3d[n, 1, 1] for n > 3. Figure 7.15: The 3-dimensional astroid ast3d[3, 1, 1] 7.8. EXERCISES 215 M 8. An elliptical helix is given by (7.38) helix[a, b, c](t) = a cos t, b sin t, ct . Plot an elliptical helix, its curvature and torsion. Show that an elliptical helix has constant curvature if and only if a2 = b2 . [Hint: Compute the derivative of the curvature of an elliptical helix and evaluate it at π/4.] M 9. The Darboux9 vector ﬁeld along a unit-speed curve β : (a, b) → R3 is defined by D = τ T + κ B. Show that   T′ = D × T,   N′ = D × N,    ′ B = D × B. Plot the curves traced out by the Darboux vectors of an elliptical helix, Viviani’s curve , and a bicylinder parametrized in Exercise 6. M 10. A curve constructed from the Bessel functions Ja , Jb , Jc is defined by besselcurve[a, b, c](t) = Ja (t), Jb (t), Jc (t) Plot besselcurve[0, 1, 2]. M 11. A generalization of the twisted cubic (defined on page 202) is the curve twistedn[n, a]: R → Rn given by twistedn[n, a](t) = t, t2 , . . . , tn . Compute the curvature of twistedn[n, a] for 1 6 n 6 6. M 12. Plot the torus knot with p = 3 and q = 2 and graph its curvature and torsion. Plot a tubular surface surrounding it. 13. Find a parametrization for the tube about Viviani’s curve illustrated in Figure 7.16. 9 Jean Gaston Darboux (1842–1917). French mathematician, who is best known for his contributions to differential geometry. His four-volume work Leçons sur la théorie general de surfaces remains the bible of surface theory. In 1875 he provided new insight into the Riemann integral, first defining upper and lower sums and then defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller. 216 CHAPTER 7. CURVES IN SPACE Figure 7.16: A tube of radius 1.5 around Viviani’s curve M 14. The definition of a spherical spiral shown in Figure 7.17 is similar to that of a torus knot, namely sphericalspiral[a][m, n](t) = a cos(mt) cos(nt), sin(mt) cos(nt), sin(nt) . Find formulas for its curvature and torsion. Figure 7.17: The spherical spiral with m = 24 and n = 1 M 15. A ﬁgure eight knot can be parametrized by eightknot(t) = 10(cos t + cos 3t) + cos 2t + cos 4t, 6 sin t + 10 sin 3t, 4 sin 3t sin Plot the tube about the figure eight knot. 5t + 4 sin 4t − 2 sin 6t . 2 Chapter 8 Construction of Space Curves The analog of the Fundamental Theorem of Plane Curves (Theorem 5.13) is the Fundamental Theorem of Space Curves. The uniqueness part of this theorem, proved in Section 8.1, states that two curves with the same torsion and positive curvature differ only by a Euclidean motion of R3 . The torsion of a space curve is not defined at a point where the curvature is zero, so it is not unreasonable to require as a hypothesis that the curvature never vanish. But this requirement turns out to be essential, in the light of a counter-example that we present (see Figure 8.1). In Section 8.2, we prove the existence part, namely that space curves exist with prescribed torsion and positive curvature. We then describe some of the examples with specified curvature and torsion that were generated by the associated computer programs in Notebook 8. Closed curves can be obtained by imposing constant curvature and periodic torsion (see Figure 8.14 on page 253). The notion of contact between two curves or between a curve and a surface is defined in Section 8.3. This part of the chapter is also relevant to the theory of plane curves, although it leads eventually to the definition of the evolute of a space curve. The evolute is traced out by the centers of spheres having third order contact with the given space curve at each of its points. Curves that lie on spheres are characterized in Section 8.4, and they furnish examples for the rest of the chapter. Curves of constant slope are generalizations of a helix, and such a curve γ can be constructed so as to project to a given plane curve β. We see in Section 8.5 that the assumption that γ lie on a sphere requires β to be an epicycloid. Loxodromes on spheres are defined, discussed and plotted in Section 8.6. 229 230 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 8.1 The Fundamental Theorem of Space Curves The curvature and torsion of a space curve determine the curve in very much the same way as the signed curvature κ2 determines a plane curve. First, we establish the invariance of curvature and torsion under Euclidean motions of R3 . For that we need a fact about the vector triple product, defined in Section 7.1. Lemma 8.1. Let A: R3 → R3 be a linear map and a, b, c ∈ R3 . Then [(Aa) (Ab) (Ac)] = det(A)[a b c]. Proof. If we treat a, b, c as column vectors then we may regard A as a matrix premultiplying the vectors. Let (a|b|c) denote the 3 × 3 matrix whose columns are a, b, c. With this notation, [(Aa) (Ab) (Ac)] = Aa · (Ab × Ac) = det(Aa|Ab|Ac) = det (A(a|b|c)) = det(A)[a b c]. Now we can determine the effect of a Euclidean motion on arc length, curvature and torsion. Theorem 8.2. The curvature and the absolute value of the torsion are invariant under Euclidean motions of R3 , though the torsion changes sign under an orientation-reversing Euclidean motion. Proof. Let α: (a, b) → R3 be a curve, and let F : R3 → R3 be a Euclidean motion. Denote by A the linear part of F , so that for all p ∈ R3 we have F (p) = Ap + F (0). We define a curve γ : (a, b) → R3 by γ = F ◦ α; then for a < t < b we have γ(t) = Aα(t) + F (0). Hence γ ′ (t) = Aα′ (t), γ ′′ (t) = Aα′′ (t) and γ ′′′ (t) = Aα′′′ (t). Let sα and sγ denote the arc length functions with respect to α and γ. Since A is an orthogonal transformation, we have s′γ (t) = γ ′ (t) = Aα′ (t) = α′ (t) = s′α (t). (8.1) We compute the curvature of γ, making use of the Lagrange identity (7.2) on page 193: κ[γ](t) = = q γ ′ (t) × γ ′′ (t) γ ′ (t) Aα′ (t) 2 3 = Aα′′ (t) 2 Aα′ (t) × Aα′′ (t) Aα′ (t) 3 2 − Aα′ (t) · Aα′′ (t) Aα′ (t) 3 8.1. FUNDAMENTAL THEOREM OF SPACE CURVES = q α′ (t) 2 α′′ (t) 2 2 − α′ (t) · α′′ (t) α′ (t) 3 = 231 α′ (t) × α′′ (t) α′ (t) 3 = κ[α](t). Similarly, Lemma 8.1 and the fact that det(A) = ±1 implies that τ [γ](t) = = [γ ′ (t) γ ′′ (t) γ ′′′ (t)] γ ′ (t) × γ ′′ (t) 2 = [(Aα′ (t))(Aα′′ (t))(Aα′′′ (t))] det(A)[α′ (t) α′′ (t) α′′′ (t)] α′ (t) × α′′ (t) 2 Aα′ (t) × Aα′′ (t) 2 = det(A)τ [α](t) = ±τ [α](t). Inherent in the above proof, and a consequence of (8.1), is the fact that the arc length of a curve is itself invariant under Euclidean motions. In Theorem 5.13, page 136, we showed that a plane curve is determined up to a Euclidean motion of the plane by its signed curvature. This leads us to the conjecture that a space curve is determined up to a Euclidean motion of R3 by its curvature and torsion. We prove that this is true, with one additional (but essential) assumption. Theorem 8.3. (Fundamental Theorem, Uniqueness) Let α and γ be unit-speed curves in R3 defined on the same interval (a, b), and assume they have the same torsion and the same positive curvature. Then there is a Euclidean motion F of R3 that maps α onto γ. Proof. Since both α and γ have nonzero curvature, both of the Frenet frames {Tα , Nα , Bα } and {Tγ , Nγ , Bγ } are well-defined. Fix s0 with a < s0 < b. There exists a translation of R3 taking α(s0 ) into γ(s0 ). Choose a rotation A that maps α′ (s0 ) onto γ ′ (s0 ). By rotating again in the plane perpendicular to this vector, we may assume that A also maps Nα (s0 ) onto Nγ (s0 ). Lemma 8.1 then guarantees that A maps Bα (s0 ) onto Bγ (s0 ). Thus there exists a Euclidean motion F of R3 such that F α(s0 ) = γ(s0 ) A Tα (s0 ) = Tγ (s0 ), A Nα (s0 ) = Nγ (s0 ), A Bα (s0 ) = Bγ (s0 ). To show that F ◦ α coincides with γ, we define a real-valued function f by f (s) = (A◦Tα )(s)−Tγ (s) 2 + (A◦Nα )(s)−Nγ (s) 2 + (A◦Bα )(s)−Bγ (s) 2 for a < s < b. A computation similar to that of (5.5) on page 137 (using the assumptions that κ[α] = κ[γ] and τ [α] = τ [γ], see Exercise 1) shows that the derivative of f is 0. Since f (s0 ) = 0, we conclude that f (s) = 0 for all s. Hence (F ◦ α)′ (s) = (A ◦ Tα )(s) = γ ′ (s) for all s, and so there exists q ∈ R3 such that (F ◦ α)(s) = γ(s) + q for all s. By the choice of F we have q = 0, and F maps α onto γ. 232 CHAPTER 8. CONSTRUCTION OF SPACE CURVES We now discuss what can go wrong when the assumption of nowhere-zero curvature is dropped. The first thing to note is that it is not possible to distinguish a unit normal vector N(t0 ) at a point where κ(t0 ) = 0, and consequently τ (t0 ) is undefined. Nonetheless, if γ : (a, b) → R3 is a regular curve whose torsion vanishes everywhere except for a single t0 with a < t0 < b, it is reasonable to say that γ has zero torsion. With this convention, we shall explain that the assumption in Theorem 8.3 that α and γ have nonzero curvature is essential. There are indeed two space curves that have the same curvature and torsion for which it is impossible to find a Euclidean motion mapping one onto the other. For example, consider curves α and γ defined by α(t) = ( 0, (t, 0, 5e if t = 0, −1/t2 ), if t 6= 0,  2  (t, 5e−1/t , 0)   γ(t) = 0,   2  (t, 0, 5e−1/t ), if t < 0, and if t = 0, if t > 0. Figure 8.1: The ‘very flat’ curves α and γ 2 The function f (t) = e−1/t (with f (0) = 0) is infinitely differentiable, and (n) f (0) = 0 for all n > 0. It follows that α and γ are regular, infinitely differentiable curves, which have identical curvature functions vanishing at t = 0. Furthermore, their torsion vanishes for both t > 0 and t < 0 since within these parameter ranges the curves are planar. Any Euclidean motion mapping α onto γ would have to be the identity on one portion of R3 and a rotation on another portion of R3 , which is impossible. Despite the nonexistence of an isometry between them, α and γ have the same curvature, and the same torsion function, wherever the latter can legitimately be defined. 8.2. ASSIGNED CURVATURE AND TORSION 233 8.2 Assigned Curvature and Torsion Now we turn to the question of the existence of curves with prescribed curvature and torsion. Theorem 8.4. (Fundamental Theorem of Space Curves, Existence) Suppose that κ : (a, b) → R and τ : (a, b) → R are differentiable functions with κ > 0. Then there exists a unit-speed curve β : (a, b) → R3 whose curvature and torsion are κ and τ . For a < s0 < b the value β(s0 ) can be prescribed arbitrarily. Also, the values of T(s0 ) and N(s0 ) can be prescribed subject to the conditions that kT(s0 )k = kN(s0 )k = 1 and T(s0 ) · N(s0 ) = 0. Proof. Consider the following system of 12 differential equations   x′i (s) = ti (s),      t′ (s) = κ(s)ni (s), i (8.2) ′  n  i (s) = −κ(s)ti (s) + τ (s)bi (s),     ′ bi (s) = −τ (s)ni (s), all for 1 6 i 6 3, together with the  xi (s0 )       ti (s0 )     ni (s0 ) (8.3)   b1 (s0 )      b2 (s0 )    b3 (s0 ) initial conditions = pi , = qi , = ri , = q2 r3 − q3 r2 , = q3 r1 − q1 r3 , = q1 r2 − q2 r1 . In (8.3) we require that 3 X qi2 = 1 = i=1 3 X i=1 ri2 and 3 X qi ri = 0. i=1 From the theory of systems of differential equations, we know that this linear system has a unique solution. If we put β = (x1 , x2 , x3 ) and T = (t1 , t2 , t3 ), N = (n1 , n2 , n3 ), B = (b1 , b2 , b3 ), we see that (8.2) can be compressed into the equations β ′ = T and  ′  κ N,  T = (8.4) N′ = −κ T +τ B,   ′ B = −τ N. Comparison with the Frenet formulas (7.12) shows that κ is the curvature of the curve β, and τ is the torsion. 234 CHAPTER 8. CONSTRUCTION OF SPACE CURVES In practice, Theorem 8.4 is implemented by a program in Notebook 8 which solves the system of differential equations numerically, and then plots the resulting space curve. The following initial examples of its use are designed to investigate cases in which one of κ, τ is constant. If κ = 0 then the definition of τ is irrelevant, as a straight line results. If τ = 0 then the curve is planar. If both κ, τ are constant, then the curve is a helix, and this will be oriented according to the assignment of the initial Frenet frame {T(s0 ), N(s0 ), B(s0 )}. Figure 8.2: Curve with κ(s) = s and τ (s) = 1 Less familiar examples are formed by taking at least one of κ, τ to be the identity function s 7→ s, or s 7→ cs with c > 0 a constant. Figure 8.2 is a 3-dimensional analogue of a clothoid, though a closer resemblance with a helix over a clothoid is obtained by setting κ(s) = |s| and τ (s) = 1. Figure 8.3 shows a ‘double corkscrew’ characterized by both κ, τ being linear. Figure 8.3: A curve with κ(s) = s and τ (s) = s 8.2. ASSIGNED CURVATURE AND TORSION 235 Figure 8.4 displays a straightforward example of a space curve with κ constant. It starts off at the origin resembling a semicircle, but becomes less planar as |τ | increases. There are many other types of curves with constant curvature in space, characterized by different assignments of the torsion τ (s) as a function of arc length. Figure 8.4: A curve with κ(s) = 1 and τ (s) = s Figure 8.5 consists of frames of an animation detailing the evolution of a space curve with constant curvature and periodic torsion proportional to sin s. It begins with a circle traversed twice, which is then broken open and eventually joined up to become a simple closed curve admitting one point with τ (s) = 1. A similarly-formed closed curve is shown in Figure 8.14. Figure 8.5: Curves with κ(s) = 1 and τ (s) = 1 12 n sin s, n = 1, . . . 12 236 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 8.3 Contact In Section 4.1, we defined the evolute of a plane curve as the locus of its centers of curvature. In this section we show how to define the evolute of a space curve. This requires that we generalize the notion of center of curvature to space curves. For this purpose, we need the notion of contact. The latter is also important for plane curves, so we discuss it in a generality that will apply to both cases. We begin by extending Definition 3.1. Definition 8.5. An implicitly-deﬁned hypersurface in Rn is the set of zeros of a differentiable function F : Rn → R. We denote the set of zeros by F −1 (0). Clearly, an implicitly-defined hypersurface in R2 is a curve, and an implicitlydefined hypersurface in R3 is a surface. These are the main cases that we need. The simplest hypersurfaces are hyperplanes and hyperspheres. A hyperplane in Rn consists of the points represented by the set (8.5) { p ∈ Rn | (p − u) · v = 0 } of vectors whose difference from a fixed one u is always perpendicular to another fixed vector v (normal to the plane). A hyperplane in R2 is a line (see the analogous Lemma 6.23), and a hyperplane in R3 is a plane. A hypersphere in Rn of radius r centered at q ∈ Rn is the set of points (8.6) { p ∈ Rn | kp − qk = r } at a distance r from q. In particular, a hypersphere in R2 is a circle and a hypersphere in R3 is a sphere. We leave it to the reader to find differentiable functions whose zero sets are precisely (8.5) and (8.6); recall (3.1) on page 75. Among all lines passing through a point p on a curve α, the tangent line affords the best approximation, though other curves through p may approximate the curve even more closely. The mathematics that makes this idea precise arises from Definition 8.6. Let α: (a, b) → Rn be a regular curve, and let F : Rn → R be a differentiable function. We say that a parametrically-defined curve α and an implicitly-defined hypersurface F −1 (0) have contact of order k at α(t0 ) provided (8.7) but (8.8) (F ◦ α)(t0 ) = (F ◦ α)′ (t0 ) = · · · = (F ◦ α)(k) (t0 ) = 0, (F ◦ α)(k+1) (t0 ) 6= 0. To check that contact is a geometric concept, we prove: 8.3. CONTACT 237 Lemma 8.7. The definition of contact between α and F −1 (0) is independent of the parametrization of α. Proof. Let γ : (c, d) → Rn be a reparametrization of α; then there exists a differentiable function h: (c, d) → (a, b) such that γ(u) = α(h(u)) for c < u < d. Let u0 be such that h(u0 ) = h0 . Then (F ◦ γ)(u0 ) = (F ◦ α)(h0 ) = 0, (F ◦ γ)′ (u0 ) = (F ◦ α)′ (h0 )h′ (u0 ) = 0, (F ◦ γ)′′ (u0 ) = (F ◦ α)′′ (h0 )h′ (u0 )2 + (F ◦ α)′ (h0 )h′′ (u0 ) = 0, (F ◦ γ)′′′ (u0 ) = (F ◦ α)′′′ (h0 )h′ (u0 )3 + 3(F ◦ α)′′ (h0 )h′ (u0 )h′′ (u0 ) +(F ◦ α)′ (h0 )h′′′ (u0 ) = 0, and so forth. Such Leibniz formulas were computed in Notebook 1. In any case, we see that (8.7) implies that (F ◦ γ)(u0 ) = (F ◦ γ)′ (u0 ) = · · · = (F ◦ γ)(k) (u0 ) = 0, and (8.8) implies that (F ◦ γ)(k+1) (u0 ) 6= 0. For example, the x-axis is the set of zeros of the function F : R2 → R given by F (x, y) = y. If n is a positive integer, the generalized parabola t 7→ (t, tn ) has contact of order n − 1 with this horizontal line at (0, 0). A number of these curves are plotted in Figure 8.6. 1.5 1 0.5 -1 -0.5 0.5 -0.5 -1 -1.5 Figure 8.6: Generalized parabolas 1 238 CHAPTER 8. CONSTRUCTION OF SPACE CURVES Next, we discuss contact between hyperplanes and curves in Rn . Lemma 8.8. Let β : (a, b) → Rn be a unit-speed curve, let a < s0 < b and let v ∈ Rn . Then the hyperplane { q | (q − β(s0 )) · v = 0 } has at least order 1 contact with β at β(s0 ) if and only if v is perpendicular to β′ (s0 ). Proof. The hyperplane in question is the zero set of the function F : Rn → R given by F (q) = (q − β(s0 )) · v, so we define f (s) = (β(s) − β(s0 )) · v. Then f ′ (s) = β ′ (s) · v. Hence f ′ (s0 ) = 0 if and only if β ′ (s0 ) · v = 0. Thus at each point on a regular curve, there are hyperplanes with at least order 1 contact with the curve. The tangent line to a plane curve is such a hypersurface, even though it only has dimension 1. As we have just seen, higher order contact with a hyperplane is possible for special curves; in this regard, see Exercise 11 and Figure 7.3. We turn to contact between hyperspheres and curves. Intuitively, it should be the case that a hypersphere can be chosen to have higher order contact with a curve at a given point than is possible with a hyperplane. We first consider plane curves. Lemma 8.9. Let β : (a, b) → R2 be a unit-speed curve, and choose s0 such that a < s0 < b. Let v ∈ R2 be distinct from β(s0 ) so that r = kβ(s0 ) − vk > 0. Let C = { q ∈ R2 | kq − vk = r }. (i) The circle C has at least order 1 contact with β at β(s0 ) if and only if its center v lies on the normal line to β at β(s0 ). (ii) Suppose that κ2(s0 ) 6= 0. Then C has at least order 2 contact with β at β(s0 ) if and only if 1 JT(s0 ). v = β(s0 ) + κ2(s0 ) When this is the case, the radius r of C equals 1/|κ2(s0 )|. Proof. Define g(s) = kβ(s) − vk2 . Then g ′ = 2β′ · (β − v) = 2T · (β − v), so that g ′ (s0 ) = 0 if and only if β ′ (s0 ) · (β(s0 ) − v) = 0. This proves (i). It follows from (i) that there is a number λ such that β(s0 ) − v = λJ T(s0 ). Also, we have g ′′ = 2 T · T + T′ · (β − v) = 2 1 + κ2J T · (β − v) ; hence g ′′ (s0 ) = 2 1 + κ2(s0 )λ . Thus g ′′ (s0 ) = 0 if and only if λ = −1/κ2(s0 ), proving (ii). 8.3. CONTACT 239 Part (ii) of Lemma 8.9 yields a characterization of the osculating circle defined in Section 4.4: Corollary 8.10. Let α : (a, b) → R2 be a regular plane curve, and let a < t0 < b be such that κ2[α](t0 ) 6= 0. Then the osculating circle of α at α(t0 ) is the unique circle which has at least order 2 contact with α at α(t0 ). Next, we consider contact between space curves and spheres. In many cases contact of order 3 is possible. Theorem 8.11. Let β : (a, b) → R3 be a unit-speed curve, and choose s0 such that a < s0 < b. Let v ∈ R3 be distinct from β(s0 ) so that r = kβ(s0 ) − vk > 0. We abbreviate κ[β] and τ [β] to κ and τ , and let S = { q ∈ R3 | kq − vk = r }. (i) The sphere S has at least order 1 contact with β at β(s0 ) if and only if its center v lies on a line perpendicular to β ′ (s0 ) at β(s0 ). (ii) Suppose that κ(s0 ) 6= 0. Then S has at least order 2 contact with β at β(s0 ) if and only if v − β(s0 ) = 1 N(s0 ) − b B(s0 ) κ(s0 ) for some number b. (iii) Suppose that κ(s0 ) 6= 0 and τ (s0 ) 6= 0. Then S has at least order 3 contact with β at s0 if and only if v − β(s0 ) = 1 κ′ (s0 ) N(s0 ) − B(s0 ). κ(s0 ) κ(s0 )2 τ (s0 ) Proof. Define g(s) = β(s) − v 2 . We calculate as in part (i) of Lemma 8.9: g ′ = 2β ′ · (β − v) = 2T · (β − v), so that g ′ (s0 ) = 0 if and only if β ′ (s0 ) · β(s0 ) − v = 0, proving (i). It follows from (i) that there are numbers a and b such that β(s0 ) − v = aN(s0 ) + b B(s0 ). Also, computing as in part (ii) of Lemma 8.9, we find that (8.9) 1 ′′ 2g = T · T + T′ · (β − v) = 1 + κN · (β − v). Hence 1 ′′ 2 g (s0 ) = 1 + κ(s0 )N(s0 ) · aN(s0 ) + b B(s0 ) = 1 + κ(s0 )a. 240 CHAPTER 8. CONSTRUCTION OF SPACE CURVES Thus g ′′ (s0 ) = 0 if and only if a = −1/κ(s0 ), proving (ii). For (iii) we use (8.9) to compute 1 ′′′ 2g = κ′ N + κ N′ · (β − v) + κ N · T = κ′ N − κ2 T + κ τ B · (β − v). In particular, g ′′′ (s0 )/2 equals N(s0 ) κ′ (s0 )N(s0 ) − κ(s0 )2 T(s0 ) + κ(s0 )τ (s0 )B(s0 ) · + b B(s0 ) κ(s0 ) κ′ (s0 ) + κ(s0 )τ (s0 )b. =− κ(s0 ) Thus g ′′′ (s0 ) = 0 if and only if b = κ′ (s0 )/ κ(s0 )2 τ (s0 ) . The space curve analog of an osculating circle is an osculating sphere, first considered by Fuss1 in his paper [Fuss]. It is defined precisely as follows: Definition 8.12. Let α : (a, b) → R3 be a regular space curve, and let a < t0 < b be such that κ[α](t0 ) 6= 0 and τ [α](t0 ) 6= 0. Then the osculating sphere of α at α(t0 ) is the unique sphere which has at least order 3 contact with α at α(t0 ). Corollary 8.13. Let α : (a, b) → R3 be a regular space curve whose curvature and torsion do not vanish at t0 . Then the osculating sphere at α(t0 ) is a sphere of radius s 1 κ[α]′ (t0 )2 (8.10) + κ[α](t0 )2 τ [α](t0 )2 κ[α](t0 )4 and center (8.11) v = α(t0 ) + 1 κ[α]′ (t0 ) B(t0 ). N(t0 ) − κ[α](t0 ) α′ (t0 ) κ[α](t0 )2 τ [α](t0 ) Proof. Let β be a unit-speed reparametrization of α with α(t) = β(s(t)); then by the chain rule κ[β]′ (s(t))s′ (t) = κ[α]′ (t). From this fact and part (iii) of Theorem 8.11 we get (8.11). Since the radius of the osculating sphere at α(t0 ) is kv − α(t0 )k, equation (8.10) follows from (8.11). 1 Nicolaus Fuss (1755–1826). Born in Basel, Fuss went to St. Petersburg, where he became Euler’s secretary. He edited Euler’s works and wrote papers on insurance and many textbooks. From 1800 to his death he was permanent secretary of the St. Petersburg Academy. 8.3. CONTACT 241 Figure 8.7: Dual pairs helix[1, 1], helix[−1, 1] (left) and helix[ 32 , 12 ], helix[− 16 , 21 ] (right) In Section 4.1, we defined the evolute of a plane curve to be the locus of the centers of the osculating circles to the curve. It is natural to define the evolute of a space curve to be the locus of the centers of the osculating spheres. Definition 8.14. The evolute of a regular space curve α is the curve given by evolute3d[α](t) = α(t) + κ[α]′ (t) 1 N(t) − B(t). κ[α](t) α′ (t) κ[α](t)2 τ [α](t) Consider for example α√ = helix[a, b]. From (7.16), we have κ[α]′ = 0. Using (7.17), with t in place of s/ a2 + b2 , yields 2 b2 b evolute3d[α](t) = − cos t, − sin t, b t . a a This is merely helix[−b2/a, b] (it is now convenient to allow the radial parametrer to be negative). The original helix can be recovered by applying taking the evolute again, so the two helices are ‘dual’ to each other; see Figure 8.7. As a second exmple, the evolute of part of a twisted cubic is shown in Figure 8.8. 242 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 2 0 -2 2 1 0 -1 -1 0 1 Figure 8.8: The evolute of a segment (thickened underneath) of the twisted cubic t 7→ (t, t3 , t2 ) 8.4 Space Curves that Lie on a Sphere If a space curve (such as Viviani’s curve) lies on a sphere, then it has contact of order n with that sphere for any positive integer n. In this section we show that certain relations between the curvature and torsion of a space curve are necessary and sufficient for the space curve to lie on a sphere. Theorem 8.15. Let β : (a, b) → R3 be a unit-speed curve, with curvature κ and torsion τ . Suppose that β lies on the sphere of radius c > 0 centered at q ∈ R3 . Then: (i) κ > 1/c; (ii) κ and τ are related by (8.12) ′ 2 1 κ τ 2 c2 − 2 = ; κ κ2 (iii) κ′ (t0 ) = 0 implies τ (t0 ) = 0 or κ(t0 ) = 1/c; ′ ′ κ τ = ; (iv) κ τ κ2 (v) κ′ (t0 ) = κ′′ (t0 ) = 0 implies τ (t0 ) = 0; 8.4. CURVES THAT LIE ON A SPHERE 243 (vi) κ ≡ 1/a if and only if β is a circle of radius a, necessarily contained in some plane. Proof. Differentiation of the equation kβ − qk2 = c2 yields (β − q) · T = 0. Another differentiation results in 1 + (β − q) · T′ = 0. Hence T′ is nonzero, which implies the Frenet frame {T, N, B} is well-defined on β. Thus (8.13) 1 + (β − q) · (κ N) = 0. The Cauchy–Schwarz inequality implies that 1 = |(β − q) · (κ N)| 6 κkβ − qk kNk = κ c, proving (i). Differentiating (8.13), we obtain (8.14) 0 = T · (κ N) + (β − q) · κ′ N + κ(−κ T + τ B) = κ′ (β − q) · N + κ τ (β − q) · B = − κ′ + κ τ (β − q) · B. κ We may now write (8.15) τ (β − q) = τ (β − q) · T T + τ (β − q) · N N + τ (β − q) · B B κ′ τ = − N + 2 B, κ κ the last line from (8.13) and (8.14). Taking norms, τ 2 κ′ 2 + τ 2 c2 = τ 2 kβ − qk2 = , κ κ2 from which we obtain (ii). The latter implies (iii). To prove (iv), we first rearrange (8.14) as (8.16) τ (β − q) · B = κ′ . κ2 Differentiation of (8.16) together with (8.13) yields ′ ′ κ (8.17) = τ ′ (β − q) · B + τ T · B + (β − q) · (−τ N) 2 κ κ′ τ ′ τ2 = 2 + . κ τ κ 244 CHAPTER 8. CONSTRUCTION OF SPACE CURVES This implies (iv). Furthermore, (iv) implies (v). Finally, if κ ≡ 1/a, then (v) implies that τ ≡ 0. Hence by Lemma 7.11 we see that α is a plane curve of constant curvature. From Theorem 1.22, page 16 it follows that α is a circle. It is interesting to check that for Viviani’s curve (which lies on a sphere of radius 2a), τ and κ′ have the same zeros. This is already suggested by Figure 7.7. Instead, Figure 8.9 shows the curve q 9 α(t) = 43 cos t, 12 sin t, 1 − 16 cos2 t − 41 sin2 t lying on a sphere of radius 1, projecting to an ellipse in the xy-plane. Figure 8.10 plots κ[α]′ (with maximum value in excess of 4) and τ [α]. Figure 8.9: The elliptical space curve α 4 2 1 2 3 4 5 -2 -4 Figure 8.10: κ[α]′ and τ [α] 6 8.4. CURVES THAT LIE ON A SPHERE 245 Notice that equation (8.12) also holds for a helix of slope b over a circle of radius a where ca = a2 + b2 , but that conditions (iv) and (v) fail. Indeed, a helix with nonzero torsion lies on no sphere, and we cannot therefore expect to characterize spherical curves by (8.12) alone. Nevertheless, we have the following result, which uses the technique of Theorem 1.22. Theorem 8.16. Let β : (a, b) → R3 be a unit-speed curve whose curvature κ and torsion τ satisfy (8.12). Assume that κ, τ and κ′ vanish only at isolated points. Then β lies on a sphere of radius c centered at some q ∈ R3 . Proof. Let γ =β+ 1 N− κ κ′ τ κ2 B. The Frenet formulas (7.10) imply that κ′ 1 γ = T − 2 N + (−κ T + τ B) − κ κ (8.18) ′ ′ ! κ τ B. − = κ τ κ2 ′ κ′ τ κ2 = κ′ . κ3 ′ B+ κ′ τ κ2 τN Differentiation of (8.12) yields (8.19) κ′ τ κ2 κ′ τ κ2 ′ From (8.18) and (8.19), we get ′ τ κ′ κ′ κ ′ γ = B=0 − τ κ2 κ τ κ2 κ3 or κ′ γ ′ = 0. Thus γ ′ (t) = 0 whenever t is such that κ′ (t) 6= 0. Since γ ′ is continuous, γ ′ (t) = 0 for all t. Hence there exists q ∈ R3 such that γ(t) = q for all t. In other words, (8.20) β(t) − q = − κ′ (t) 1 B(t) N(t) + κ(t) τ (t)κ(t)2 for all t. Then (8.12) implies that kβ(t) − qk2 = c2 . Hence β lies on a sphere of radius c centered at q. The conclusion of Theorem 8.16 can be expressed by saying that the osculating spheres of β all coincide. 246 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 8.5 Curves of Constant Slope Since a helix has constant curvature κ and torsion τ , the ratio τ /κ is constant. Are there other curves with this property? In order to answer this question, we first show that the constancy of τ /κ for a curve is equivalent to the constancy of the angle between the curve and a fixed vector. More precisely: Definition 8.17. A curve γ : (a, b) → R3 is said to have constant slope with respect to a unit vector u ∈ R3 , provided the angle φ between u and the unit tangent vector T of γ is constant. The analytical condition is T · u = cos φ, and we call cot φ the slope of γ. With this convention, if the trace of γ is a straight line lying in the yz-plane x = 0 and u = (0, 0, 1), then the slope of γ with respect to u equals its gradient dz/dy. Lemma 8.18. If a curve γ : (a, b) → R3 has nonzero curvature κ and constant slope with respect to some unit vector u, then τ /κ has the constant value ± cot φ. Proof. Without loss of generality, γ has unit speed. We differentiate T · u = cos φ, obtaining (κ N) · u = 0. Therefore, u is perpendicular to N, and so we can write u = T cos φ ± B sin φ. When we differentiate this equation, we obtain 0 = N(κ cos φ ∓ τ sin φ), and the lemma follows. The converse of Lemma 8.18 is true. Lemma 8.19. Let γ : (a, b) → R3 be a curve for which the ratio τ /κ is constant. Write τ /κ = cot φ. Then γ has constant slope cot φ with respect to some unit vector u ∈ R3 . Proof. Without loss of generality, γ has unit speed. We have 0 = N(κ cos φ − τ sin φ) = d (T cos φ + B sin φ). ds It follows that there exists a constant unit vector u ∈ R3 such that u = T cos φ + B sin φ. In particular, u · T is constant. 8.5. CURVES OF CONSTANT SLOPE 247 Returning to the helix, it is easily verified that (7.15) has constant slope b/a with respect to (0, 0, 1) (Exercise 3). Next, we show that any unit-speed plane curve gives rise to a constant-slope space curve in the same way that a circle gives rise to a helix. Lemma 8.20. Let β : (a, b) → R2 be a unit-speed plane curve, and write β(s) = b1 (s), b2 (s) . Define a space curve γ by γ(s) = b1 (s), b2 (s), s cos ψ , p where ψ is constant. Then γ has constant speed 1 + cos2 ψ and constant slope cos ψ with respect to (0, 0, 1). p Proof. The speed of γ is the norm of γ ′ and is obviously 1 + cos2 ψ. Using this fact, the angle φ between γ ′ (t) and (0, 0, 1) is determined by cos φ = whence cos ψ γ ′ (t) · (0, 0, 1) =p , kγ ′ (t)k 1 + cos2 ψ sec2 φ = 1 + 1 , cos2 ψ and cot φ = cos ψ as stated. The reverse construction also works: a space curve of constant slope projects to a plane curve in a natural way. Lemma 8.21. Let γ : (a, b) → R3 be a unit-speed curve that has constant slope cot φ with respect to a unit vector u ∈ R3 . Assume that cot φ 6= 1. Let β be the projection of γ on a plane perpendicular to u; that is, β(s) = γ(s) − (γ(s) · u)u for a < s < b. Then β has constant speed | sin φ|; furthermore, the curvature of γ and the signed curvature of β are related by (8.21) κ2[β] = ±κ[γ] csc2 φ. Proof. Let v denote the speed of β, and let Tβ and Tγ denote the unit tangent vectors of β and γ. We have vTβ = β ′ (s) = Tγ − (Tγ · u)u = Tγ − (cos φ)u. Thus 1 = kTγ k2 = v 2 + cos2 φ, proving that v = | sin φ|. Also, we have JTβ = ±Nγ . Hence κ[γ]Nγ = T′γ = vT′β = v 2 κ2[β]JTβ = ±(sin2 φ)κ2[β]Nγ . Since sin φ 6= 0, we get (8.21). 248 CHAPTER 8. CONSTRUCTION OF SPACE CURVES To find interesting curves of constant slope other than a helix or a plane curve, we look for curves of constant slope that lie on some sphere. Figure 8.11 represents a typical example of such a curve (viewed from above and from the sides). First, we determine what relations must exist between the curvature and torsion of such curves. Figure 8.11: Four views of a curve on a sphere with constant slope Lemma 8.22. Let γ : (a, b) → R3 be a unit-speed curve that has constant slope cot φ with respect to a unit vector u ∈ R3 , where 0 < φ < π/2. Assume also that γ lies on a sphere of radius c > 0. (i) The curvature and torsion of γ are given by (8.22) κ[γ](s)2 = c2 1 − s2 cot2 φ and τ [γ](s)2 = c2 1 . tan φ − s2 2 (ii) Let β be the projection of γ on a plane perpendicular to u. Then the signed curvature of β satisfies (8.23) κ2[β](s1 )2 = c2 1 , sin φ − s12 cos2 φ 4 where s1 = s sin φ is the arc length function of β. (iii) Let c = a + 2b and cos φ = a/(a + 2b). Then the natural equation (8.23) of β is the same as that of epicycloid[a, b] defined on page 144. Proof. Since τ = κ cot φ, from (8.12) we get ′ 2 κ 1 2 2 = τ c − 2 = (κ2 c2 − 1) cot2 φ, κ2 κ or κ′ √ = ± cot φ. κ2 κ2 c2 − 1 Integrating (8.24), we obtain (8.24) (8.25) 1p 2 2 κ c − 1 = ±s cot φ. κ 8.5. CURVES OF CONSTANT SLOPE 249 When we solve (8.25) for κ, we obtain (8.22), proving (i). Then (ii) follows from (8.21) and (i). For (iii), we first compute 2 16b2 (a + b)2 a2 4 . = sin φ = 1 − (a + 2b)2 (a + 2b)4 Hence (8.23) becomes (8.26) 16b2 (a + b)2 a2 s21 1 = − . 2 2 κ2[β](s1 ) (a + 2b) (a + 2b)2 Then (8.26) is the same as (5.13) with s replaced by s1 . Lemma 8.22 suggests the definition of a new kind of space curve. Definition 8.23. Write epicycloid[a, b](t) = x(t), y(t) . The spherical helix of parameters a, b is defined by p at sphericalhelix[a, b](t) = x(t), y(t), 2 ab + b2 cos . 2b From Lemma 8.22 we have immediately Lemma 8.24. The space curve sphericalhelix[a, b] has constant slope with respect to (0, 0, 1) and lies on a sphere of radius a + 2b centered at the origin. Figure 8.12 illustrates two the following two examples. On the unit sphere S 2 (1), the curves of constant slope that project to a nephroid (see Exercise 6 on page 86) and a cardioid are respectively sphericalhelix[ 12 , 41 ] and sphericalhelix[ 13 , 13 ]. Figure 8.12: Spherical nephroid (left) and cardioid (right) 250 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 8.6 Loxodromes on Spheres A meridian on a sphere is a great circle that passes through the north and south poles. A parallel on a sphere is a circle parallel to the equator. More generally: Definition 8.25. A spherical loxodrome or rhumb line is a curve on a sphere which meets each meridian of the sphere at the same angle, which we call the pitch of the loxodrome. A loxodrome has the north and south poles as asymptotic points. In the early years of terrestrial navigation, many sailors thought that a loxodrome was the same as a great circle, but Nuñez2 distinguished the two. To explain how to obtain the parametrization of a spherical loxodrome, we need the the so-called stereographic map Υ: R2 → S 2 (1), defined by Υ(p1 , p2 ) = (2p1 , 2p2 , p21 + p22 − 1) . p21 + p22 + 1 In fact, Υ is the inverse of stereographic projection stereo, that we shall define in Chapter 22 using complex numbers (see page 730). In the meantime, the reader may check that the straight line joining (p1 , p2 ) and the ‘north pole’ n = (0, 0, 1) intersects the sphere in Υ(p1 , p2 ). It is easy to see that Υ is differentiable and that kΥ(p)k = 1 for all p ∈ R2 . Lemma 8.26. The map Υ has the following properties: (i) Υ maps circles in the plane onto circles on the sphere. (ii) Υ maps straight lines in the plane onto circles on the sphere that pass through (0, 0, 1). (iii) Υ maps straight lines through the origin onto meridians. (iv) Υ preserves angles. Proof. Any circle or line in the plane is given implicitly by an equation of the form (8.27) a(x2 + y 2 ) + b x + cy + d = 0. Let W = 1 + x2 + y 2 , X = 2x/W , Y = 2y/W and Z = 1 − 2/W . Under Υ, the equation (8.27) is transformed into (8.28) b X + cY + (a − d)Z + (d + a) = 0, 2 Pedro Nuñez Salaciense (1502–1578). The first great Portuguese mathematician. Professor at Coimbra. 8.6. LOXODROMES ON SPHERES 251 which is the equation of a plane in R3 . This plane meets the sphere S 2 (1) in a great circle. This proves (i). In the case of a straight line in the plane, a = 0 in (8.27). From (8.28), we see that (0, 0, 1) is on the image curve, proving (ii). If in addition the straight line passes through the origin, then a = d = 0 in (8.27). Then from (8.28) we see that the plane containing the image curve also contains the Z-axis. Hence the curve is a meridian, proving (iii). Statement (iv) will be proved in Lemma 12.14 on page 371. Lemma 8.27. A spherical loxodrome is the image under a stereographic map of a logarithmic spiral. Proof. Lemma 1.29, page 24, implies that a logarithmic spiral meets every radial line from the origin at the same angle. The stereographic map Υ maps each of these radial lines into a meridian of the sphere. Since Υ preserves angles, it must map each logarithmic spiral onto a spherical loxodrome. This leads us to the definition sphericalloxodrome[a, b](t) = 2aebt cos t 2aebt sin t a2 e2bt − 1 , , , 1 + a2 e2bt 1 + a2 e2bt 1 + a2 e2bt a particular case of which can be seen in Figure 8.13. Figure 8.13: A spherical loxodrome of pitch 0.15 252 CHAPTER 8. CONSTRUCTION OF SPACE CURVES 8.7 Exercises 1. Fill in the details of the proof of Theorem 8.3. 2. What is the evolute of Viviani’s curve? No calculation is necessary! 3. Show that a helix has constant slope with respect to (0, 0, 1). 4. Show that any curve in the xy-plane has zero slope with respect to (0, 0, 1). M 5. Consider a space curve γ that projects to a curve α in the xy-plane so that the third component of γ equals κ2[α], as explained in Section 7.5. Plot the result when γ is a figure eight, cardioid, astroid, cycloid. M 6. Another twisted generalization of the plane curve α is the space curve writhe[α](t) = a1 (t), a2 (t) cos t, a2 (t) sin t where α = (a1 , a2 ). Compute the curvature and torsion of writhe[circle[a]]. a Verify that writhe[circle[a]] and viviani[ ](2t + π) have the same curvature 2 and torsion. Conclude that there is a Euclidean motion that takes one curve into the other. Then plot the curves. M 7. Compute the curvature and torsion of writhe[lemniscate[1]], and plot the curve. 8. Let θ : (a, b) → R be an arbitrary differentiable function and define a curve α : (a, b) → R3 by Z t Z t α(t) = cos θ(u)du, sin θ(u)du, ct , a a where c is a constant. Compute the curvature and torsion of α and show that α has constant slope. M 9. Compute the curvature and torsion of sphericalhelix[a, 2b]. Check that the ratio of the curvature to torsion is constant and that sphericalhelix[a, 2b] is a curve on a sphere. Plot several spherical helices. 10. A plane curve is defined by teardrop[a, b, c, d, n](t) = a sin t, d(b sin t + c sin 2t)n . Find the order of contact between teardrop[8, 1, 2, 1, n] and the x-axis at the origin (0, 0). 8.7. EXERCISES 253 11. Let β : (a, b) → Rn be a unit-speed curve, let a < s0 < b and let v ∈ Rn . Then the hyperplane { q | (q−β(s0 )) · v = 0 } has at least order 2 contact with β at β(s0 ) if and only if v is perpendicular to β′ (s0 ) and the curvature of β vanishes at the point of contact β(s0 ). 2 1 0 4 -2 3 0 2 2 1 4 0 Figure 8.14: Space curves with κ(s) = 1, τ (s) = sin and κ(s) = 2, τ (s) = 2 sin s (small) s (large) 2 12. Suppose that a space curve C has arc length s, curvature κ(s) > 0 and torsion τ (s). Let c > 0 be a constant, and set κ̃(s) = cκ(cs), τ̃ (s) = cτ (cs). Prove that a space curve with arc length s, curvature κ̃(s) and torsion τ̃ (s) is a rescaled version of C . An example of this phenomenon can be seen in Figure 8.14. Chapter 9 Calculus on Euclidean Space More than the algebraic operations on Rn are needed for differential geometry. We need to know how to differentiate various geometric objects, and we need to know the relationship between differentiation and the algebraic operations. This chapter establishes the theoretical foundations of the theory of differentiation in Rn that will be the basis for subsequent chapters. In Section 9.1, we define the notion of tangent vector to Rn . The interpretation of a tangent vector to Rn as a directional derivative is given in Section 9.2. This allows us to define tangent maps as the derivatives of differentiable maps between Euclidean spaces Rn and Rm in Section 9.3. It is important to understand how tangent vectors vary from point to point, so we begin the study of vector fields on Rn in Section 9.4. Derivatives of vector fields on Rn with respect to tangent vectors are considered in Section 9.5, in which we also prove some elementary facts about the linear map on vector fields induced by a diffeomorphism. The classical notions of gradient, divergence and Laplacian, are defined in Sections 9.4 and 9.5. Finally, in Section 9.6 we return to our study of curves, using tangent vectors to provide a different perspective and make more rigorous some previous definitions. n 9.1 Tangent Vectors to R In Section 1.1, we defined the notion of a vector v in Rn . So far, we have generally considered vectors in Rn to be points in Rn ; however, a vector v can be regarded as an arrow or displacement which starts at the origin and ends at the point v. From this point of view, and with the theory of surfaces in mind, it makes sense to study displacements that start at points other than the origin, and to retain the starting point in the definition of the tangent vector. 263 264 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Definition 9.1. A tangent vector vp to Euclidean space Rn consists of a pair of elements v, p of Rn ; v is called the vector part and p is called the point of application of vp . v v p Figure 9.1: Vectors applied at different points The tangent vector can be identified with the arrow from p to p + v. Furthermore, two tangent vectors vp and wq are declared to be equal if and only if both their vector parts and points of application coincide: v = w and p = q. Definition 9.2. Let p ∈ Rn . The tangent space of Rn at p is the set Rnp = vp | v ∈ Rn . The tangent space Rnp is a carbon copy of the vector space Rn ; in fact, there is a canonical isomorphism between Rnp and Rn given by vp 7→ v. This isomorphism allows us to turn Rnp into a vector space; we can add elements in Rnp and multiply by scalars in the obvious way, retaining the common point of application: vp + wp = (v + w)p , λvp = (λv)p . To sum up, Rn has a tangent space Rnp attached to each of its points, and the tangent space looks like Rn itself. For this reason, it is possible to transfer the dot product and norm to each tangent space. We simply put vp · wp = v · w for vp , wp ∈ Rnp . It follows that there is a Cauchy–Schwarz inequality for tangent vectors, namely, |vp · wp | 6 kvp k kwp k 9.2. TANGENT VECTORS AS DIRECTIONAL DERIVATIVES 265 for vp , wp ∈ Rnp . Angles between tangent vectors can now be defined in exactly the same way as angles between ordinary vectors (see page 3). vp p+v p Figure 9.2: A tangent vector vp at p To finish this introductory section, consider tangent vectors to R2 and R3 . The complex structure of R2 that we defined on page 3 can be naturally extended to tangent vectors by putting J vp = (J v)p for a tangent vector vp to R2 . It is easy to see that J vp is also a tangent vector to R2 and that −J 2 is the identity map on tangent vectors. Furthermore, J vp · J wp = vp · wp for vp , wp ∈ R2p . Similarly, on R3 we extend the vector cross product to tangent vectors via vp × wp = (v × w)p , for vp , wp ∈ R3p . The usual identities, such as Lagrange’s kvp × wp k2 = kvp k2 kwp k2 − (vp · wp )2 (see page 193) will now hold for tangent vectors. 9.2 Tangent Vectors as Directional Derivatives Frequently, it is important to know how a real-valued function f : Rn → R varies in different directions. For example, if we write the values of f as f (u1 , . . . , un ), 266 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE the partial derivative ∂f /∂ui measures how much f varies in the ui direction, for i = 1, . . . , n. However, it is possible to measure the variation of f in other directions as well. To measure the variation of f at p ∈ Rn along the straight line t 7→ p + tv, we need to make the tangent vector vp operate on functions in a suitable way. Throughout this section, the words ‘f is a differentiable function’ or ‘f is differentiable’ will mean that f has partial derivatives of all orders on its domain. For details on why this is a reasonable definition, see the introduction to Chapter 24. Definition 9.3. Let f : Rn → R be a differentiable function, and let vp be a tangent vector to Rn at p ∈ Rn . We put (9.1) vp [f ] = d f (p + tv) dt . t=0 In elementary calculus, (9.1) goes under the name of directional derivative in the vp direction (at least if kvp k = 1), but we prefer to consider (9.1) as defining how a tangent vector operates on functions. A more explicit way to write (9.1) is vp [f ] = lim t→0 f (p + tv) − f (p) . t For example, let n = 2, f (x, y) = sin(xy), p = (1, 2) and v = (2, 3). Then f (p + tv) = f (1, 2) + t(2, 3) = f (2t + 1, 3t + 2) so that Thus = sin (2t + 1)(3t + 2) = sin 6t2 + 7t + 2 , d f (p + tv) = (12t + 7) cos 6t2 + 7t + 2 . dt vp [f ] = (12t + 7) cos 6t2 + 7t + 2 t=0 = 7 cos 2 = −2.913. Although it is possible to compute vp [f ] directly from the definition, it is usually easier to use a general formula, which we now derive. Lemma 9.4. Let vp = (v1 , . . . , vn )p be a tangent vector to Rn , and f : Rn → R a differentiable function. Then (9.2) vp [f ] = n X j=1 vj ∂f (p). ∂uj 9.2. TANGENT VECTORS AS DIRECTIONAL DERIVATIVES 267 Proof. Let p = (p1 , . . . , pn ), so that p + tv = (p1 + tv1 , . . . , pn + tvn ). The chain rule implies that d d f (p + tv) = f (p1 + tv1 , . . . , pn + tvn ) dt dt n X d(pj + tvj ) ∂f (p1 + tv1 , . . . , pn + tvn ) = ∂uj dt j=1 = n X ∂f (p1 + tv1 , . . . , pn + tvn )vj . ∂u j j=1 When we put t = 0 in this equation, we get (9.2). Next, we list some algebraic properties of the operation (9.1). Lemma 9.5. Let f, g : Rn → R be differentiable functions, vp , wp tangent vectors in Rnp , and a, b real numbers. Then (9.3) (avp + bwp )[f ] = avp [f ] + bwp [f ], (9.4) vp [af + bg] = avp [f ] + bvp [g], (9.5) vp [f g] = f (p)vp [g] + g(p)vp [f ]. Proof. For example, to prove (9.5): d (f g)(p + tv) dt t=0 d d = f (p + tv) g(p + tv) + g(p + tv) f (p + tv) dt dt vp [f g] = t=0 = f (p)vp [g] + g(p)vp [f ]. There is a chain rule for tangent vectors: Lemma 9.6. Let g1 , . . . , gk : Rn → R and h: Rk → R be differentiable functions and write f = h ◦ (g1 , . . . , gk ) for the composed function Rn → R. Let vp ∈ Rnp where p ∈ Rn . Then vp [f ] = k X ∂h g1 (p), . . . , gk (p) vp [gj ]. ∂uj j=1 Proof. Let vp = (v1 , . . . , vn )p . We use the ordinary chain rule to compute vp [f ] = n X i=1 vi n X k X ∂gj ∂f ∂h vi (p) = (p) g1 (p), . . . , gk (p) ∂ui ∂uj ∂ui i=1 j=1 = k X ∂h g1 (p), . . . , gk (p) vp [gj ]. ∂uj j=1 268 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE 9.3 Tangent Maps or Differentials We shall need to deal with functions F : Rn → Rm other than linear maps. For any such function, we can write (9.6) F (p) = f1 (p), . . . , fm (p) , where fj : Rn → R for j = 1, . . . , m, and p ∈ Rn . It will be convenient to use the notation F = (f1 , . . . , fm ) as a shorthand for (9.6). Clearly, (9.6) also makes sense for a function F : U → Rm if U is an open subset of Rn . Recall that a subset U of Rn is said to be open if for any p ∈ U it is possible to ‘squeeze’ a small ball with center p into U, meaning that there exists ε > 0 such that {q ∈ Rm | kq − pk < ε} ⊆ U. Definition 9.7. Let U ⊆ Rn be open. A function F : U → Rm is said to be differentiable provided that each fj is differentiable. More generally, if A is any subset of Rn , we say that F : A → Rm is differentiable if there exists an open set U containing A and a differentiable map Fe : U → Rm such that the restriction of Fe to A is F . A diffeomorphism F between open subsets U and V of Rn is a differentiable map F : U → V possessing an inverse F −1 : V → U that is also differentiable. It is easy to prove that the composition of differentiable functions is differentiable: Lemma 9.8. Suppose that there are differentiable maps U ∩ Rℓ F −→ V ∩ Rm G −→ Rn . Then the composition G ◦ F : U → Rn is differentiable. Any differentiable map F : U → Rm gives rise in a natural way to a linear map between tangent spaces. To establish this, first note that if F : U → Rm is differentiable, then so is t 7→ F (p + tv) for p + tv ∈ U, because it is the composition of differentiable functions. Therefore, the following makes sense: Definition 9.9. Let F : U → Rm be a differentiable map, where U is an open subset of Rn , and let p ∈ U. For each tangent vector vp to Rn set F∗p (vp ) = F (p + tv)′ (0). Then F∗p : Rnp → Rm F (p) is called the tangent map or differential of F at p. Note that F∗p (vp ) is the initial velocity of the curve t 7→ F (p + tv). 9.3. TANGENT MAPS OR DIFFERENTIALS 269 One often abbreviates F∗p to F∗ if the identity of the point in question is clear. For example, let p = (1, 2, 3), v = (1, 0, −1) and F (x, y, z) = (yz, zx, xy). Then F (p + tv) = 6 − 2t, 3 + 2t − t2 , 2 + 2t , so that F∗ (vp ) = d 6 − 2t, 3 + 2t − t2 , 2 + 2t dt t=0 F (p) = (−2, 2, 2)(6,3,2). The following lemma provides a useful way of computing tangent maps. Lemma 9.10. Let F : U → Rm be a differentiable map defined on an open subset U ⊆ Rn , and write F = (f1 , . . . , fm ). If p ∈ U and vp is a tangent vector to Rn at p, then (9.7) F∗ (vp ) = vp [f1 ], . . . , vp [fm ] F (p) . Proof. Replace p by p + tv in (9.6) and differentiate with respect to t: (9.8) F (p + tv)′ = f1 (p + tv)′ , . . . , fm (p + tv)′ . When we evaluate both sides of (9.8) at t = 0 and use (9.1), we obtain (9.7). The next result is an easy consequence of (9.7) and (9.3): Corollary 9.11. If F : U → Rm is differentiable, then at each point p ∈ U the tangent map F∗p : Rnp → Rm F (p) is linear. The tangent map of a differentiable map F can be thought of as the best linear approximation to F at a given point p. We have already seen examples of tangent maps; for example, the velocity of a curve is a tangent map in disguise. To explain this, we let 1p be the tangent vector to R whose vector part is 1 and whose point of application is p. This makes perfect logical sense, and it follows from the definitions that (9.9) 1p [f ] = d f (p + t1) dt = f ′ (p) t=0 for any differentiable function f : R → R. Lemma 9.12. Let α: (a, b) → Rn be a curve. Then for p ∈ (a, b) we have α∗ (1p ) = α′ (p). 270 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Proof. If we write α = (a1 , . . . , an ), then from (9.9) and (9.7) we get α∗ (1p ) = 1p [a1 ], . . . , 1p [an ] = a′1 (p), . . . , a′n (p) = α′ (p). If α: (a, b) → U ⊆ Rn is a differentiable curve, and F : U → Rm a differentiable map, then F ◦ α: (a, b) → Rm is a differentiable curve by Lemma 9.8. It is reasonable to suspect that F∗ maps the velocity of α into the velocity of F ◦ α. We show that this is indeed the case. Lemma 9.13. Let F : U → Rm be a differentiable map, where U is an open subset of Rn , and let α : (a, b) → U be a curve. Then (F ◦ α)′ (t) = F∗ α′ (t) for a < t < b. Proof. Write F = (f1 , . . . , fm ) and α = (a1 , . . . , an ). Lemmas 9.4, 9.10 and the chain rule imply that F∗ α′ (t) = α′ (t)[f1 ], . . . , α′ (t)[fm ]   n n X X ∂fm  ∂f1 a′j (t) a′j (t) , ... , =  ∂u ∂uj j j=1 j=1 α(t) ′ ′ = (f1 ◦ α) (t), . . . , (fm ◦ α) (t) = (F ◦ α)′ (t). Corollary 9.14. Let F : U → Rm be a differentiable map, where U is an open subset of Rn . Let p ∈ U, vp ∈ Rnp , and let α : (a, b) → U be a curve such that α′ (0) = vp . Then F∗p (vp ) = (F ◦ α)′ (0). We originally defined F∗ (vp ) = F∗p (vp ) to be the velocity vector at 0 of the image under F of the straight line t → p + tv. Corollary 9.14 says that F∗ (vp ) equals the velocity vector at 0 of the image under F of any curve α such that α(0) = p and α′ (0) = vp . Corollary 9.15. Suppose that there are differentiable maps U ∩ Rℓ Then (G ◦ F )∗ = G∗ ◦ F∗ . F −→ G V −→ Rn ∩ Rm . 9.3. TANGENT MAPS OR DIFFERENTIALS 271 Proof. Let vp be a tangent vector to Rℓ at p ∈ Rℓ . Then t 7→ F (p + tv) is a curve in Rm such that F∗ (vp ) = F (p + tv)′ (0). Hence ′ ′ (G ◦ F )∗ (vp ) = (G ◦ F )(p + tv) (0) = G F (p + tv) (0) = G∗ F (p + tv)′ (0) = G∗ F∗ (vp ) . Recall from Chapter 1 that if L : Rn → Rm is a linear map, then the matrix associated with L is the m × n matrix A = (aij ) such that L(v) = Av for all v ∈ Rn . (Here, for clarity, we use different symbols for the map and the matrix.) Let {e1 , . . . , en } be the standard basis of Rn , that is, ei is the n-tuple with 1 in the ith spot and zeros elsewhere; similarly, let {e′1 , . . . , e′m } be the standard basis of Rm . Then the formula we are adopting to relate A to L is L(ej ) = m X aij e′i i=1 for j = 1, . . . , n. Thus, it is the j th column of A that represents the image of the j th basis element. If F : U → Rm is a differentiable map, where U is an open subset of Rn , then for each p ∈ U the tangent map L = F∗p is a linear map between the vector n n m spaces Rnp and Rm F (p) . Since Rp is canonically isomorphic to R , and RF (p) is m R , we can associate to F∗p an m × n matrix A, which has the property that F∗p (vp ) = Avp for all vp ∈ Rnp . More precisely, Definition 9.16. Let F : U → Rm be a differentiable map, where U is an open subset of Rn , and write F = (f1 , . . . , fm ). The Jacobian matrix1 is the matrixvalued function J(F ) of F given by   ∂f1 ∂f1 (p) . . . (p)   ∂u1 ∂un     .. ..   .. J(F )(p) =   . . .       ∂fm ∂fm (p) . . . (p) ∂u1 ∂un for p ∈ U. 1 Carl Gustav Jacob Jacobi (1804–1851). Professor of Mathematics at Königsberg and Berlin. He is remembered for his work on dynamics and elliptic functions. His Fundamenta Nova Theoria Functionum Ellipticarum, in which elliptic function theory is based on four theta functions, was published in 1829. 272 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Recall that the rank of an m × n matrix A can be defined as the dimension ρ of the space spanned by the columns of A. A fundamental theorem of linear algebra asserts that it is also equal to the dimension of the space spanned by the rows of A, and is therefore no larger than the minimum of m and n. Accordingly, if ρ equals min(m, n), then A is said to have maximum rank. The situation described by the next result can only occur when m > n. Lemma 9.17. Let F : U → Rm be a differentiable map, where U is an open subset of Rn . Then each tangent map F∗p : Rnp → Rm F (p) is injective if and only if J(F )(p) has rank equal to n. Proof. Let vp ∈ Rnp and write v = (v1 , . . . , vn ); also put F = (f1 , . . . , fm ). Suppose that F∗ (vp ) = 0 and v 6= 0. Then Lemma 9.10 implies that vp [f1 ] = · · · = vp [fm ] = 0, and by Lemma 9.4 we have n X i=1 vi ∂fj (p) = 0 ∂ui for j = 1, . . . , m. It follows that n X ∂f1 ∂fm vi (p), . . . , (p) = 0. ∂ui ∂ui i=1 Therefore, the n columns of the Jacobian matrix J(F )(p) are linearly dependent, and consequently J(F )(p) has rank less than n. Conversely, if the rank of J(F )(p) is less than n, the steps in the above proof can be reversed to conclude that F∗ (vp ) = 0 for some nonzero vp ∈ Rnp . n 9.4 Vector Fields on R It will be necessary to consider tangent vectors as they vary from point to point. Definition 9.18. A vector field V on an open subset U of Rn is a function that assigns to each p ∈ U a tangent vector V(p) ∈ Rnp . If f : U → R is differentiable, we let V act on f via V[f ](p) = V(p)[f ]. The vector field V is said to be differentiable provided that V[f ]: U → R is differentiable whenever f is. It is important to distinguish between the ith coordinate of a point p ∈ Rn and the function that assigns to p its ith coordinate: 9.4. VECTOR FIELDS 273 Definition 9.19. The natural coordinate functions of Rn are the functions ui defined by ui (p) = pi , i = 1, . . . , n, for p = (p1 , . . . , pn ). In the special cases R or R2 or R3 , we shall often denote the natural coordinate functions by symbols such as t or u, v or x, y, z, respectively. Next, we need some ‘standard’ vector fields on Rn . Definition 9.20. The vector field Uj on Rn is defined by Uj (p) = (0, . . . , 0, 1, 0, . . . , 0)p , where 1 occurs in the j th spot. We shall call {U1 , . . . , Un } the natural frame field of Rn . It is obvious from Lemma 9.4 that Uj [f ] = ∂f /∂uj for any differentiable function f , and it is common practice to write ∂/∂uj in place of Uj . Note that the restrictions of U1 , . . . , Un to any open subset U ⊆ Rn are vector fields on U. The proofs of the following algebraic properties of vector fields are easy. Lemma 9.21. Let X and Y be vector fields on an open subset U ⊆ Rn , let a ∈ R and let g : Rn → R. Define X + Y, aX, g X and X · Y by (X + Y)[f ] = X[f ] + Y[f ], (aX)[f ] = a X[f ], (g X)[f ] = g X[f ], (X · Y)(p) = X(p) · Y(p), where f : U → R is differentiable and p ∈ U. Then X + Y, aX, g X are vector fields on U and X · Y : U → R is a function. If X, Y and g are differentiable, so are the vector fields X + Y, aX, g X, and the function X · Y. It is not hard to express a vector field in terms of the Uj ’s: Lemma 9.22. If V is any vector field on an open subset U ⊆ Rn , there exist functions vj : U → R for j = 1, . . . , n such that (9.10) V= n X vj Uj . j=1 Proof. Since V(p) ∈ Rnp , we can write V(p) = v1 (p), . . . , vn (p) p . In this way the functions vj : U → R are defined. On the other hand, v1 (p), . . . , vn (p) p = v1 (p)(1, 0, . . . , 0)p + · · · + vn (p)(0, . . . , 0, 1)p = n X vj (p)Uj (p). j=1 Since this equation holds for arbitrary p ∈ U, we get (9.10). 274 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE The next lemma yields a important practical criterion for the differentiability of a vector field: Corollary 9.23. A vector field V on an open subset U ⊆ Rn is differentiable if and only if the natural coordinate functions v1 , . . . , vn of V given in (9.10) are differentiable. There is a natural way to construct vector fields from functions on Rn . Definition 9.24. Let g : Rn → R be a differentiable function. The gradient of g is the vector field grad g given by grad g = n X ∂g Uj . ∂uj j=1 This operator is characterized by Lemma 9.25. The gradient of a function g is the unique differentiable vector field grad g on Rn for which (grad g) · V = V[g] for any differentiable vector field V on Rn . Proof. Write V = n X vj Uj . Then j=1 grad g · V = n X ∂g ∂g Uj vj ·V = ∂uj ∂uj j=1 j=1 X n = n X vj Uj [g] = V[g]. j=1 Conversely, let Z be a vector field for which Z · V = V[g] for any vector field V on Rn . We can write n X fi Ui , Z= i=1 n where fi : R → R is differentiable for i = 1, . . . , n. Then for j = 1, . . . , n we have ∂g fj = Z · Uj = Uj [g] = , ∂uj so that Z = grad g. Finally, we note some special operations on vector fields on R2 and R3 , which extend the pointwise ones defined at the end of Section 9.1. 9.5. DERIVATIVES OF VECTOR FIELDS 275 Lemma 9.26. For a vector field Y on an open subset U ⊆ R2 , define J Y by (J Y)(p) = J Y(p) for p ∈ U. Similarly, for vector fields Y, Z on U, define their cross product in a pointwise fashion, namely (Y × Z)(p) = Y(p) × Z(p) for p ∈ U. It is immediate that J Y and Y×Z are differentiable vector fields on U whenever Y and Z are differentiable. 9.5 Derivatives of Vector Fields In Section 9.2, we learned how to differentiate functions by means of tangent vectors. A key concept in differential geometry is the notion of how to differentiate a vector field. For vector fields on open subsets of Rn , the definition is natural. If W is a vector field on an open subset U ⊆ Rn , one frequently does not distinguish the element W(p) inside the tangent space Rnp from its vector part v ∈ Rn . Strictly speaking this is incorrect, since W(p) = vp and one loses knowledge of the point p. It is however a convenient abuse of notation, which (as will be explained in Section 9.6) was carried out repeatedly in Chapter 1. Restrict W to that portion of the straight line t 7→ p + tv contained in U so that, with the convention above, t 7→ W(p + tv) is now a curve in Rn . Definition 9.27. Let W be a differentiable vector field on an open subset U of Rn , and let vp ∈ Rnp . The derivative of W with respect to vp is the element of Rnp given by W(p + tv) − W(p) . W(p + tv)′ (0)p = lim t→0 t p This is denoted by DvpW, or DvW if the identity of p is understood. It is possible to compute the tangent vector DvW directly from its definition, but there is an alternative formula (similar to equation (9.7)), which is usually more convenient. Lemma 9.28. Let W be a differentiable vector field on an open subset U ⊆ Rn , and write W= n X i=1 wi Ui . 276 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Then for p ∈ U and vp ∈ Rnp , (9.11) DvW = n X i=1 vp [wi ]Ui (p) = vp [w1 ], . . . , vp [wn ] p . Proof. We have W(p + tv) = n X wi (p + tv)Ui (p + tv) i=1 = w1 (p + tv), . . . , wn (p + tv) p+tv . Hence W(p + tv)′ = w1 (p + tv)′ , . . . , wn (p + tv)′ . But by the definition of tangent vector, we have wj (p + tv)′ (0) = vp [wj ] for j = 1, . . . , n, so as required. W(p + tv)′ (0) = vp [w1 ], . . . , vp [wn ] , It is an easy matter to compute the derivatives of the standard vector fields U1 , . . . , Un . Corollary 9.29. We have DpUj = 0 for any tangent vector vp to Rn , and j = 1, . . . , n. Proof. If we use (9.10) to express Uj in terms of U1 , . . . , Un , then each of the coefficients v1 , . . . , vn in (9.10) is either 0 or 1, and is in any case constant. The result follows from (9.11). Next, we show that the operator D behaves naturally with respect to the operations on vector fields given in Lemma 9.21. Lemma 9.30. Let Y, Z be differentiable vector fields on an open subset U of Rn , and vp , wp tangent vectors to Rn at p ∈ U. Then Dav+bwY = aDvY + b DwY, Dv(aY + b Z) = aDvY + b DvZ, Dv(f Y) = vp [f ]Y(p) + f (p)DvY, (9.12) vp [Y · Z] = DvY · Z(p) + Y(p) · DvZ, where a, b ∈ R and f : U → R is a differentiable function. 9.5. DERIVATIVES OF VECTOR FIELDS 277 Proof. We prove (9.12); the proofs of the other formulas are similar. Write Y= n X yi Ui and Z = n X zi Ui . i=1 i=1 From (9.4) and (9.5) it follows that vp [Y · Z] = = n X vp [yi zi ] i=1 n X i=1 = X n i=1 vp [yi ]zi (p) + yi (p)vp [zi ] X n zi (p)Ui (p) vp [yi ]Ui (p) · + X n i=1 i=1 X n vp [zi ]Ui (p) yi (p)Ui (p) · i=1 = DvY · Z(p) + Y(p) · DvZ. The operator D also behaves naturally with respect to the complex structure J of R2 and the vector cross product of R3 : Lemma 9.31. Let Y and Z be differentiable vector fields on an open subset U ⊆ Rn , where n = 2 or 3, and let vp , wp be tangent vectors to Rn . Then (9.13) Dv(J Y) = J DvY (for n = 2), (9.14) Dv(Y × Z) = DvY × Z(p) + Y(p) × DvZ (for n = 3). 2 Proof. Given a vector field Y on R , we can write Y = y1 U1 + y2 U2 ; then J Y = −y2 U1 + y1 U2 . It follows from (9.11) that Dv(J Y) = − vp [y2 ], vp [y1 ] p = J vp [y1 ], vp [y2 ] p = J (DvY), giving (9.13). The proof of (9.14) is similar. It is also possible to define the derivative of one vector field by another. Lemma 9.32. Let V and W be vector fields defined on an open subset U ⊆ Rn , and assume that W is differentiable. Define DVW by (DVW)(p) = DV(p)W for p ∈ U. Then DVW is a vector field on U which is differentiable if V and W are differentiable. Moreover, X X n n (9.15) V[wi ]Ui . wi Ui = DV i=1 i=1 Proof. Equation (9.15) is an immediate consequence of (9.11). 278 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE Let F : U → V be a diffeomorphism between open subsets U and V of Rn , and let V be a differentiable vector field on U. We want to define the image F∗ (V) of V under F , in a manner consistent with (9.7). The following does the job: Definition 9.33. Let F : U → V be a diffeomorphism between open subsets U and V of Rn and let V be a differentiable vector field on U. Write F = (f1 , . . . , fn ). Then F∗ (V) is the vector field on V given by F∗ (V) = n X i=1 V[fi ] ◦ F −1 Ui . It is easy to prove Lemma 9.34. Suppose that F : U → V is a diffeomorphism between open subsets U and V of Rn . Let Y and Z be differentiable vector fields on an open subset U. Then (9.16) (9.17) F∗ (aY + b Z) = aF∗ (Y) + b F∗ (Z), F∗ (f Y) = (f ◦ F −1 )F∗ (Y), F∗ (Y)[f ◦ F −1 ] = Y[f ] ◦ F −1 , where a, b are real numbers and f : U → R is a differentiable function. Next, we show that certain maps of Rn into itself have a desirable effect on the derivatives of vector fields. Lemma 9.35. Suppose that F : Rn → Rn is an affine transformation. Then (9.18) DF∗ (V) F∗ (W) = F∗ DVW . Proof. By assumption, F (p) = Ap + q, where A: Rn → Rn is linear. We can therefore write n X (9.19) aij Ui , j = 1, . . . , n F∗ (Uj ) = i=1 where each aij is constant. From (9.16) and (9.17), we get X n (9.20) wj Uj DF∗ (V)F∗ (W) = DF∗ (V) F∗ j=1 X n = DF∗ (V) j=1 = n X j=1 (wj ◦ F −1 )F∗ (Uj ) DF∗ (V) (wj ◦ F −1 )F∗ (Uj ) . 9.5. DERIVATIVES OF VECTOR FIELDS 279 Then (9.15) and (9.19) imply that (9.21) DF∗ (V) (wj ◦ F −1 n X −1 aij Ui )F∗ (Uj ) = DF∗ (V) (wj ◦ F ) i=1 = n X i=1 F∗ (V) (wj ◦ F −1 )aij Ui . Since each aij is constant, (9.22) F∗ (V) (wj ◦ F −1 )aij = F∗ (V) wj ◦ F −1 aij . From (9.20)–(9.22), we get DF∗ (V)F∗ (W) n X = i,j=1 (9.23) F∗ (V) wj ◦ F −1 aij Ui = n X (V[wj ] ◦ F −1 )F∗ (Uj ) = n X F∗ V[wj ]Uj . j=1 j=1 Now (9.18) follows from (9.23) and (9.15). We conclude this section by defining some well-known operators from vector analysis. Definition 9.36. Let V be a differentiable vector field defined on an open subset U ⊆ Rn . The divergence of V is the function div V = n X DUiV · Ui . i=1 If f : U → R is differentiable, then the Laplacian of f is the function ∆f = div(grad f ). Lemma 9.37. If f : U → R is a differentiable function, where U ⊆ Rn is open, then ∆f = n X Ui2 f = n X ∂2f i=1 i=1 ∂u2i . Proof. We have ∆f = div(grad f ) = n X i=1 DUi(grad f ) · Ui 280 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE = = n X i=1 Ui grad f · Ui n X Ui Ui [f ] . i=1 9.6 Curves Revisited In this section, we assume for simplicity that all curves are differentiable in the manner explained on page 266, so that partial derivatives to all orders exist. In the study of curves and surfaces, it is common practice to make no distinction between a tangent vector vp and its vector part. For example, let α : (a, b) → Rn be a curve. On page 6, we defined its velocity α′ (t) to be a vector in Rn . More properly, it should be a tangent vector at the point α(t): Definition 9.38. (Revised) The velocity of a curve α is the function t 7→ α′ (t)α(t) , and the acceleration of α is the function t 7→ α′′ (t)α(t) . Since α′ (t)α(t) is a tangent vector, it can be applied to a differentiable function f : Lemma 9.39. Let α : (a, b) → Rn be a curve and f : Rn → R a differentiable function. Then α′ (t)α(t) [f ] = (f ◦ α)′ (t). Proof. We write α′ (t) = a′1 (t), . . . , a′n (t) . It follows from Lemma 9.4 that (9.24) ′ α (t)α(t) [f ] = n X j=1 a′j (t) ∂f α(t) . ∂uj But the chain rule says that the right-hand side of (9.24) is (f ◦ α)′ (t). In Section 1.2 we defined the notion of vector field along a curve α in Rn . A vector field W on Rn gives rise to a vector field on α; it is simply W ◦ α. Note that for each t, (W ◦ α)(t) is a vector in Rn . A closely related concept is the restriction of W to α, which is the function t 7→ W α(t) α(t) . Frequently, it is useful to make no distinction between the vector (W ◦ α)(t) in Rn , and the tangent vector W α(t) = W α(t) α(t) in Rnα(t) , mainly because it is too much trouble to write the subscript. Similarly, unless complete clarity is required, we shall not distinguish between α′ (t) and α′ (t)α(t) . 9.7. EXERCISES 281 In Section 1.2, we also defined the derivative of a vector field along a curve. In particular, we know how to compute (W ◦ α)′ . Let us make precise the relation between (W ◦ α)′ and the derivative of W with respect to α′ (t). Lemma 9.40. If W is a differentiable vector field on Rn and α : (a, b) → Rn is a curve, then Dα′ (t)W = (W ◦ α)′ (t)α(t) . Proof. By Lemma 9.28 we have Dα′ (t)W = n X α′ (t)[wi ]Ui . i=1 On the other hand, Lemma 9.39 implies that α′ (t)[wi ] = (wi ◦ α)′ (t). Thus, Dα′ (t)W = n X α′ (t)[wi ]Ui = n X (wi ◦ α)′ (t)Ui i=1 i=1 = (W ◦ α)′ (t)α(t) . 9.7 Exercises 1. Let v = (v1 , v2 , v3 ) and p = (2, −1, 4). Define f : R3 → R by f (x, y, z) = xy 2 z 4 . Compute vp [f ]. 2. Fill in the details of the proofs of Lemma 9.5, Lemma 9.8, Corollary 9.11 and Corollary 9.14. 3. Define F : R3 → R3 by F (x, y, z) = (xy, yz, zx). Determine the following sets: A = B = p ∈ R3 kpk = 1, F (p) = 0 , p ∈ R3 kpk = 1, F∗ (vp ) = 0 for some vp ∈ R3p . 4. Compute by hand the gradients of the following functions: (a) (x, y, z) 7→ eaz cos ax − cos ay. (b) (x, y, z) 7→ sin z − sinh x sinh y. 282 CHAPTER 9. CALCULUS ON EUCLIDEAN SPACE y2 x2 − − c z. a2 b2 (d) (x, y, z) 7→ xm + y n + z p . (c) (x, y, z) 7→ M 5. Compute the gradient of (a) g1 (x, y, z) = x2 + 2y 2 + 3z 2 − xz + yz − xy. (b) g2 (x, y, z) = x3 + y 3 + z 3 − 3axyz. (c) g3 (x, y, z) = xyz ex+y+z . (d) g4 (x, y, z) = arctan x + y + z − xyz . 1 − xy − yz − xz 6. Compute by hand the divergence of each of the following vector fields: (a) (x, y, z) 7→ ayz cos xy, b yz sin xy, cxy . (b) (x, y, z) 7→ sin(xy), cos(yz), −3 cos2(yz) sin(xy) + sin3(xy) . (c) (x, y, z) 7→ (xyz)a , (xyz)b , (xyz)c . (d) (x, y, z) 7→ log(x + y − z), ex−y+z , (−x + y + z)5 . M 7. Compute the divergence of (a) V1 (x, y, z) = (xyz, 2x + 3y + z, x2 + z 2 ). (b) V2 (x, y, z) = (6x2 y 2 − z 3 + yz − 5, 4x3 y + xz + 2, xy − 3xz 2 − 3). 8. Compute the Laplacian of the functions in Exercises 4 and 5. 9. Let f : Rn → R be a differentiable function. A point p ∈ Rn is said to be a critical point of f if f∗ : Rnp → RF (p) maps some nonzero tangent vector to zero. If p is a critical point of f , define the Hessian of f at p by Hessian[f ] vp , wp = vp W[f ] , for vp , wp ∈ Rnp . Here W is any vector field on Rn such that W(p) = wp . n X ∂2f . vi wj (a) Show that Hessian[f ] vp , wp = ∂ui ∂uj i,j=1 (b) Conclude that the definition of the Hessian is independent of the choice of the vector field W. 10. Fill in the details of the proofs of Lemmas 9.30, 9.31 and 9.34. Chapter 10 Surfaces in Euclidean Space The 2-dimensional analog of a curve is a surface. However, surfaces in general are much more complicated than curves. In this introductory chapter, we give basic definitions that will be used throughout the rest of the book. The intuitive idea of a surface is a 2-dimensional set of points. Globally, the surface may be rather complicated and not look at all like a plane, but any sufficiently small piece of the surface should look like a ‘warped’ portion of a plane. The most straightforward 2-dimensional generalization of a curve in Rn is a patch or local surface, which we define in Section 10.1. A given partial derivative of the patch function constitutes a column of the associated Jacobian matrix, which can then be used to characterize a regular patch. Patches in R3 are discussed in Section 10.2, and the definition of an associated unit normal vector leads to a first encounter with the Gauss map of a surface, which is an analog of the mapping used in Chapters 1 and 6 to define the turning angle and turning number of a curve. A second generalization of a curve in Rn is a regular surface in Rn ; this is a notion defined in Section 10.3 that adopts a more global perspective. A selection of surfaces in R3 is presented in Sections 10.4 and 10.6. As well as discussing familiar cases, we shall plot a few interesting but less well-known examples. In later chapters, we shall be showing how to compute associated geometric quantities, such as curvature, for all the surfaces that we have introduced. In Section 10.5, we define a tangent vector to a regular surface, and a surface mapping; these are the surface analogs of a tangent vector to Rn and a mapping from Rn to Rn . As well as providing examples, Section 10.6 is devoted to the nonparametric representation of level surfaces as the zero sets of differentiable functions on R3 . The Gauss map of such a surface is determined by the gradient of the defining function. 287 288 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE n 10.1 Patches in R Since a curve in Rn is a vector-valued function of one variable, it is reasonable to consider vector-valued functions of two variables. Such an object is called a patch. First, we give the precise definition. Definition 10.1. A patch or local surface is a differentiable mapping x: U −→ Rn , where U is an open subset of R2 . More generally, if A is any subset of R2 , we say that a map x: A → Rn is a patch provided that x can be extended to a differentiable mapping from U into Rn , where U is an open set containing A. We call x(U) (or more generally x(A)) the trace of x. Although the domain of definition of a patch can in theory be any set, more often than not it is an open or closed rectangle. The standard parametrization of the sphere S 2 (a) of radius a is the mapping (10.1) π π sphere[a]: [0, 2π] × [− , ] −→ R3 2 2 (u, v) 7→ (a cos v cos u, a cos v sin u, a sin v). This is chosen so that u measures longitude and v measures latitude. Although sphere[a] is defined on the closed rectangle R = [0, 2π] × [−π/2, π/2], it has a differentiable extension to an open set containing R. Any such extension will multiply cover substantial parts of S 2 (a), while the image of R multiply covers only a semicircle from the north pole (0, 0, a) to the south pole (0, 0, −a). By restricting the parametrization to an open subset of R, it is possible to visualize the inside of the sphere. Figure 10.1: The image of sphere[1] for 0 6 u 6 3π 2 10.1. PATCHES IN R N 289 Since a patch can be written as an n-tuple of functions (10.2) x(u, v) = x1 (u, v), . . . , xn (u, v) , we can define the partial derivative xu of x with respect to u by ∂xn ∂x1 (10.3) (u, v), . . . , (u, v) . xu (u, v) = ∂u ∂u The other partial derivatives of x, namely xv , xuu , xuv , . . ., are defined similarly. Frequently, we abbreviate (10.2) and (10.3) to ∂xn ∂x1 . ,..., x = (x1 , . . . , xn ) and xu = ∂u ∂u The partial derivatives xu and xv can be expressed in terms of the tangent map of the patch x. Lemma 10.2. Let x: U → Rn be a patch, and let q ∈ U. Then x∗ e1 (q) = xu (q) and x∗ e2 (q) = xv (q), where x∗ denotes the tangent map of x, and {e1 , e2 } denotes the natural frame field of R2 . Proof. By Lemma 9.10, we have x∗ e1 (q) = e1 (q)[x1 ], . . . , e1 (q)[xn ] x(q) = ∂x1 ∂xn (q), . . . , (q) ∂u ∂u x(q) = xu (q), and similarly for x∗ e2 (q) . On page 271, we defined the Jacobian matrix of a differentiable map. We now specialize to the case of a patch. Definition 10.3. The Jacobian matrix of a patch x: U → Rn is the matrix-valued function J(x) given by  (10.4)      J(x)(u, v) =      ∂x1 (u, v) ∂u .. . ∂x1 (u, v) ∂v .. . .. . .. . ∂xn (u, v) ∂u ∂xn (u, v) ∂v       =     xu (u, v) xv (u, v) !T . 290 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Observe that differentiation with respect to each of the two coordinates u, v corresponds to a column of (10.4), or a row of its transpose indicated with the superscript T . This is slightly inconvenient typographically, but consistent with the definition on page 271. An equivalent definition of the rank of a matrix A is that it is the largest integer m such that a has an m × m submatrix whose determinant is nonzero. The following lemma is a consequence of well-known facts from linear algebra. Lemma 10.4. Let p, q ∈ Rn . The following conditions are equivalent: (i) p and q are linearly dependent; ! p·p p·q (ii) det = 0; q·p q·q (iii) the n × 2 matrix p, q has rank less than 2. Proof. If p and q are linearly dependent, then either p is a multiple of q, or vice versa. For example, if p = λq, then ! ! ! p·p p·q λq · λq λq · q q·q q·q 2 det = det = λ det = 0. q·p q·q λq · q q·q q·q q·q Thus (i) implies (ii). Next, suppose that (ii) holds. Write p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ). Then 2 X X X n n n pi qi qj2 − p2i 0 = kpk2 kqk2 − (p · q)2 = i=1 = X i=1 j=1 2 (pi qj − pj qi ) . 16i<j6n It follows that pi qj = pj qi for all i and j. Hence (ii) implies (iii). Finally, suppose (iii) holds. Then pi qj = pj qi for all i and j. Without loss of generality, we can suppose that qi 6= 0 for some i. Then pj = (pi /qi )qj for all j, and pi pi p = (p1 , . . . , pn ) = (q1 , . . . , qn ) = q, qi qi and p and q are linearly dependent. Corollary 10.5. Let x: U → Rn be a patch. Then the following conditions are equivalent: (i) xu (u0 , v0 ) and xv (u0 , v0 ) are linearly dependent; 10.1. PATCHES IN R (ii) det xu · xu xv · xu N 291 xu · xv xv · xv ! vanishes at (u0 , v0 ); (iii) the Jacobian matrix J(x) has rank less than 2 at (u0 , v0 ). We shall need a stronger notion of patch in order that x(U) resemble the open set U more closely. Definition 10.6. A regular patch is a patch x: U → Rn for which J(x)(u, v) has rank 2 for all (u, v) ∈ U. An injective patch is a patch such that x(u1 , v1 ) = x(u2 , v2 ) implies that u1 = u2 and v1 = v2 . Figure 10.2: (u, v) 7→ (cos u, sin u, v) There are regular patches which are not injective. Consider, for example, the circular cylinder of Figure 10.2 defined by x(u, v) = (cos u, sin u, v), where u ∈ R and −2 < v < 2. There are also injective patches which are not regular. This is illustrated by the function y(u, v) = (u3 , v 3 , uv) of Figure 10.3, with −1 < u, v < 1. Figure 10.3: (u, v) 7→ (u3 , v 3 , uv) 292 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE The patch sphere[a] fails to be regular when v = ±π/2, that is, at the north and south poles. This is typical: often it is necessary to deal with patches that fail to be regular at just a few points. To handle such cases, we record the following obvious variant of Definition 10.6. Definition 10.7. A patch x: U → Rn is regular at a point (u0 , v0 ) ∈ U (or sometimes we say at x(u0 , v0 )) provided J(x)(u0 , v0 ) has rank 2. As a consequence of Lemma 10.2 and Corollary 10.5 we have Lemma 10.8. A patch x: U → Rn is regular at q ∈ U if and only if its tangent map x∗ : R2q → Rnx(q) is injective. The following lemma is a consequence of the inverse function theorem (as given, for example, in [Buck, page 276]), and refers to Definition 9.7 on page 268. Lemma 10.9. Let x: U → Rn be a regular patch and let q ∈ U. There exists a neighborhood Uq of q such that x: Uq → x(Uq ) is the restriction of a diffeomorphism between open sets of Rn . Proof. Write x = (x1 , . . . , xn ). Since x is regular, its Jacobian matrix has a 2 × 2 submatrix with nonzero determinant. By renaming x1 , . . . , xn if necessary, we can suppose that   ∂x1 ∂x1 (u, v) (u, v)   ∂u ∂v   (10.5) det ∂x  ∂x2 2  (u, v) (u, v)  ∂u ∂v is nonzero for (u, v) ∈ U. We extend x: U → Rn to a map (10.6) by defining e : U × Rn−2 −→ Rn x x(u, v, t3 , . . . , tn ) 7→ x1 (u, v), x2 (u, v), x3 (u, v)+t3 , . . . , xn (u, v)+tn . e is differentiable; moreover, the determinant It is clear that x  ∂x1 ∂x2 ∂x3 (u, v) (u, v) (u, v) · · ·  ∂u ∂u ∂u   ∂x ∂x2 ∂x3 1  (u, v) (u, v) (u, v) · · ·  ∂v ∂v  ∂v  T det J(e x) = det 0 0 1 ···    .. .. .. ..  . . . .   0 0 0 ··· ∂xn (u, v) ∂u ∂xn (u, v) ∂v 0 .. . 1                10.1. PATCHES IN R N 293 equals (10.5). Because det J(e x) (q) 6= 0, the inverse function theorem says e has a differentiable that there is a neighborhood Ueq of (q, 0) upon which x e : Ueq → x e(Ueq ) is a diffeomorphism, and we may take inverse. It follows that x Uq = Ueq ∩ U to complete the proof. The conclusion of Lemma 10.9 can be expressed by saying that x: Uq → x(Uq ) is itself a diffeomorphism, anticipating Definition 10.42 on page 309. We may also state Corollary 10.10. Let x: U → Rn be an injective regular patch. Then x maps U diffeomorphically onto x(U). Associated with any patch are some naturally-defined curves. Definition 10.11. Let x: U → Rn be a patch, and fix (u0 , v0 ) ∈ U. The curves u 7→ x(u, v0 ) and v 7→ x(u0 , v) are called u- and v- parameter curves or coordinate curves. These are the curves that are usually displayed by computer graphics. There is also a convenient way to represent a general curve whose trace is contained in the trace of a patch. Lemma 10.12. Let α : (a, b) → Rn be a curve whose trace lies on the image x(U) of a regular patch x: U → Rn such that x: U → x(U) is a homeomorphism. Then there exist unique differentiable functions u, v : (a, b) → R such that (10.7) α(t) = x u(t), v(t) for a < t < b. Proof. For a < t < b we can write (x−1 ◦ α)(t) = u(t), v(t) ; this equation is equivalent to (10.7). It is clear that u and v are unique; by Lemma 10.9 they are also differentiable. Now we can define the notion of tangent vector to a patch. Definition 10.13. Let x: U → Rn be an injective patch, and let p ∈ x(U). A tangent vector to x at p is a tangent vector vp ∈ Rnp for which there exists a curve α: (a, b) → Rn that can be written as (10.8) α(t) = x u(t), v(t) (a < t < b), such that α(0) = p and α′ (0) = vp . We denote the set of tangent vectors to x at p by x(U)p . 294 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Lemma 10.14. The set x(U)p of all tangent vectors to a patch x: U → Rn at a regular point p = x(u0 , v0 ) ∈ x(U) forms a vector space that is spanned by xu (u0 , v0 ) and xv (u0 , v0 ). Proof. By definition of tangent vector vp to x, there exists a curve α of the form (10.8) such that α(0) = p and α′ (0) = vp . The chain rule for curves (Lemma 1.9, page 7) implies that α′ (t) = u′ (t)xu (u(t), v(t)) + v ′ (t)xv (u(t), v(t)). In particular, vp = α′ (0) = u′ (0)xu (u0 , v0 ) + v ′ (0)xv (u0 , v0 ). Conversely, if vp = c1 xu (u0 , v0 ) + c2 xv (u0 , v0 ), then vp is the velocity vector at x(u0 , v0 ) = p of the curve t 7→ x(u0 + tc1 , v0 + tc2 ). The proof of Lemma 10.14 yields Corollary 10.15. Let α : (a, b) → Rn be a curve whose trace lies on the image x(U) of a regular patch x: U → Rn such that x: U → x(U) is a homeomorphism. Then there exist unique differentiable functions u, v : (a, b) → R such that (10.9) α′ = u′ xu + v ′ xv . Definition 10.16. Let x: U → Rn be an injective patch, and let zp ∈ Rnp with p ∈ x(U). We say that zp is normal or perpendicular to x at p, provided zp · vp = 0 for all vectors vp tangent to x at p. Let x(U)⊥ p denote the orthogonal complement of x(U)p , namely the space of vectors in Rnp perpendicular to x(U)p . It is easy to verify that there is a direct sum (10.10) Rnp = x(U)p ⊕ x(U)⊥ p of vector spaces. We shall also need vector fields on patches: Definition 10.17. A vector ﬁeld V on a patch x: U → Rn is a function that assigns to each q ∈ U a tangent vector V(q) ∈ Rnp , where p = x(q). We say that V is tangent to x if V(q) ∈ x(U)p for all q ∈ U. Similarly, a vector field W on x is normal or perpendicular to x if W(q) · vp = 0 for all vp ∈ x(U)p and q ∈ U. 10.2. PATCHES IN R N 295 3 10.2 Patches in R and the Local Gauss Map We now restrict our attention to patches in R3 because they are the ones that are easiest to visualize. Computations are also simpler because there is a vector cross product on R3 . Here is a useful criterion for the regularity of a patch: Lemma 10.18. A patch x: U → R3 is regular at (u0 , v0 ) ∈ U if and only if xu × xv is nonzero at (u0 , v0 ). Proof. This follows immediately from the equation   ∂x1 ∂x1 i   ∂u ∂v     ∂x2 ∂x2  xu × xv = det j  , ∂v   ∂u   ∂x ∂x3 3 k ∂u ∂v confirming that (xu × xv )(u0 , v0 ) = 0 if and only if the rank of J(x)(u0 , v0 ) is less than 2. The vector cross product gives rise to a convenient way of finding a vector satisfying Definition 10.16 when n = 3. Lemma 10.19. Let x: U → R3 be an injective regular patch. Then the vector field (u, v) 7→ xu × xv is everywhere perpendicular to x(U). Proof. Certainly, xu × xv is perpendicular to both xu and xv . Since any tangent vector vp to x is a linear combination of xu and xv at p, it follows from Lemma 10.14 that (u, v) 7→ xu × xv is perpendicular to x(U). Now we can define a perpendicular vector field of unit length. Definition 10.20. For an injective patch x: U → R3 the unit normal vector ﬁeld or surface normal U is defined by (10.11) U(u, v) = xu × xv (u, v) kxu × xv k at those points (u, v) ∈ U at which xu × xv does not vanish. The notion of regularity of a patch has the following geometric interpretation. Corollary 10.21. Let x: U → R3 be an injective patch. Then x is regular if and only if the unit normal vector field U is everywhere well defined. The points for which xu × xv vanishes will be called singular in the next section. 296 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE One of the key elements necessary for the study of surfaces is the map that assigns to each point p on a surface M ⊂ R3 the point on the unit sphere S 2 (1) ⊂ R3 that is parallel to the unit normal U(p). It is called the Gauss1 map. It is an analog of the mapping illustrated in Figure 6.2 on page 157, that associates to a point on a curve its unit tangent vector. It is not always possible to define this map at all points of a surface, for reasons that we take up in Chapter 11, but it is defined for any patch. Definition 10.22. Let x: U → R3 be an injective patch. Then the unit normal U to x viewed as a mapping from U to the unit sphere S 2 (1) ⊂ R3 is called the local Gauss map of x. Note that U is undefined at singular points of x. We shall illustrate this concept for a well-known surface, that will be discussed again later in this chapter. The hyperbolic paraboloid is the trace of the patch (10.12) x(u, v) = (u, v, u v). Figure 10.4 shows the image of the rectangle [−1, 1] × [−1, 1], superimposed with the corresponding portion of the unit sphere S 2 (1) determined by the unit normal U(u, v). The patch (10.12) is closely related to the more general one (10.16) given below, and the hyperbolic paraboloid is an example of a ruled surfaces (see page 434 in Chapter 16). 1 Carl Friedrich Gauss (1777–1855). Gauss’s Disquisitiones Generales Circa Superficies Curvas (published in 1828) revolutionized differential geometry as his Disquisitiones Arithmeticae had revolutionized number theory. In particular, his approach depended only on the intrinsic properties of surfaces, not on their embedding in Euclidean 3-space. Gauss’ genius extended into nearly every branch of mathematics. While a student at Göttingen he made his first major original discovery – the constructibility of the 17-sided regular polygon, closely followed by the first of his four proofs of the fundamental theorem of algebra (his doctoral dissertation) and his calculation of the orbit of the newly discovered asteroid Ceres. Gauss’s interest in geodesy led to the invention of the heliotrope and the development of least squares approximation, a technique he also used in his investigation of the distribution of prime numbers. Gauss’ work on terrestrial magnetism led to his construction, in collaboration with Wilhelm Weber, of the first operating electric telegraph. Other applied discoveries of significance were those in optics, potential theory, and astronomy. Gauss spent his entire career in Brunswick, under the patronage of the local royalty; for many years he directed the observatory there. Perhaps due to his deeply ingrained conservatism or to his distaste for controversy, Gauss did not publish his development of the theory of non-Euclidean geometry nor his work anticipating Hamilton’s investigation of quaternions. Although he worked for the most part in mathematical isolation, his results laid the basis for new departures in number theory and statistics, as well as in differential geometry. 10.3. DEFINITION OF REGULAR SURFACE 297 Figure 10.4: Gauss image of the hyperbolic paraboloid 10.3 The Definition of a Regular Surface There are some subsets of Rn that we would like to call surfaces, but they do not fit into the framework of Section 10.1. Such a subset cannot be expressed as the image of a single regular one-to-one patch defined on an open set; spheres and tori are examples. In this section we define the notion of regular surface; roughly speaking, what we shall do is to combine several patches. The resulting definition is more complicated, but it is what is needed in many situations. Definition 10.23. A subset M ⊂ Rn is a regular surface, provided that for each point p ∈ M there exist a neighborhood V of p in Rn and a map x: U → Rn of an open set U ⊂ R2 onto V ∩ M such that: (i) x: U → M is a regular patch; (ii) x: U → V ∩ M is a homeomorphism; thus x has a continuous inverse x−1 : V ∩ M → U such that x−1 is the restriction to V ∩ M of a continuous mapping F : W → R2 , where W is an open subset of Rn that contains V ∩ M. Each map x: U → M is called a local chart or system of local coordinates in a neighborhood of p ∈ M. 298 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE We shall need the following fact. Lemma 10.24. Let W be an open subset of Rn , and suppose that G: W → Rm is a map such that G: W → G(W) is a diffeomorphism. If M ⊆ W is a regular surface, then G(M) is also a regular surface. Proof. If x is a patch on M satisfying (i) and (ii) in the definition of regular surface, then G ◦ x: W → Rm also satisfies (i) and (ii). There is an easy way to find new regular surfaces inside a given regular surface M. A subset V of a regular surface M ⊂ Rn is said to be open provided it is the intersection of an open subset of Rn with M. Lemma 10.25. An open subset W of a regular surface M is also a regular surface. Proof. Let x be a patch on M satisfying (i) and (ii) in the definition of regular surface. Then x is continuous, and the inverse image of an open set is open, so U = x−1 (W) is open. If U is nonempty, the restriction x|U becomes a patch on W. It is clear that x|U satisfies (i) and (ii) in the definition of regular surface. Thus W becomes a regular surface. Figure 10.5: The patch (u, v) 7→ (sin u, sin 2u, v) with − π 5π 6u6 3 4 Let us clarify the relation between patches and regular surfaces. Certainly an arbitrary patch will fail to be a regular surface if its trace is 0- or 1-dimensional. Figure 10.5 shows that even a regular patch can fail to be a regular surface if it has self-intersections (which are precluded by condition (ii) in Definition 10.23). Nevertheless, let us prove that when we restrict the domain of definition of a regular patch we obtain a regular surface. 10.3. DEFINITION OF REGULAR SURFACE 299 Lemma 10.26. Let x: U → Rn be a regular patch. For any q ∈ U there exists a neighborhood Uq of q such that x(Uq ) is a regular surface. Proof. Lemma 10.9 states that q has a neighborhood Uq such that x is a diffeomorphism between Uq and x(Uq ). The neighborhood Uq is an open set in the plane, and hence a regular surface by Lemma 10.25. Then Lemma 10.24 implies that x(Uq ) is a regular surface. Corollary 10.27. Let x: U → Rn be an regular injective patch. Then x(Uq ) is a regular surface. Definition 10.28. Let M be a regular surface in Rn . If there exists a single regular injective patch x: U → Rn such that x(U) = M, we say that M is parametrized by x. This definition applies to the case of a Monge patch defined in Section 10.4 below. On the other hand, a sphere is an example of a regular surface which needs at least two patches to cover it. Figure 10.6 shows that the patches (u, v) 7→ cos u cos v, sin u cos v, sin v (u, v) 7→ cos 35 π−u cos v, sin v, sin 53 π−u cos v , both defined for 0 < u < 7π/4 and −π/2 < v < π/2, cover the sphere. For plotting purposes, the second patch is best regarded as the composition of the first with a rotation. Figure 10.6: Two local charts covering the sphere Next, we establish a fact about differentiable maps into a regular surface. Theorem 10.29. Let M ⊂ Rn be a regular surface, and let V ⊂ Rm be an open subset. Suppose that F : V → Rn is a differentiable mapping such that F (V) ⊆ M, and that x: U → M is a regular patch on M such that F (V) ⊆ x(U). Then x−1 ◦ F : V → U is differentiable. 300 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Proof. It is tempting to say that since x−1 and F are differentiable, so is x−1 ◦ F . Unfortunately, this is not valid, because we do not know at this point what it means for a function defined on a regular surface to be differentiable. Therefore, we are forced to prove the theorem in a more roundabout fashion, modifying the proof of Lemma 10.9. Write x = (x1 , . . . , xn ). Since x is regular, its Jacobian matrix has a 2 × 2 submatrix with nonzero determinant. By renaming x1 , . . . , xn if necessary, we can suppose that (10.5) holds, and we define (10.6) in exactly the same way. e is differentiable, and Once again, x det J(e x)(u, v) 6= 0. Let p ∈ V; then F (p) ∈ x(U) ⊆ M. Because det J(e x) 6= 0 on U, the e has an inverse x e−1 on a neighborhood inverse function theorem says that x e Since F is continuous, e−1 is differentiable on U. Ue ⊆ Rn of F (p), and that x e e e Moreover, there exists a neighborhood V of p such that F (V) ⊂ U. e = x−1 ◦ F | V. e e−1 ◦ F | V x e is differentiable at p, so is x−1 ◦ F | V. e Since p ∈ V is arbitrary, e−1 ◦ F | V Since x −1 it follows that x ◦ F is differentiable on all of V. An important special case of Theorem 10.29 is: Corollary 10.30. Let M be a regular surface, and suppose we are given regular patches x: U → M and y : V → M such that x(U) ∩ y(V) = W is nonempty. Then the change of coordinates (10.13) x−1 ◦ y : y−1 (W) −→ x−1 (W) is a diffeomorphism between open subsets of R2 . Proof. That x−1 ◦ y and y−1 ◦ x are differentiable is a consequence of Theorem 10.29. Since both x and y are homeomorphisms, so are x−1 ◦ y and y−1 ◦ x. Moreover, these maps are inverses of each other. Hence x−1 ◦ y and y−1 ◦ x are diffeomorphisms. We can write (10.13) more explicitly as follows: there exist differentiable functions ū, v̄ such that y(u, v) = x ū(u, v), v̄(u, v) . Note that (ū, v̄) = (x−1 ◦ y)(u, v). Lemma 10.31. Let x: U → M and y : V → M be patches on a regular surface M with x(U) ∩ y(V) nonempty. Let x−1 ◦ y = (ū, v̄): U ∩ V → U ∩ V be the associated change of coordinates, so that (10.14) y(u, v) = x ū(u, v), v̄(u, v) . 10.3. DEFINITION OF REGULAR SURFACE Then (10.15) yu = ∂v̄ ∂ ū xū + xv̄ ∂u ∂u and 301 yv = ∂ ū ∂v̄ xū + xv̄ . ∂v ∂v Proof. (10.15) is an immediate consequence of (10.14) and the chain rule. Usually there are only a few points on a general surface which cannot be in the image of a regular patch. We refer to these as nonregular points or singular points of the surface. All other points of the surface are called regular points. Note that by Corollary 10.30, the definition of regular point and singular point does not depend on the choice of patch. Furthermore, if we remove all of the nonregular points from a general surface, we obtain a surface. We need to extend the calculus that we developed in Chapter 9 for Rn to regular surfaces. Our first task is to define what it means for a real-valued function on a regular surface to be differentiable. Definition 10.32. Let f : W → R be a function defined on an open subset W of a regular surface M. We say that f is differentiable at p ∈ W provided that for some patch x: U ⊂ R2 → M with p ∈ x(U) ⊂ W, the composition f ◦ x: U ⊂ R2 −→ R is differentiable at x−1 (p). If f is differentiable at all points of W, we say that f is differentiable on W. It is important to realize that the definition of differentiability of a realvalued function on a regular surface does not depend on the choice of patch. If x and y are patches on a regular surface M, then y−1 ◦ x is differentiable by Corollary 10.30, and the composition of differentiable functions is differentiable. Lemma 10.33. Let M be a regular surface in Rn . Then the restriction f |M of any differentiable function f : Rn → R to M is differentiable. Proof. If x is any patch on M, then (f |M) ◦ x = f ◦ x is differentiable, since it is the composition of differentiable functions. Thus by definition f |M is differentiable. As a simple consequence of Lemma 10.33, we see that the restrictions to any regular surface M ⊂ Rn of the natural coordinate functions u1 , . . . , un of Rn are differentiable. Here are two important examples of differentiable functions. Definition 10.34. Let M be a regular surface and let v ∈ Rn . The height function and the square of the distance function of M relative to v are the functions h, f : M → R defined by h(p) = p · v and f (p) = kp − vk2 . 302 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Both the height and the square of the distance function are algebraic combinations of the natural coordinate functions of Rn . Their differentiability therefore follows from Lemma 10.33. We shall also need the notion of a curve on a regular surface. Definition 10.35. A curve on a regular surface M ⊂ Rn is simply a curve α : (a, b) → Rn such that α(t) ∈ M for a < t < b. The curve α is said to be differentiable provided that x−1 ◦ α : (a, b) → U is differentiable for any patch x: U → M, where U ⊆ R2 . 10.4 Examples of Surfaces An important class of surfaces in R3 consists of those that are graphs of a real-valued function of two variables. Definition 10.36. A Monge2 patch is a patch x: U → R3 of the form x(u, v) = u, v, h(u, v) , where U is an open set in R2 and h: U → R is a differentiable function. This is a regular patch, because the Jacobian matrix  1    J(x) =  0   ∂h ∂u 0    1    ∂h  ∂v obviously has rank 2. Its trace is the graph of h, whose nonparametric form is simply z = h(x, y). 2 Gaspard Monge (1746–1818). French mathematician, erstwhile secretary of the navy, founding director of the École Polytechnique, which played a leading role in the development and organization of scientific research and education in France. Monge’s work on fortifications led him to descriptive geometry, and from there he went on to a broad exposition of the differential geometry of space curves. His major contribution to the development of differential geometry was the integration of geometrical facts and intuition with the use of partial differential equations. A fervent supporter first of the Revolution and then of Napoleon, Monge was expelled from the Institut de France following the Battle of Waterloo. Many consider Monge to be the father of French differential geometry. 10.4. EXAMPLES OF SURFACES 303 Paraboloids and Monkey Saddles The concept of Monge patch is well illustrated by taking h(u, v) proportional to au2 + bv 2 . The trace of the parametrization (10.16) paraboloid[a, b](u, v) = u, v, au2 + b v 2 , or equivalently the graph of z = ax2 + by 2 , is a paraboloid provided both a, b are nonzero. The paraboloid is circular if a = b 6= 0. Figure 10.7: Subsets of the circular paraboloid paraboloid[a, a] A clearer representation of a circular paraboloid is obtained by using polar coordinates (r, θ) instead of the rectangular coordinates (u, v). The relation between the two systems is of course u = r cos θ, v = r sin θ, and the polar parametric representation is merely (r, θ) 7→ (r cos θ, r sin θ, ar2 ). Plotting this with a bound on the radius r provides the more symmetrical effect visible on the right of Figure 10.7. Because of its shape, the hyperbolic paraboloid defined by taking a, b to have different signs in (10.16), is a type of a saddle surface. A person can sit comfortably on a hyperbolic paraboloid because there are indentations for two legs. The monkey saddle, which also takes account of a tail, is represented by the patch (10.17) monkeysaddle(u, v) = (u, v, u3 − 3uv 2 ). 304 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Figure 10.8: The monkey saddle The plane, hyperbolic paraboloid ((10.16) with a = 1, b = −1) and monkey saddle are the first three members of a series of surfaces, namely the graphs of the real part of the function z 7→ z n , where z is a complex number and n a positive integer. Explicitly, we set monkey[n](u, v) = u, v, Re [(u + iv)n ] , so that monkey[2] = paraboloid[1, −1]. The imaginary part of z n gives rise to a similar surface, namely the image of monkey[n] under a rotation by π/(2n) radians. The surface defined when n = 5 is illustrated on the left in Figure 10.9. On the right is the corresponding contour plot, in which the lighter the shading, the greater the value of the function monkey[5]. Figure 10.9: The monkey[5] surface 10.4. EXAMPLES OF SURFACES 305 Elliptical Tori The patch (10.18) torus[a, b, c]: [0, 2π) × [0, 2π) −→ R3 (u, v) 7→ (a + b cos v) cos u, (a + b cos v) sin u, c sin v is a parametrization of an elliptical torus in R3 . Examples were encountered in Section 7.6. When b = c < a, we get a standard round torus, where a is the wheel radius and b the tube radius of the torus. A torus with a > b and a > c is called a ring torus, and is a regular surface in the sense of Section 10.3. The domain in (10.18) can be extended to any subset of R2 , and including the rectangles 3π (0, 3π 2 ) × (0, 2 ) (0, 3π 2 ) × (π, 5π 2 ) 3π (π, 5π 2 ) × (0, 2 ) (π, 5π 2 ) × (π, 5π 2 ). Their images are displayed in the same order in Figure 10.10 for an elliptical torus with a = 8, b = 3 and c = 7. Figure 10.10: Four local charts covering torus[8, 3, 7] If the wheel radius is less than the tube radius, the resulting torus becomes a self-intersecting surface and appears ‘inside-out’. It is usually called a horn torus, though the intermediate case when the wheel radius equals the tube radius is called a spindle torus. Figure 10.11 shows both a horn and spindle torus. The number of self-intersections is 0 for a ring torus, 1 for a spindle torus and 2 for a horn torus. These facts are checked analytically in Notebook 10, using the determinant of the matrix in Exercise 1. See also [Fischer, page 28]. 306 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Figure 10.11: torus[3, 8, 8] and torus[4, 4, 4] Patches with Singularities From the discussion in the previous section, we know that a surface fails to be regular at points of self-intersection, though the patch may or may not be singular on a 1-dimensional array of such points. Figure 10.11 shows isolated points of self-intersection where the patch is indeed singular, and we now give two more examples of this phenomenon. Figure 10.12: The eight surface The eight surface consists of a figure eight revolved about the z-axis, and is defined as the trace of the patch (10.19) eightsurface(u, v) = cos u cos v sin v, sin u cos v sin v, sin v , The eight surface becomes a regular surface only when the central ‘vertex’ is excluded. In spite of the existence of a singular point, there is no difficulty in plotting the surface. 10.5. TANGENT VECTORS AND SURFACE MAPPINGS 307 An example of a surface with a pinch point is the Whitney umbrella, defined by whitneyumbrella(u, v) = (u v, u, v 2 ). The Jacobian matrix (10.4) of this patch has rank 2 unless (u, v) = (0, 0), though whitneyumbrella maps (0, ±v) to the same point for all v 6= 0. Figure 10.13: Two views of the Whitney umbrella For a discussion of this surface see [Francis, pages 8–9]. 10.5 Tangent Vectors and Surface Mappings We are now ready to discuss the important notion of tangent vector to a regular surface. This is the next step in the extension to regular surfaces of the calculus of Rn that we developed in Chapter 9. Definition 10.37. Let M be a regular surface in Rn and let p ∈ M. We say that vp ∈ Rnp is tangent to M at p provided there exists a curve α: (a, b) → Rn such that α(0) = p, α′ (0) = vp and α(t) ∈ M for a < t < b. The tangent space to M at p is the set Mp = vp ∈ Rnp | vp is tangent to M at p . On page 289, we defined xu (u, v) and xv (u, v) to be vectors in Rn . Sometimes it is useful to modify this definition so that xu (u, v) and xv (u, v) are tangent vectors at x(u, v). Thus if x(u, v) = x1 (u, v), . . . , xn (u, v) we can redefine ∂xn ∂x1 (u, v), . . . , (u, v) , xu (u, v) = ∂u ∂u x(u,v) ∂x1 ∂xn xv (u, v) = (u, v), . . . , (u, v) . ∂v ∂v x(u,v) 308 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Figure 10.14: Tangent plane at the saddle point of a hyperbolic paraboloid Similar remarks hold for xuu and other higher partial derivatives of x. However, such fine distinctions are unnecessary in practice, since when we calculate the partial derivatives of a patch explicitly (either by hand or computer) we consider them to be vectors in Rn . In summary, we regard xu and the other partial derivatives of x to be vectors in Rn unless for some theoretical reason we require them to be tangent vectors at the point of application. Lemma 10.38. Let p be a point on a regular surface M ⊂ Rn . Then the tangent space Mp to M at p is a 2-dimensional vector subspace of Rnp . If x: U → M is any regular patch on M with p = x(q), then x∗ (R2q ) = Mp . Proof. It follows from Lemma 10.2 that x∗ (R2q ) is spanned by xu (q) and xv (q). The regularity of x at q implies that x∗ : R2q → Rnp is injective; consequently, xu (q) and xv (q) are linearly independent and dim x∗ (R2q ) = 2 by Lemma 10.8. On the other hand, Lemma 10.14 implies that any tangent vector to M at q is a linear combination of xu (q) and xv (q). Frequently, we shall need the notion of tangent vector perpendicular to a surface, in complete analogy to Definition 10.16. 10.5. TANGENT VECTORS AND SURFACE MAPPINGS 309 Definition 10.39. Let M be a regular surface in Rn and let zp ∈ Rnp with p ∈ M. We say that zp is normal or perpendicular to M at p, provided zp · vp = 0 for all tangent vectors vp ∈ Mp . We denote the set of normal vectors to M at p by M⊥ p , so that Rnp = Mp ⊕ M⊥ p, just as in (10.10). The notions of tangent and normal vector fields also make sense. Definition 10.40. A vector ﬁeld V on a regular surface M is a function which assigns to each p ∈ M a tangent vector V(p) ∈ Rnp . We say that V is tangent to M if V(p) ∈ Mp for all p ∈ M and that V is normal or perpendicular to M if V(p) ∈ M⊥ p for all p ∈ M. The definition of differentiability of a mapping between regular surfaces is similar to that of a real-valued function on a regular surface. Definition 10.41. A function F : M → N from one regular surface to another is differentiable, provided that, for any two regular injective patches x of M and y of N , the composition y−1 ◦ F ◦ x is differentiable. When this is the case, we call F a surface mapping. The simplest example of a surface mapping is the identity map 1M : M → M, defined by 1M (p) = p for p ∈ M. Definition 10.42. A diffeomorphism between regular surfaces M and N is a differentiable map F : M → N which has a differentiable inverse, that is, a surface mapping G: N → M such that G ◦ F = 1M and F ◦ G = 1N , where 1M and 1N denote the identity maps of M and N . Definition 10.43. Let M, N be regular surfaces in Rn , and W an open subset of M. We shall say that a mapping F : W −→ N is a local diffeomorphism at p ∈ W, if there exists a neighborhood W ′ ⊂ W of p such that the restriction of F to W ′ is a diffeomorphism of W ′ onto an open subset F (W ′ ) ⊆ N . Just as a regular surface has a tangent space, a surface mapping has a tangent map. 310 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Definition 10.44. Let M, N be regular surfaces in Rn , and let p ∈ M. Let F : M → N be a surface mapping. Then the tangent map of F at p is the map F∗ : Mp −→ NF (p) given as follows. For vp ∈ Mp , choose a curve α : (a, b) → M such that α′ (0) = vp . Then we define F∗ (vp ) to be the initial velocity of the image curve F ◦ α : → N ; that is, F∗ (vp ) = (F ◦ α)′ (0). Lemma 10.45. Let F : M → N be a surface mapping, and let vp ∈ Mp . The definition of F∗ (vp ) is independent of the choice of curve α with α′ (0) = vp . Furthermore, F∗ : Mp → NF (p) is a linear map. e be curves in M with α(0) = α(0) e Proof. Let α and α = p and α′ (0) = ′ n e (0) = vp . Let x: U → R be a regular patch on M such that x(u0 , v0 ) = p. α Write e (t) = x u α(t) = x u(t), v(t) and α e(t), ve(t) . Just as in the proof of Lemma 10.14, the chain rule for curves implies that (F ◦ α)′ (t) = u′ (t) F ◦ x)u (u(t), v(t) + v ′ (t) F ◦ x)v (u(t), v(t) and e ′ (t) = u e(t), ve(t) . e(t), ve(t) + ve′ (t)(F ◦ x)v u (F ◦ α) e′ (t)(F ◦ x)u u Since u(0) = u e(0) = u0 , v(0) = ve(0) = v0 , u′ (0) = u e′ (0) and v ′ (0) = ve′ (0), it follows that e )′ (0). F∗ (vp ) = (F ◦ α)′ (0) = (F ◦ α This proves that F∗ (vp ) does not depend on the choice of α. To prove the linearity of F∗ , we note that F∗ (vp ) = u′ (0)(F ◦ x)u u(0), v(0) + v ′ (0)(F ◦ x)v u(0), v(0) . Since u′ (0), v ′ (0) depend linearly on vp , it follows that F∗ must be linear. We have the following consequence of the inverse function theorem for R2 . Theorem 10.46. If M, N are regular surfaces and F : W → N is a differentiable mapping of an open subset W ⊆ M such that the tangent map F∗ of F at p ∈ W is an isomorphism, then F is a local diffeomorphism at p. Proof. Let x: U → M and y : V → N be injective regular patches on M and N such that (F ◦ x)(U) ∩ y(V) is nonempty. By restricting the domains of definition of x and y if necessary, we can assume that (F ◦ x)(U) = y(V). Then the tangent map of y−1 ◦ F ◦ x is an isomorphism, and the inverse function theorem for R2 implies that the map y−1 ◦ F ◦ x possesses a local inverse. Hence F also has a local inverse. 10.6. LEVEL SURFACES IN R N 311 Examples of Surface Mappings 1. Let y : U → M be a regular patch on a regular surface M. Theorem 10.29 implies that y is a diffeomorphism between the regular surfaces U and y(U). 2. Let y : U → M be a regular patch on a regular surface M. Then the local Gauss map U ◦ y−1 is a surface mapping from y(U) to the unit sphere S 2 (1) (see Definition 10.22 and Exercise 5). 3. Let S 2 (a) = { p | kpk = a} be the sphere of radius a in R3 . Then the antipodal map is the function S 2 (a) → S 2 (a) defined by p 7→ −p. It is a diffeomorphism. 4. Let F : Rn → Rn be a diffeomorphism, and let M ⊂ Rn be a regular surface. Then the restriction F |M: M → F (M) is a diffeomorphism of regular surfaces. In particular, any Euclidean motion of Rn gives rise to a surface mapping. 10.6 Level Surfaces in R 3 So far we have been considering regular surfaces defined by patches. Such a description is called a parametric representation. Another way to describe a regular surface M is by means of a nonparametric representation. For a regular surface in R3 , this means that M is the set of points mapped by a differentiable function g : R3 → R into the same real number. Definition 10.47. Let g : R3 → R be a differentiable function and c a real number. Then the set M(c) = p ∈ R3 | g(p) = c is called the level surface of g corresponding to c. Theorem 10.48. Let g : R3 → R be a differentiable function and c a real number. Then the level surface M(c) of g is a regular surface if it is nonempty and the gradient grad g is nonzero at all points of M(c). When these conditions are satisfied, grad g is everywhere perpendicular to M(c). Proof. For each p ∈ M(c), we must find a regular patch on a neighborhood of p. The hypothesis that (grad g)(p) 6= 0 is equivalent to saying that at least one of the partial derivatives ∂g/∂x, ∂g/∂y, ∂g/∂z does not vanish at p. Let us suppose that ∂g/∂z)(p 6= 0. The implicit function theorem states that the equation g(x, y, z) = c can be solved for z. More precisely, there exists a function h such that g x, y, h(x, y) = c. 312 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Then the required patch is defined by x(u, v) = u, v, h(u, v) . Furthermore, let vp = (v1 , v2 , v3 )p ∈ M(c)p . Then there exists a curve α on M(c) with α(0) = p and α′ (0) = vp . Write α(t) = (a1 (t), a2 (t), a3 (t)). Since α lies on M(c), we have g(a1 (t), a2 (t), a3 (t)) = c for all t. The chain rule implies that 3 X dai ∂g ◦α = 0. ∂u dt i i=1 In particular, 0= 3 3 X X dai ∂g ∂g (p)vi = (grad g)(p) · vp . (0) = α(0) ∂ui dt ∂ui i=1 i=1 Hence (grad g)(p) is perpendicular to M(c)p for each p ∈ M(c). To conclude this final section, we return to some examples. Ellipsoids The nonparametric equation that defines the ellipsoid is (10.20) x2 y2 z2 + 2 + 2 = 1. 2 a b c Here, a, b and c are the lengths of the semi-axes of the ellipsoid. Ellipsoids with only two of a, b and c distinct are considerably simpler than general ellipsoids. Such an ellipsoid is called an ellipsoid of revolution, and can be obtained by rotating an ellipse about one of its axes. Figure 10.15: Regions of ellipsoid[1, 1, 2] and ellipsoid[1, 2, 3] 10.6. LEVEL SURFACES IN R N 313 We shall study surfaces of revolution in detail in Chapter 15. Figure 10.15 illustrates the difference between an ellipsoid of revolution and a general ellipsoid. A parametric form of (10.20) is ellipsoid[a, b, c](u, v) = a cos v cos u, b cos v sin u, c sin v . The patch is regular except at the north and south poles, which are (0, 0, ±c). We next define a different parametrization of an ellipsoid, the stereographic ellipsoid. It is a generalization of the stereographic projection of the sphere that can be found in books on complex variables. It is often useful to change parametrization in order to highlight a particular property of a surface, and we shall see several instances of this procedure in the sequel. The ellipsoid, for reasons of physics and cartography, is one of the surfaces most amenable to the procedure (see Exercise 6 and Section 19.6). Figure 10.16: stereographicellipsoid[5, 3, 1] For viewing convenience, the stereographic ellipse has been rotated so that the ‘north pole’ appears at the front of the picture. Its definition is: stereographicellipsoid[a, b, c](u, v) = 2a u 2b v c(u2 + v 2 − 1) . , , 1 + u2 + v 2 1 + u2 + v 2 1 + u2 + v 2 Hyperboloids Consider the following patches: (10.21) hyperboloid1[a, b, c](u, v) = a cosh v cos u, b cosh v sin u, c sinh v , hyperboloid2[a, b, c](u, v) = a sinh v cos u, b sinh v sin u, c cosh v . 314 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE They describe surfaces whose shape is altered by varying the three parameters a, b, c. Assuming all three are positive, hyperboloid1 describes a hyperboloid of one sheet, whereas hyperboloid2 describes the upper half of a hyperboloid of two sheets (the lower half is obtained by taking c < 0). As |v| becomes large, the surfaces are asymptotic to ellipticalcone[a, b, c](u, v) = av cos u, bv cos u, cv , and rotational symmetry is present if a = b. In this case, the surfaces are formed by revolving a branch of a hyperbola (or a line) about the z-axis, as is the case in Figure 10.17. Figure 10.17: Hyperboloids in and outside of a cone The nonparametric equations y2 z2 x2 + − = 1, a2 b2 c2 y2 z2 x2 + − = −1, a2 b2 c2 describe a hyperboloid of one sheet and two sheets respectively. The equations can easily be distinguished, as the second has no solutions for z = 0, indicating that no points of the surface lie in the xy-plane. 10.6. LEVEL SURFACES IN R N 315 Higher Order Surfaces Consider the function gn : R3 → R defined by gn (x, y, z) = xn + y n + z n , where n > 2 is an even integer. Using Definition 9.24, the gradient of gn is given by (10.22) (grad gn )(x, y, z) = nxn−1 , ny n−1 , nz n−1 ; it vanishes if and only if x = y = z = 0, so the set { p ∈ R3 | gn (p) = c } is a regular surface for any c > 0. It is a sphere for n = 2, but as n becomes larger and larger, (10.22) becomes more and more cube-like; see Figure 10.18. Figure 10.18: The regular surface x6 + y 6 + z 6 = 1 There are functions g for which {p ∈ R3 | g(p) = c} has several components. An obvious example is the regular surface defined by (x2 + y 2 + z 2 − 1) (x − 3)2 + y 2 + z 2 − 1 = 0, which consists of two disjoint spheres. Because the gradient of a function g : R3 → R is always perpendicular to each of its level surfaces, there is an easy way to obtain the Gauss map for such surfaces. 316 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE Lemma 10.49. Let g : R3 → R be a differentiable function and c a number such that grad g is nonzero on all points of the level surface M(c) = { p ∈ R3 | g(p) = c }. Then the vector field U defined by U(p) = grad g(p) grad g(p) is a globally defined unit normal on M(c). Hence we can define the Gauss map of M(c) to be U considered as a mapping U: M(c) → S 2 (1). Steven Wilkinson’s program ImplicitPlot3D is extremely effective in plotting surfaces such as those in Figures 10.18, 10.19 and 10.20. Details are given in Notebook 10. The reader is invited to test it out in Exercises 10–13. Figure 10.19: (yz)2 + (zx)2 + (xy)2 = 1 10.7 Exercises 1. Show that the Jacobian matrix of a patch x: U → Rn is related to xu and xv by the formula ! xu · xu xu · xv T , J(x) J(x) = xv · xu xv · xv where AT denotes the transpose of a matrix A (see Corollary 10.5(ii)). Show that when n = 3 then det J(x)TJ(x) = kxu × xv k2 . 10.7. EXERCISES 317 2. Determine where the patch (u, v) 7→ cos u cos v sin v, sin u cos v sin v, sin v is regular. 3. A torus can be defined as a level surface M(b2 ) of the function f : R3 → R defined by p 2 x2 + y 2 − a , f (x, y, z) = z 2 + where a > b > 0. Show that such a torus is a regular surface and compute its unit normal. 4. Show that the Gauss map of a regular surface M ⊂ R3 is a surface mapping from M to the unit sphere S 2 (1) ⊂ R3 in the sense of Definition 10.41. 5. Find unit normals to the monkey saddle, the sphere, the torus and the eight surface. 6. Define the Mercator3 parametrization by mercatorellipsoid[a, b, c](u, v) = a sech v cos u, b sech v sin u, c tanh v . Verify that its trace is an ellipsoid. 7. Determine appropriate domains for the patch (10.19) so as to obtain two local charts (satisfying Definition 10.23) that cover the eight surface illustrated in Figure 10.5. M 8. Given a patch x(u, v), let x[n] (u, v) denote the surface whose x-, y- and z-entries are the nth powers of those of x(u, v). Plot the ‘cubed surface’ torus[3] [8, 3, 8]. M 9. Verify that the following quartic factorizes into two quadratic polynomials, and hence plot the surface described by setting it equal to zero: x4 y4 82x2 y 2 10x2 z 2 10y 2 z 2 10x2 10y 2 + + z4 − − − − + + 2z 2 + 1. 9 9 81 9 9 9 9 3 Gerardus Mercator (Latinized name of Gerhard Kremer) (1512–1594). Flemish cartographer. In 1569 he first used the map projection which bears his name. 318 CHAPTER 10. SURFACES IN EUCLIDEAN SPACE M 10. Use ImplicitPlot3D to plot Kummer’s surface4 , defined implicitly by x4 + y 4 + z 4 − (y 2 z 2 + z 2 x2 + x2 y 2 ) − (x2 + y 2 + z 2 ) + 1 = 0. Figure 10.20: Part of Kummer’s surface M 11. Goursat’s surface5 is defined by x4 + y 4 + z 4 + a(x2 + y 2 + z 2 )2 + b(x2 + y 2 + z 2 ) + c = 0. Plot it for a = −1/5, b = 0 and c = −1 in the range −1 6 x, y, z 6 1. 4 Ernst Eduard Kummer(1810–1893). Kummer is remembered for his work in hypergeometric functions, number theory and algebraic geometry. 5 Édouard Jean Babtiste Goursat (1858–1936). Goursat was a leading analyst of his day. His Cours d’analyse mathématique has long been a classic text in France and elsewhere. 10.7. EXERCISES 319 M 12. Plot the two-cusp surface (z − 1)2 (x2 − z 2 ) − (x2 − z)2 − y 4 − y 2 (2x2 + z 2 + 2z − 1) = 0 and verify that it does indeed possess two singular points. M 13. Plot the sine surface defined by sinsurface(u, v) = sin u, sin v, sin(u + v) . Find a nonparametric form of the surface. Describe the singular points of the sine surface. 14. Let V be an open subset of the monkey saddle containing its planar point (0, 0, 0). Describe the image of V under the Gauss map. Figure 10.21: The seashell[helix[1, 0.6]][0.1] M 15. In Section 7.6 we showed how to construct tubes about curves in R3 . The construction can be easily modified to allow the radius of the tube to change from point to point. For example, given a curve γ : (a, b) → R3 , let us define a sea shell seashell[γ, r](t, θ) = γ(t) + rt −cos θ N(t) + sin θ B(t) , where N and B are the normal and binormal to γ. Explain why the sea shell of a helix resembles Figure 10.21. Chapter 11 Nonorientable Surfaces An important discovery of the nineteenth century was that nonorientable surfaces exist. The goal of the present chapter is to gain an understanding of such surfaces. We begin in Section 11.1 by saying precisely what it means for a regular surface to be orientable, and defining the associated Gauss map. Examples of nonorientable surfaces are first described in Section 11.2 by means of various identifications of the edges of a square. We obtain topological descriptions of some of the simpler nonorientable surfaces, without reference to an underlying space like Rn with n > 3. The analytical theory of surfaces described in such an abstract way is developed in Chapter 26. The rest of the chapter is devoted to realizing nonorientable surfaces inside 3dimensional space. Section 11.3 is devoted to the Möbius strip, a building block for all nonorientable surfaces. Having parametrized the Möbius strip explicitly, we are able to depict its Gauss image in Figures 11.8 and 11.9. The Möbius strip has become an icon that has formed the basis of many artistic designs, including one on the Mall in Washington DC. Sections 11.4 and 11.5 describe two basic compact nonorientable surfaces, namely the Klein bottle and real projective plane. These necessarily have points of self-intersection in R3 , and two different models of the Klein bottle are discussed. Different realizations of the projective plane are described using the notion of a map with the antipodal property (Definition 11.5). 11.1 Orientability of Surfaces If V is a 2-dimensional vector space, we call a linear map J : V → V such that J 2 = −1 a complex structure on V . We used this notion from the beginning of our study of curves, and observed on page 265 that each tangent space to R2 has a complex structure. More generally, because all 2-dimensional vector spaces are isomorphic, each tangent space Mp to a regular surface M admits 331 332 CHAPTER 11. NONORIENTABLE SURFACES a complex structure J p : Mp → Mp . However, it may or may not be possible to make a continuous choice of J p . Continuity here means that the vector field p 7→ J p X is continuous for each continuous vector field X tangent to M. This leads to Definition 11.1. A regular surface M ⊂ R3 is called orientable provided each tangent space Mp has a complex structure J p : Mp → Mp such that p 7→ J p is a continuous function. An oriented regular surface M ⊂ R3 is an orientable regular surface together with a choice of the complex structure p 7→ J p . We shall see in Chapter 26 that this definition works equally well for surfaces not contained in R3 . For regular surfaces contained in R3 , there is a more intuitive way to describe orientability using the vector cross product. Theorem 11.2. A regular surface M ⊂ R3 is orientable if and only if there is a continuous map p 7→ U(p) that assigns to each p ∈ M a unit normal vector U(p) ∈ M⊥ p. Proof. Suppose we are given p 7→ U(p). Then for each p ∈ M we define J p : Mp → Mp by (11.1) J p vp = U(p) × vp . It is easy to check that J p maps Mp into Mp and not merely into R3p , and that p 7→ J p is continuous. From (7.3), page 193, it follows that J p2 vp = U(p) × U(p) × vp = U(p) · vp U(p) − U(p) · U(p) vp = −vp . Conversely, if we are given a regular surface M ⊂ R3 with a globally-defined continuous complex structure p 7→ J p , we define U(p) ∈ R3p by (11.2) U(p) = vp × J p vp kvp × J p vp k for any nonzero vp ∈ Mp . Then U(p) is perpendicular to both vp and J p vp . Since vp and J p vp form a basis for Mp , it follows that U(p) is perpendicular to Mp . To check that the U(p) defined by (11.2) is independent of the choice of vp , let wp be another nonzero tangent vector in Mp . Then wp = avp + bJ p vp is a linear combination of vp , and wp × J p wp = (avp + b J p vp ) × (−b vp + a J p vp ) = (a2 + b2 )(vp × J p vp ). Hence wp × J p wp (a2 + b2 )(vp × J p vp ) vp × J p vp = = , kwp × J p wp k k(a2 + b2 )(vp × J p vp )k kvp × J p vp k and (11.2) is unambiguous. Since p 7→ J p is continuous, so is p 7→ U(p). 11.1. ORIENTABILITY OF SURFACES 333 Theorem 11.2 permits us to define the Gauss map of an arbitrary orientable regular surface in R3 , rather than just a patch. Definition 10.22 on page 296 extends in an obvious fashion to Definition 11.3. Let M be an oriented regular surface in R3 , and let U be a globally defined unit normal vector field on M that defines the orientation of M. Let S 2 (1) denote the unit sphere in R3 . Then U, viewed as a map U: M −→ S 2 (1), is called the Gauss map of M. It is easy to find examples of orientable regular surfaces in R3 . Lemma 11.4. Let U ⊆ R2 . The graph Mh of a function h: U → R is an orientable regular surface. Proof. As in Section 10.4, we have the Monge patch x: U → Mh by x(u, v) = u, v, h(u, v) . Then x covers all of Mh ; that is, x(U) = Mh . Furthermore, x is regular and injective. The surface normal U to Mh is given by U◦x= xu × xv (−hu , −hv , 1) = p . kxu × xv k 1 + h2u + h2v The unit vector U is everywhere nonzero, and it follows from Theorem 11.2 that Mh is orientable. More generally, the method of proof of Lemma 11.4 yields: Lemma 11.5. Any surface M ⊂ R3 which is the trace of a single injective regular patch x is orientable. Figure 11.1: Gauss image of a quarter torus 334 CHAPTER 11. NONORIENTABLE SURFACES As an example, Figure 11.1 shows that the Gauss map of a quarter of a torus covers half the unit sphere. Here is another generalization of Lemma 11.4. Lemma 11.6. Let g : R3 → R be a differentiable function and c a number such 3 that grad g is nonzero on all points of M(c) = p ∈ R | g(p) = c . Then M(c), as well as every component of M(c), is orientable. Proof. Theorem 10.48, page 311, implies that M(c) is a regular surface and that grad g is everywhere perpendicular to M(c). We are assuming that grad g never vanishes on M(c). Putting these facts together we see that grad g k grad gk is a unit vector field that is well defined at all points of M(c) and everywhere perpendicular to M(c). Then Theorem 11.2 says that M(c) and each of its components are orientable. Recall that a subset X of Rn is said to be closed if it contains all its limit points. Equivalently, X is closed if and only if its complement Rn \X is open (see page 268). A subset X is said to be connected if any decomposition X = X1 ∪X2 of X into closed subsets X1 , X2 with X1 ∩ X2 empty must be trivial; that is, either X = X1 or X = X2 . Lemma 11.7. Let M be a connected orientable regular surface in R3 . Then M has exactly two globally-defined unit normal vector fields. Proof. Since M is orientable, it has at least one globally-defined unit normal vector field p 7→ U(p). Let p 7→ v(p) be any other globally-defined unit normal vector field on M. The sets W± = p ∈ M | U(p) = ±v(p) are closed, because U and v are continuous. Also M = W+ ∪ W− . The connectedness of M implies that M coincides with either W+ or W− . Consequently, the globally-defined unit vector fields on M are p 7→ U(p) and p 7→ −U(p). Definition 11.8. Let M be an orientable regular surface in R3 . An orientation of M is the choice of a globally-defined unit normal vector field on M. In general, it may not be possible to cover a regular surface with a single patch. If we have a family of patches, we must know how their orientations are related in order to define the orientation of a regular surface. Lemma 11.9. Let M ⊂ R3 be a regular surface. If x: U → M and y : V → M are patches such that x(U) ∩ y(V) is nonempty, then yu × yv = det J(x−1 ◦ y) xū × xv̄ , where J(x−1 ◦ y) denotes the Jacobian matrix of x−1 ◦ y. 11.1. ORIENTABILITY OF SURFACES 335 Proof. It follows from Lemma 10.31, page 300, that ∂ ū ∂v̄ ∂v̄ ∂ ū xū + xv̄ × xū + xv̄ yu × yv = ∂u ∂u ∂v ∂v   ∂ ū ∂ ū  ∂u ∂v   xū × xv̄ = det   ∂v̄ ∂v̄  ∂u ∂v = det J(x−1 ◦ y) xū × xv̄ . Definition 11.10. We say that patches x: U → M and y : V → M on a regular surface M with x(U) ∩ y(V) nonempty are coherently oriented, provided the determinant of the Jacobian matrix J(x−1 ◦ y) is positive on x(U) ∩ y(V). Theorem 11.11. A regular surface M ⊂ R3 is orientable if and only if it is possible to cover M with a family B of regular injective patches such that any two patches (x, U), (y, V) with x(U) ∩ y(V) 6= ∅ are coherently oriented. Proof. Suppose that M is orientable. Theorem 11.2 implies that M has a globally-defined surface normal p 7→ U(p). Let A be a family of regular injective patches whose union covers M. Without loss of generality, we may suppose that the domain of definition of each patch in A is connected. We must construct from A a family B of coherently-oriented patches whose union covers M. To this end, we first note that if x is any regular injective patch on M, then e defined by x e (u, v) = x(v, u) x e and x are oppositely oriis also a regular injective patch on M. Moreover, x eu × x ev = −xu × xv . Now it is clear how to choose the patches ented, because x that are to be members of the family B. If x is a patch in A and xu × xv =U kxu × xv k we put x into B, but if xu × xv = −U kxu × xv k e into B. Then B is a family of coherently-oriented patches that covers we put x M. Conversely, suppose that M is covered by a family B of coherently-oriented patches. If x is a patch in B defined on U ⊂ R2 , we define Ux on x(U) by xu × xv Ux x(u, v) = (u, v). kxu × xv k 336 CHAPTER 11. NONORIENTABLE SURFACES Let y : V → M be another patch in B with x(U) ∩ y(V) 6= ∅. Lemma 11.9 implies that on x(U) ∩ y(V) we have det J(x−1 ◦ y) xu × xv yu × yv (u, v) (u, v) = Uy y(u, v) = kyu × yv k | det J(x−1 ◦ y)| kxu × xv k = xu × xv (u, v) = Ux x(u, v) . kxu × xv k Thus we get a well-defined surface normal U on M by putting U = Ux on x(U) for any patch x in B. By Theorem 11.2, M is orientable. The proof of Theorem 11.11 establishes Corollary 11.12. A family of coherently-oriented regular injective patches on a regular surface in R3 defines globally a unit normal vector field on a surface M in R3 , that is, an orientation of M. The image under the Gauss map of the equatorial region hyperboloid1[1, 1, 1](u, v) | 0 6 u 6 2π, −1 6 v 6 1 of a hyperboloid of one sheet is illustrated in Figure 11.2. The image of the whole hyperboloid is also a bounded equatorial region, because the normals to the surface approach those of the asymptotic cone (shown in Figure 10.17). The image of the whole hyperboloid of two sheets under the Gauss map consists of two antipodal disks (see Exercise 2). Figure 11.2: Gauss image of a hyperboloid 11.2 Surfaces by Identification There are useful topological descriptions of some elementary surfaces that are obtained from identifying edges of a square. For example, if we identify the 11.2. SURFACES BY IDENTIFICATION 337 top and bottom edges, we obtain a cylinder. It is conventional to describe this identification by means of an arrow along the top edge and an arrow pointing in the same direction along the bottom edge, as in Figure 11.3 (left). Figure 11.3: Models of the cylinder and Möbius strip Now consider a square with the top and bottom edges identified, but in reverse order. This identification is indicated by means of an arrow along the top edge pointing in one direction and an arrow along the bottom edge pointing in the opposite direction. The resulting surface is called a Möbius1 strip or a Möbius band. It is easy to put a cylinder into Euclidean space R3 , but to find the actual parametrization of a Möbius strip in R3 is more difficult. We shall return to this problem shortly. Now let us see what happens when we identify the vertical as well as the horizontal edges of a square. There are three possibilities. If the vertical arrows point in the same direction, and if the direction of the horizontal arrows is also the same, we obtain a torus. We have already seen on page 305 how to parametrize a torus in R3 using the function torus[a, b, c]. For different values of a, b, c, we get tori which are different in shape, but provided a < b and a < c they are topologically all the same. Figure 11.4: Models of the torus and Klein bottle 1 August Ferdinand Möbius (1790–1868). Professor at the University of Leipzig. Möbius’ name is attached to several important mathematical objects such as the Möbius function, the Möbius inversion formula and Möbius transformations. He discovered the Möbius strip at age 71. 338 CHAPTER 11. NONORIENTABLE SURFACES The Klein2 bottle is the surface that results when the edges of the square are identified with the vertical arrows pointing in the same direction, but the horizontal arrows pointing in opposite directions. Clearly, the same surface results if we interchange horizontal with vertical, to obtain the right-hand side of Figure 11.4. Finally, the surface that results when we identify the edges of the square with the two vertical arrows pointing in different directions and the two horizontal arrows pointing in different directions is called the real projective plane, and sometimes denoted RP2 . This surface can also be thought of as a sphere with antipodal points identified. It is one of a series of important topological objects, namely the real projective spaces RPn , that are defined in Exercise 12 on page 798. The superscript indicates dimension, a concept that will be defined rigorously in Chapter 24. Whilst RP1 is equivalent to a circle, it turns out that RP3 can be identified with the set of rotations R3 discussed in Chapter 23. We have described the Möbius strip, the Klein bottle and the real projective plane topologically. All these surfaces turn out to be nonorientable. It is quite another matter, however, to find effective parametrizations of these surfaces in R3 . That is the subject of the rest of this chapter. Figure 11.5: Model of the projective plane 2 Christian Felix Klein (1849–1925). Klein made fruitful contributions to many branches of mathematics, including applied mathematics and mathematical physics. His Erlanger Programm (1872) instituted research directions in geometry for a half century; his subsequent work on Riemann surfaces established their essential role in function theory. In his writings, Klein concerned himself with what he saw as a developing gap between the increasing abstraction of mathematics and applied fields whose practitioners did not appreciate the fundamental rôle of mathematics, as well as with mathematics instruction at the secondary level. Klein discussed nonorientable surfaces in 1874 in [Klein]. 11.3. MÖBIUS STRIP 339 11.3 The Möbius Strip Another useful description of the Möbius strip is as the surface resulting from revolving a line segment around an axis, putting one twist in the line segment as it goes around the axis. Figure 11.6: Lines twisting to form a Möbius strip Figure 11.7: Möbius strip We can use this model to get a parametrization of the strip: u (11.3) moebiusstrip[a](u, v) = a cos u + v cos cos u, 2 u u a sin u + v cos sin u, v sin . 2 2 We see that the points moebiusstrip[a](u, 0) = a cos u, a sin u, 0 , 0 6 u < 2π, constitute the central circle which therefore has radius a. It is convenient to retain this parameter a in order to plot Möbius strips of different shape, and our notation reflects that of Notebook 11. 340 CHAPTER 11. NONORIENTABLE SURFACES The points moebiusstrip[a](u0 , v), as v varies for each fixed u0 , form a line segment meeting the central circle. As u0 increases from 0 to 2π, the angle between this line and the xy-plane changes from 0 to π. The nonorientability of the Möbius strip means that the Gauss map is not well defined on the whole surface: any attempt to define a unit normal vector on the entire Möbius strip is doomed to failure. However, the Gauss map is defined on any orientable portion, for example, on a Möbius strip minus a line orthogonal to the central circle. Figure 11.8: Gauss image of a Möbius strip traversed once Figure 11.9: Gauss image of a Möbius strip traversed twice It is possible to compute a normal vector field to the patch moebiusstrip[1]; the result is u u 3u −v cos + 2 cos u + v cos sin , 2 2 2 (11.4) u 3u u u 2 . cos − cos + v(cos u + sin u), −2 cos 1 + v cos 2 2 2 2 11.4. KLEIN BOTTLE 341 The Gauss map is determined by normalizing this vector. Although (11.4) is complicated, it is used in Notebook 11 to plot the accompanying figures. If one tries to extend the definition of the unit normal so that it is defined on all of the Möbius strip by going around the center circle, then the unit normal comes back on the other side of the surface (Figure 11.8). If we repeat this operation to return to where we started on the sphere, we have effectively associated two unit normal vectors to each point of the strip (Figure 11.9). Figure 11.10: Parallel surface to the Möbius strip Figure 11.10 displays a Möbius strip together with the surface formed by moving a small fixed distance along both of the unit normals emanating from each point of the strip. This construction is considered further in Notebook 11 and Section 19.8; here we merely observe that the resulting surface parallel to the Möbius strip is orientable. 11.4 The Klein Bottle The Klein bottle is also a nonorientable surface, but in contrast to the Möbius strip it is compact, equivalently closed without boundary, like the torus. An elementary account of the theory of compact surfaces can be found in [FiGa]. One way to define the Klein bottle is as follows. It is the surface that results from rotating a figure eight about an axis, but putting a twist in it. The rotation and twisting are the same as a Möbius strip but, instead of a line segment, one uses a figure eight. Here is a parametrization of the Klein bottle that uses this construction. u u kleinbottle[a](u, v) = a + cos sin v − sin sin 2v cos u, 2 2 u u u u a + cos sin v − sin sin 2v sin u, sin sin v + cos sin 2v . 2 2 2 2 The central circle of the Klein bottle is traversed twice by the curve u 7→ kleinbottle[a](u, 0), 0 6 u < 4π. 342 CHAPTER 11. NONORIENTABLE SURFACES Each of the curves v 7→ kleinbottle[a](u0 , v) is a figure eight. As u varies from 0 to 2π, the figure eights twist from 0 to π; this is the same twisting that we encountered in the parametrization (11.3) of the Möbius strip. Figure 11.11: Figure eights twisting to form a Klein bottle The Klein bottle is not a regular surface in R3 because it has self-intersections. However, the Klein bottle can be shown to be an abstract surface, a concept to be defined in Chapter 26. There are two kinds of Klein bottles in R3 . The first one, K1 illustrated in Figure 11.11, has the feature that a neighborhood V1 of the self-intersection curve is nonorientable. In fact, V1 is formed from an ‘X’ that rotates and twists about an axis in the same way that the figure eights move when they form the surface. Figure 11.12: Klein bottle with an orientable neighborhood of the self-intersection curve 11.5. REAL PROJECTIVE PLANE 343 The original description of Klein (see [HC-V, pages 308-311]) of his surface was much different, and is described as follows. Consider a tube T of variable radius about a line. Topologically, a torus is formed from T by bending the tube until the ends meet and then gluing the boundary circles together. Another way to glue the ends is as follows. Let one end of T be a little smaller than the other. We bend the smaller end, then push it through the surface of the tube, and move it so that it is a concentric circle with the larger end, lying in the same plane. We complete the surface by adjoining a torus on the other side of the plane. The result is shown in Figures 11.12 and 11.13. Figure 11.13: Open views displaying the self-intersection To see that this new Klein bottle K2 is (as a surface in R3 ) distinct from the Klein bottle K1 formed by twisting figure eights, consider a neighborhood V2 of the self-intersection curve of K2 This neighborhood is again formed by rotating an ‘X’, but in an orientable manner. Thus the difference between K1 and K2 is that V1 is nonorientable, but V2 is orientable. Nevertheless, K1 and K2 are topologically the same surface, because each can be formed by identifying sides of squares, as described in Section 11.2. A parametrization of K2 is given in Notebook 11. 11.5 Realizations of the Real Projective Plane Let S 2 = S 2 (1) = { p | kpk = 1} be the sphere of unit radius in R3 . Recall that the antipodal map of the sphere is the diffeomorphism S 2 −→ S 2 p 7→ −p. The real projective plane RP2 can be defined as the set that results when antipodal points of S 2 are identified; thus RP2 = {p, −p} | kpk = 1 . 344 CHAPTER 11. NONORIENTABLE SURFACES To realize the real projective plane as a surface in R3 let us look for a map of R3 into itself with a special property. Definition 11.13. A map F : R3 → R3 such that (11.5) F (−p) = F (p) is said to have the antipodal property. That we can use a map with the antipodal property to realize the real projective plane is a consequence of the following easily-proven lemma: Lemma 11.14. A map F : R3 → R3 with the antipodal property gives rise to a map Fe : RP2 → F S 2 ⊂ R3 defined by Fe ({p, −p}) = F (p). Thus we can realize RP2 as the image of S 2 under a map F which has the antipodal property. Moreover, any patch x: U → S 2 will give rise to a patch e : U → F S 2 defined by x e (u, v) = F x(u, v) . x Ideally, one should choose F so that its Jacobian matrix is never zero. It turns out that this can be accomplished with quartic polynomials (see page 347), but the zeros can be kept to a minimum with well-chosen quadratic polynomials. We present three examples of maps with the antipodal property and associated realizations of RP2 . The first is illustrated in Figure 11.14. Figure 11.14: Steiner’s Roman surface 11.5. REAL PROJECTIVE PLANE 345 Steiner’s Roman Surface When Jakob Steiner visited Rome in 1844 he developed the concept of a surface that now carries his name (see [Apéry, page 37]). It is a realization of the real projective plane. To describe it, we first define the map (11.6) romanmap(x, y, z) = (yz, zx, xy) from R3 to itself. It is obvious that romanmap has the antipodal property; it therefore induces a map RP2 → romanmap S 2 . We call this image Steiner’s Roman surface3 of radius 1. Moreover, we can plot a portion of romanmap(S 2 ) by composing romanmap with any patch on S 2 , for example, the standard parametrization sphere[1]: (u, v) 7→ cos v cos u, cos v sin u, sin v . Then the composition romanmap ◦ sphere[1] parametrizes all of Steiner’s Roman surface. Figure 11.15: Two cut views of the Roman surface Therefore, we define roman(u, v) = 1 2 sin u sin 2v, cos u sin 2v, sin 2u cos2 v Figure 11.15 cuts open Steiner’s surface and displays the inside and outside in different colors to help visualize the intersections. Notice that the origin (0, 0, 0) of R3 can be represented in one of the equivalent ways romanmap(±1, 0, 0) = romanmap(0, ±1, 0) = romanmap(0, 0, ±1). 3 Jakob Steiner (1796–1863). Swiss mathematician who was professor at the University of Berlin. Steiner did not learn to read and write until he was 14 and only went to school at the age of 18, against the wishes of his parents. Synthetic geometry was revolutionized by Steiner. He hated analysis as thoroughly as Lagrange hated geometry, according to [Cajori]. He believed that calculation replaces thinking while geometry stimulates thinking. 346 CHAPTER 11. NONORIENTABLE SURFACES This confirms that the origin is a triple point of Steiner’s surface, meaning that three separate branches intersect there. In addition, there are six singularities of the umbrella type illustrated in Figure 10.14. These occur at either end of each of the three ‘axes’ visible in Figure 11.15, and one is clearly visible on the left front in Figure 11.14. The Cross Cap A mapping with the antipodal property formed from homogeneous quadratic polynomials, and similar to (11.6), is given by crosscapmap(x, y, z) = yz, 2xy, x2 − y 2 . We can easily get a parametrization of the cross cap, just as we did for Steiner’s Roman surface. An explicit parametrization is (11.7) crosscap(u, v) = 21 sin u sin 2v, sin 2u cos2 v, cos 2u cos2 v . Figure 11.16: A cross cap and a cut view The cross cap is perhaps the easiest realization of the sphere with antipodal points identified. Given a pair of points ±p in S 2 , either (i) both points belong to the equator (meaning z = 0), or (ii) they correspond to a unique point in the ‘southern hemisphere’ (for which z < 0). To obtain RP2 from S 2 , it therefore suffices to remove the open northern hemisphere (for which z > 0), and then deform the equator upwards towards where the north pole was, and ‘sew’ it to itself so that opposite points on (what was) the equator are placed next to each other. This requires a segment in which the surface intersects itself, but is the idea behind Figure 11.16. 11.5. REAL PROJECTIVE PLANE 347 Boy’s Surface A more complicated mapping F = F1 (x, y, z), F2 (x, y, z), F3 (x, y, z) , with the antipodal property is one defining the remarkable surface discovered in 1901 by Boy4 , whose components are homogeneous quartic polynomials. This description of the surface was found by Apéry [Apéry], and is obtained by taking 4F1 (x, y, z) = F2 (x, y, z) = √1 F3 (x, y, z) 3 = (x + y + z) (x + y + z)3 + 4(y − x)(z − y)(x − z) , (2x2 − y 2 − z 2 )(x2 + y 2 + x2 ) + 2yz(y 2 − z 2 ) +zx(x2 − z 2 ) + xy(y 2 − x2 ), (y 2 − z 2 )(x2 + y 2 + z 2 ) + zx(z 2 − x2 ) + xy(y 2 − x2 ). Figure 11.17: A view of Boy’s surface Figure 11.17 displays another realization of the surface, described in Notebook 11, using the concept of inversion from Section 20.4. Boy’s surface can be covered by patches without singularities, in contrast to the two previous examples. Figure 11.18 helps to understand that curves of self-intersection meet in a triple point, at which the surface has a 3-fold symmetry, but there are no pinch points. 4 Werner Boy, a student of David Hilbert (see page 602). 348 CHAPTER 11. NONORIENTABLE SURFACES Figure 11.18: Two cut views of Boy’s surface, revealing a triple self-intersection point 11.6 Twisted Surfaces In this section we define a class of ‘twisted’ surfaces that generalize the Klein bottle and Möbius strip. Definition 11.15. Let α be a plane curve with the property that (11.8) α(−t) = −α(t) and write α(t) = (ϕ(t), ψ(t)). The twisted surface with profile curve α and parameters a and b is defined by twist[α, a, b](u, v) = a + cos(bu)ϕ(v) − sin(bu)ψ(v) (cos u, sin u, 0) + sin(bu)ϕ(v) + cos(bu)ψ(v) (0, 0, 1). To understand the significance of this definition, take a = 0 and b = 1/2. The coordinate curve of the surface defined for each fixed value of u consists of a copy of α mapping to the plane Πu generated by the vectors (cos u, sin u, 0) and (0, 0, 1). This is the image of the xz-plane Π0 under a rotation by u, and the curve α in Πu is rotated by the same angle u. Note that the curve α starts from the xz-plane (corresponding to u = 0), and by the time it returns to the same plane (u = 2π) it has been rotated by only 180o , though the two traces coincide by (11.8). The resulting surface may or may not be orientable, depending on the choice of α. We shall now show that both the Möbius strip and Klein bottle can be constructed in this way. 11.6. TWISTED SURFACES 349 The Möbius Strip Define α by α(t) = (at, 0). Then u u u twist α, a, 21 (u, v) = a cos u + v cos cos u, sin u + v cos sin u, v sin 2 2 2 = moebiusstrip[a](u, v). The Klein Bottle Define γ by γ(t) = (sin t, sin 2t). Then u u (a + cos sin v − sin sin 2v) cos u, twist γ, a, 12 (u, v) = 2 u 2 2 u 2 u 2 u 2 (a + cos sin v − sin sin 2v) sin u, sin sin v + cos sin 2v = kleinbottle[a](u, v). Twisted Surfaces of Lissajous Curves Instead of twisting a figure eight around a circle we can twist a Lissajous curve. The latter is parametrized by lissajous[n, d, a, b](t) = a sin(nt + d), b sin t . For example, lissajous[2, 0, 1, 1] resembles a figure eight, and the associated twisted surface is a Klein bottle. Figure 11.19 (left) displays the trace of the curve β = lissajous[4, 0, 1, 1], whilst the right-hand side consists of the points twist[β, 2, 12 ](u, v) 0 6 u, v < 2π, describing another self-intersecting surface. Figure 11.19: A Lissajous curve and twisted surface 350 CHAPTER 11. NONORIENTABLE SURFACES 11.7 Exercises 1. Prove Corollary 11.12. M 2. Sketch the image under the Gauss map of the region n o hyperboloid2[1, 1, 1](u, v) | 0 6 u 6 2π, 1 6 v of the 2-sheeted hyperboloid. Show that the image of the whole hyperboloid of two sheets under the Gauss map consists of two antipodal disks, and is not therefore the entire sphere. 3. The image of the Gauss map of an ellipsoid is obviously the whole unit sphere. Nonetheless, plotting the result can be of interest since one can visualize the image of coordinate curves. Explain how the curves visible in Figure 11.20, left, describe the orientation of the unit normal to the patch ellipsoid[3, 2, 1]. The associated approximation, right, is therefore an ‘elliptical’ polyhedral representation of the sphere. Figure 11.20: Gauss images of ellipsoid[3, 2, 1] M 4. Let x = moebiusstrip[1], and consider the two patches x(u, v), 0 < u < 32 π, −1 < v < 1; x(u, v), −π < u < 12 π, −1 < v < 1. Compute the unit normal (see (11.4)), and prove that the two normals have equal and opposite signs on their two respective regions of intersection. Deduce that the Möbius strip is nonorientable. 11.7. EXERCISES 351 5. Verify that Equation (11.7) is indeed the composition of sphere[a] with crosscapmap. M 6. A cross cap can be defined implicitly by the equation (ax2 + by 2 )(x2 + y 2 + z 2 ) − 2z(x2 + y 2 ) = 0. Plot this surface using ImplicitPlot3D. 7. Let us put a family of figure eight curves into R3 in such a way that the first and last figures eight reduce to points. This defines a surface, which we call a pseudo cross cap, shown in Figure 11.21. It can be parametrized as follows: pseudocrosscap(u, v) = (1 − u2 ) sin v, (1 − u2 ) sin 2v, u . Show that pseudocrosscap is not regular at the points (0, 0, ±1). Why should the pseudo cross cap be considered orientable, even though it is not a regular surface? Figure 11.21: A pseudo cross cap 8. Show that the image under crosscapmap of a torus centered at the origin is topologically a Klein bottle. To do this, explain how points of the image can be made to correspond to those in Figure 11.4 with the right-hand identifications. Chapter 12 Metrics on Surfaces In this chapter, we begin the study of the geometry of surfaces from the point of view of distance and area. When mathematicians began to study surfaces at the end of the eighteenth century, they did so in terms of infinitesimal distance and area. We explain these notions intuitively in Sections 12.1 and 12.4, but from a modern standpoint. We first define the coefficients E, F, G of the first fundamental form, and investigate their transformation under a change of coordinates (Lemma 12.4). Lengths and areas are defined by integrals involving the quantities E, F, G. The concept of an isometry between surfaces is defined and illustrated by means of a circular cone in Section 12.2. The extension of this concept to conformal maps is introduced in Section 12.3. The latter also includes a discussion of the distance function defined by a metric on a surface. In Section 12.5, we compute metrics and areas for a selection of simple surfaces, as an application of the theory. 12.1 The Intuitive Idea of Distance So far we have not discussed how to measure distances on a surface. One of the key facts about distance in Euclidean space Rn is that the Pythagorean Theorem holds. This means that if p = (p1 , . . . , pn ) and q = (q1 , . . . , qn ) are points in Rn , then the distance s from p to q is given by (12.1) s2 = (p1 − q1 )2 + · · · + (pn − qn )2 . How is this notion different for a surface? Because a general surface is curved, distance on it is not the same as in Euclidean space; in particular, (12.1) is in general false however we interpret the coordinates. To describe how to measure distance on a surface, we need the mathematically imprecise notion of infinitesimal. The infinitesimal version of (12.1) for n = 2 is 361 362 (12.2) CHAPTER 12. METRICS ON SURFACES ds2 = dx2 + dy 2 . ds dy dx We can think of dx and dy as small quantities in the x and y directions. The formula (12.2) is valid for R2 . For a surface, or more precisely a patch, the corresponding equation is (12.3) ds2 = E du2 + 2F dudv + Gdv 2 . ds !!!!! G dv !!!! E du This is the classical notation1 for a metric on a surface. We may consider (12.3) as an infinitesimal warped version of the Pythagorean Theorem. Among the many ways to define metrics on surfaces, one is especially simple and important. Let M be a regular surface in Rn . The scalar or dot product of Rn gives rise to a scalar product on M by restriction. If vp and wp are tangent vectors to M at p ∈ M, we can take the scalar product vp · wp because vp and wp are also tangent vectors to Rn at p. In this chapter we deal only with local properties of distance, so without loss of generality, we can assume that M is the image of an injective regular patch. However, the following definition makes sense for any patch. Definition 12.1. Let x: U → Rn be a patch. Define functions E, F, G: U → R by E = kxu k2 , F = xu · xv , G = kxv k2 . Then ds2 = E du2 +2F dudv+Gdv 2 is the Riemannian metric or first fundamental form of the patch x. Furthermore, E, F, G are called the coefficients of the first fundamental form of x. The equivalence of the formal definition of E, F, G with the infinitesimal version (12.3) can be better understood if we consider curves on a patch. 1 This notation (including the choice of letters E, F and G) was already in use in the early part of the 19th century; it can be found, for example, in the works of Gauss (see [Gauss2] and [Dom]). Gauss had the idea to study properties of a surface that are independent of the way the surface sits in space, that is, properties of a surface that are expressible in terms of E, F and G alone. One such property is Gauss’s Theorema Egregium, which we shall prove in Section 17.2. 12.1. INTUITIVE IDEA OF DISTANCE 363 Lemma 12.2. Let α : (a, b) → Rn be a curve that lies on a regular injective patch x: U → Rn , and fix c with a < c < b. The arc length function s of α starting at α(c) is given by (12.4) s(t) = Z ts E c du dt 2 du dv + 2F +G dt dt dv dt 2 dt. Proof. Write α(t) = x(u(t), v(t)) for a < t < b. By the definition of the arc length function and Corollary 10.15 on page 294, we have Z t Z t ′ kα (t)kdt = s(t) = u′ (t)xu u(t), v(t) + v ′ (t)xv u(t), v(t) dt c c Z tq u′ (t)2 xu = 2 + 2u′ (t)v ′ (t)xu · xv + v ′ (t)2 xv 2 dt c = Z c ts E du dt 2 + 2F du dv +G dt dt dv dt 2 dt. It follows from (12.4) that s 2 2 du dv dv ds du (12.5) + 2F . = E +G dt dt dt dt dt Now we can make sense of (12.3). We square both sides of (12.5) and multiply through by dt2 . Although strictly speaking multiplication by the infinitesimal dt2 is not permitted, at least formally we obtain (12.3). Notice that the right-hand side of (12.3) does not involve the parameter t, except insofar as u and v depend on t. We may think of ds as the infinitesimal arc length, because it gives the arc length function when integrated over any curve. Geometrically, ds can be interpreted as the infinitesimal distance from a point x(u, v) to a point x(u + du, v + dv) measured along the surface. Indeed, to first order, α(t + dt) ≈ α(t) + α′ (t)dt = α(t) + xu u′ (t)dt + xv v ′ (t)dt, so that α(t + dt) − α(t) xu u′ (t) + xv v ′ (t) dt p = E u′2 + 2F u′ v ′ + Gv ′2 dt = ds. ≈ A standard notion from calculus of several variables is that of the differential of a function f : R2 → R; it is given by df = ∂f ∂f du + dv. ∂u ∂v 364 CHAPTER 12. METRICS ON SURFACES More generally, if x: U → M is a regular injective patch and f : M → R is a differentiable function, we put df = ∂(f ◦ x) ∂(f ◦ x) du + dv. ∂u ∂v We call df the differential of f . The differentials of the functions x(u, v) 7→ u and x(u, v) 7→ v are denoted by du and dv. In spite of its appearance, ds will hardly ever be the differential of a function on a surface, since ds2 represents a nondegenerate quadratic form. But formally, dx = xu du + xv dv, so that dx · dx = xu du + xv dv · xu du + xv dv = kxu k2 du2 + 2xu · xv dudv + kxv k2 dv 2 = E du2 + 2F dudv + Gdv 2 = ds2 . Equation (12.2) represents the square of the infinitesimal distance on R2 written in terms of the Cartesian coordinates x and y. There is a different, but equally familiar, expression for ds2 in polar coordinates. Lemma 12.3. The metric ds2 on R2 , given in Cartesian coordinates as ds2 = dx2 + dy 2 , ds dy dx in polar coordinates becomes ds 2 2 2 2 r dΘ ds = dr + r dθ . dr Proof. We have the standard change of variable formulas from rectangular to polar coordinates: ( x = r cos θ, y = r sin θ. Therefore, (12.6) ( dx = −r sin θ dθ + cos θ dr, dy = r cos θ dθ + sin θ dr. 12.2. ISOMETRIES BETWEEN SURFACES 365 Hence dx2 + dy 2 = (−r sin θ dθ + cos θ dr)2 + (r cos θ dθ + sin θ dr)2 = dr2 + r2 dθ2 . We need to know how the expression for a metric changes under a change of coordinates. Lemma 12.4. Let x: U → M and y : V → M be patches on a regular surface M with x(U) ∩ y(V) nonempty. Let x−1 ◦ y = (ū, v̄): U ∩ V → U ∩ V be the associated change of coordinates, so that y(u, v) = x ū(u, v), v̄(u, v) . Suppose that M has a metric and denote the induced metrics on x and y by ds2x = Ex dū2 + 2Fx dūdv̄ + Gx dv̄ 2 Then (12.7) and ds2y = Ey du2 + 2Fy dudv + Gy dv 2 .  2 2  ∂ ū ∂ ū ∂v̄ ∂v̄   E = E + 2F , + G  y x x x  ∂u ∂u ∂u ∂u      ∂ ū ∂ ū ∂v̄ ∂v̄ ∂ ū ∂v̄ ∂ ū ∂v̄ Fy = Ex + Gx + Fx + ,  ∂u ∂v ∂u ∂v ∂v ∂u ∂u ∂v     2 2   ∂ ū ∂v̄ ∂v̄ ∂ ū   + 2Fx . + Gx  Gy = Ex ∂v ∂v ∂v ∂v Proof. We use Lemma 10.31 on page 300, to compute ∂ ū ∂ ū ∂v̄ ∂v̄ Ey = yu · yu = xū + xv̄ · xū + xv̄ ∂u ∂u ∂u ∂u 2 2 ∂v̄ ∂v̄ ∂ ū ∂ ū xū · xv̄ + xū · xū + 2 xv̄ · xv̄ = ∂u ∂u ∂v ∂v 2 2 ∂ ū ∂ ū ∂v̄ ∂v̄ = Ex + 2 Gx . Fx + ∂u ∂u ∂u ∂u The other equations are proved similarly. 12.2 Isometries between Surfaces In Section 10.5, we defined the notion of a mapping F between surfaces in Rn . Even if F is a local diffeomorphism (see page 309), F (M) can be quite different from M. Good examples are the maps romanmap and crosscapmap of Section 11.5 that map a sphere onto Steiner’s Roman surface and a cross cap. Differentiable maps that preserve infinitesimal distances, on the other hand, distort much less. The search for such maps leads to the following definition. 366 CHAPTER 12. METRICS ON SURFACES Definition 12.5. Let M1 , M2 be regular surfaces in Rn . A map Φ: M1 → M2 is called a local isometry provided its tangent map satisfies (12.8) kΦ∗ (vp )k = kvp k for all tangent vectors vp to M1 . An isometry is a surface mapping which is simultaneously a local isometry and a diffeomorphism. Lemma 12.6. A local isometry is a local diffeomorphism. Proof. It is easy to see that (12.8) implies that each tangent map of a local isometry Φ is injective. Then the inverse function theorem implies that Φ is a local diffeomorphism. Every isometry of Rn which maps a regular surface M1 onto a regular surface M2 obviously restricts to an isometry between M1 and M2 . When we study minimal surfaces in Chapter 16, we shall see that there are isometries between surfaces that do not however arise in this fashion. Let vp , wp be tangent vectors to M1 . Suppose that (12.8) holds. Since Φ∗ is linear, 1 Φ∗ (vp ) · Φ∗ (wp ) = kΦ∗ (vp + wp )k2 − kΦ∗ (vp )k2 − kΦ∗ (wp )k2 2 = 1 k(vp + wp )k2 − kvp k2 − kwp k2 2 = vp · wp . It follows that (12.8) is equivalent to (12.9) Φ∗ (vp ) · Φ∗ (wp ) = vp · wp . Next, we show that a surface mapping is an isometry if and only if it preserves Riemannian metrics. Lemma 12.7. Let x: U → Rn be a regular injective patch and let y : U → Rn be any patch. Let ds2x = Ex du2 + 2Fx dudv + Gx dv 2 and ds2y = Ey du2 + 2Fy dudv + Gy dv 2 denote the induced Riemannian metrics on x and y. Then the map Φ = y ◦ x−1 : x(U) −→ y(U) is a local isometry if and only if (12.10) ds2x = ds2y . 12.2. ISOMETRIES BETWEEN SURFACES 367 Proof. First, note that −1 Φ∗ (xu ) = (y ◦ x ∂ ∂ = y∗ = yu ; )∗ ◦ x∗ ∂u ∂u similarly, Φ∗ (xv ) = yv . If Φ is a local isometry, then Ey = kyu k2 = kΦ∗ (xu )k2 = kxu k2 = Ex . In the same way, Fy = Fx and Gy = Gx . Thus (12.10) holds. To prove the converse, consider a curve α of the form α(t) = x u(t), v(t) ; then (Φ ◦ α)(t) = y(u(t), v(t)). From Corollary 10.15 we know that α′ = u′ xu + v ′ xv Hence (12.11)   and (Φ ◦ α)′ = u′ yu + v ′ yv . kα′ k2 = Ex u′2 + 2Fx u′ v ′ + Gx v ′2 ,  k(Φ ◦ α)′ k2 = E u′2 + 2F u′ v ′ + G v ′2 . y y y Then (12.10) and (12.11) imply that (12.12) kα′ (t)k = k(Φ ◦ α)′ (t)k. Since every tangent vector to x(U) can be represented as α′ (0) for some curve α, it follows that Φ is an isometry. Corollary 12.8. Let Φ: M1 → M2 be a surface mapping. Given a patch x: U → M1 , set y = Φ ◦ x. Then Φ is a local isometry if and only if for each regular injective patch x on M1 we have ds2x = ds2y . We shall illustrate these results with a vivid example by constructing a patch on a circular cone in terms of Euclidean coordinates (u, v) in the plane. We begin with the region in the plane surrounded by a circular arc AB of radius 1 subtending an angle AOB of α radians, shown in Figure 12.4. We shall attach the edge OA to OB, in the fashion of the previous chapter, but make this process explicit by identifying the result with a circular (half) cone in 3-dimensional space. The top circle of the cone has circumference α (this being the length of the existing arc AB) and thus radius α/ 2π. It follows that if β denotes the angle (between the axis and a generator) of the cone, then sin β = α . 2π For example, if α = π, the initial region is a semicircular and the cone has angle β = π/6 or 30o . 368 CHAPTER 12. METRICS ON SURFACES O A B Figure 12.4: A planar region The planar region may be realized as a surface in R3 by merely mapping it into the xy-plane. Using polar coordinates, it is then parametrized by the patch x(r, θ) = (r cos θ, r sin θ, 0) defined for 0 6 r 6 1 and 0 6 θ 6 α. We can parametrize the resulting conical surface by the patch by realizing that a radial line in the preceding plot becomes a generator of the cone. A distance r along this generator contributes a horizontal distance of r sin β and a vertical distance r cos β. The angle about the z-axis has to run through a full turn, so a patch of the conical surface is (12.13) 2πθ 2πθ sin β, r sin sin β, r cos β , y(r, θ) = r cos α α for 0 6 r 6 1 and 0 6 θ 6 α. The final result is close to Figure 12.5, right. Let V denote the open interior of the circular region in the xy-plane. The above procedure determines a mapping Φ: V → R3 for which y(u, v) = Φ x(u, v) , in accord with Corollary 12.8. Indeed, Φ represents the ‘rigid’ folding of a piece of paper representing Figure 12.1 into the cone, and is therefore an isometry. This is confirmed by a computation of the respective first fundamental forms. The metric ds2x is given in polar coordinates (r, θ) by Lemma 12.3. But an easy computation of E, F, G for y (carried out in Notebook 12) shows that ds2y has the identical form. This means that the same norms and angles are induced on the corresponding tangent vectors on the surface of the cone. 12.3. DISTANCE AND CONFORMAL MAPS 369 Figure 12.5: Isometries between conical surfaces The following patch interpolates between x and y by representing an intermediate conical surface θ θ xt (r, θ) = r cos sin βt , r sin sin βt , r cos βt , λt where λt = 1 − t + λt αt , 2π βt = arcsin λt . For each fixed t with 0 6 t 6 1, this process determines an isometry from the planar region to the intermediate surface, for example Figure 12.5, middle. In Notebook 12, it is used to create an animation of surfaces passing from x to the closed cone y, representing the act of folding the piece of paper. 12.3 Distance and Conformal Maps A Riemannian metric determines a function which measures distances on a regular surface. Definition 12.9. Let M ⊂ Rn be a regular surface, and let p, q ∈ M. Then the intrinsic distance ρ(p, q) is the greatest lower bound of the lengths of all piecewise-differentiable curves α that lie entirely on M and connect p to q. We call ρ the distance function of M. In general, the intrinsic distance ρ(p, q) will be greater than the Euclidean distance kp − qk, since the surface will not contain the straight line joining p to q. This fact is evident in Figure 12.6. An isometry also preserves the intrinsic distance: Lemma 12.10. Let Φ: M1 → M2 be an isometry. Then Φ identifies the intrinsic distances ρ1 , ρ2 of M1 and M2 in the sense that ρ2 Φ(p), Φ(q) = ρ1 (p, q), (12.14) for p, q ∈ M1 . 370 CHAPTER 12. METRICS ON SURFACES Proof. Let α : (c, d) → M be a piecewise-differentiable curve with α(a) = p and α(b) = q, where c < a < b < d. From the definition of local isometry, it follows that Z b Z b ′ length[α] = kα (t)k dt = k(Φ ◦ α)′ (t)k dt = length[Φ ◦ α]. a a Since Φ is a diffeomorphism, there is a one-to-one correspondence between the piecewise-differentiable curves on M1 connecting p to q and the piecewisedifferentiable curves on M2 connecting Φ(p) to Φ(q). Since corresponding curves have equal lengths, we obtain (12.14). Figure 12.6: Distance on a surface Next, let us consider a generalization of the notion of local isometry. Definition 12.11. Let M1 and M2 be regular surfaces in Rn . Then a map Φ: M1 → M2 is called a conformal map provided there is a differentiable everywhere positive function λ: M1 → R such that (12.15) kΦ∗ (vp )k = λ(p)kvp k for all p ∈ M1 and all tangent vectors vp to M1 at p. We call λ the scale factor. A conformal diffeomorphism is a surface mapping which is simultaneously a conformal map and a diffeomorphism. 12.3. DISTANCE AND CONFORMAL MAPS 371 Since the tangent map Φ∗ of an isometry preserves inner products, it also preserves both lengths and angles. The tangent map of a conformal diffeomorphism in general will change the lengths of tangent vectors, but we do have Lemma 12.12. A conformal map Φ: M1 → M2 preserves the angles between nonzero tangent vectors. Proof. The proof that (12.9) is equivalent to (12.8) can be easily modified to prove that (12.15) is equivalent to (12.16) Φ∗ (vp ) · Φ∗ (wp ) = λ(p)2 vp · wp for all nonzero tangent vectors vp and wp to M1 . Then (12.16) implies that vp · wp Φ∗ (vp ) · Φ∗ (wp ) = . kΦ∗ (vp )k kΦ∗ (wp )k kvp k kwp k From the definition of angle on page 3, it follows that Φ∗ preserves angles between tangent vectors. Here is an important example of a conformal map, which was in fact introduced in Section 8.6. Definition 12.13. Let S 2 (1) denote the unit sphere in R3 . The stereographic map Υ : R2 → S 2 (1) is defined by Υ(p1 , p2 ) = (2p1 , 2p2 , p21 + p22 − 1) . p21 + p22 + 1 We abbreviate this definition to (12.17) Υ(p) = (2p; kpk2 − 1) , 1 + kpk2 for p ∈ R2 (the semicolon reminds us that the numerator is a vector, not an inner product). It is easy to see that Υ is differentiable, and that kΥ(p)k = 1. Moreover, Lemma 12.14. Υ is a conformal map. Proof. Let α : (a, b) → R2 be a curve. It follows from (12.17) that the image of α by Υ is the curve Υ◦α= (2α; kαk2 − 1) , 1 + kαk2 and this equation can be rewritten as (12.18) (1 + kαk2 )(Υ ◦ α) = (2α; kαk2 − 1). 372 CHAPTER 12. METRICS ON SURFACES Differentiating (12.18), we obtain 2(α · α′ )(Υ ◦ α) + (1 + kαk2 )(Υ ◦ α)′ = 2(α′ ; α · α′ ), and taking norms, (12.19) 2(α · α′ )(Υ ◦ α) + (1 + kαk2 )(Υ ◦ α)′ 2 = 4k(α′ ; α · α′ )k2 . Since Υ(R2 ) ⊂ S 2 (1), we have kΥ ◦ αk2 = 1 and (Υ ◦ α) · (Υ ◦ α)′ = 0. It therefore follows from (12.19) that 4(α · α′ )2 + (1 + kαk2 )2 (Υ ◦ α)′ whence = 4kα′ k2 + 4(α · α′ )2 , (Υ ◦ α)′ = 2kα′ k . 1 + kαk2 Υ∗ (vp ) = 2kvp k . 1 + kpk2 We conclude that (12.20) 2 for all p and all tangent vectors vp to R2 at p. Hence Υ is conformal with scale factor λ(p) = 2/(1 + kpk2 ). 12.4 The Intuitive Idea of Area In Rn , the infinitesimal hypercube bounded by dx1 , . . . , dxn has as its volume the product dV = dx1 dx2 · · · dxn . We call dV the infinitesimal volume element of Rn . The corresponding concept for a surface with metric (12.3), page 362, is the element of area dA, given by (12.21) dA = p E G − F 2 dudv du dv For a patch x: U → R3 , it is easy to see why this is the appropriate definition. By (7.2) on page 193, we have p (12.22) E G − F 2 = kxu × xv k = kxu k kxv k sin θ, 12.4. INTUITIVE IDEA OF AREA 373 where θ is the oriented angle from the vector xu to the vector xv . On the other hand, the quantity kxu k kxv k sin θ represents the area of the parallelogram with sides xu , xv and angle θ between the sides: ÈÈx v ÈÈ sinΘ ÈÈx u ÈÈ It is important to realize that the expression ‘dudv’ occurring in (12.21) has a very different meaning to that in the middle term 2F dudv of (12.3). The latter arises from the symmetric product xu · xv , while equation (12.22) shows that dA is determined by an antisymmetric product which logically should be written du × dv or, as is customary, du ∧ dv, satisfying (12.23) du ∧ dv = −dv ∧ du. In fact, (12.23) is an example of a differential form, an object which is subject to transformation rules to reflect (12.25) below and the associated change of variable formula in double integrals. To keep the presentation simple, we shall avoid this notation, though the theory can be developed using the approach of Section 24.6. Let us compute the element of area for the usual metric on the plane, but in polar coordinates u = r > 0 and v = θ. Lemma 12.3 tells us that in this case, E = 1, F = 0 and G = r2 . Thus, Lemma 12.15. The metric ds2 on R2 given by (12.2) has as its element of area dA = dxdy = r drdθ. dx dy r dr dΘ Equation (12.21) motivates the following definition of the area of a closed subset of the trace of a patch. Recall that a subset S of Rn is bounded, provided there exists a number M such that kpk 6 M for all p ∈ S. A compact subset of Rn is a subset which is closed and bounded. 374 CHAPTER 12. METRICS ON SURFACES Definition 12.16. Let x: U → Rn be an injective regular patch, and let R be a compact subset of x(U). Then the area of R is ZZ p (12.24) E G − F 2 dudv. area(R) = x−1 (R) That this definition is geometric is a consequence of Lemma 12.17. The definition of area is independent of the choice of patch. Proof. Let x: V → Rn and y : W → Rn be injective regular patches, and assume that R ⊆ x(V) ∩ y(W). A long but straightforward computation using Lemma 12.4 (and its notation) shows that p p ∂ ū ∂v̄ ∂ ū ∂v̄ Ey Gy − Fy2 = Ex Gx − Fx2 − (12.25) . ∂u ∂v ∂v ∂u But the last factor is the determinant of the Jacobian matrix of x−1 ◦ y. By the change of variables formulas for multiple integrals, we have ZZ ZZ p p ∂ ū ∂v̄ ∂ ū ∂v̄ 2 − dudv Ey Gy − Fy dudv = Ex Gx − Fx2 −1 −1 ∂u ∂v ∂v ∂u y (R) y (R) ZZ p = Ex Gx − Fx2 dū dv̄. x−1 (R) Surface area cannot, in general, be computed by taking the limit of the area of approximating polyhedra. For example, see [Krey1, pages 115–117]. Thus the analog for surfaces of Theorem 1.14, page 10, is false. 12.5 Examples of Metrics The Sphere The components of the metric and the infinitesimal area of a sphere S 2 (a) of radius a are easily computed. For the standard parametrization on page 288, we obtain from Notebook 12 that E = a2 cos2 v, F = 0, G = a2 . Hence the Riemannian metric of S 2 (a) can be written as ds2 = a2 cos2 v du2 + dv 2 . (12.26) The element of area is dA = p EG − F 2 dudv = a2 cos v dudv. If we note that, except for one missing meridian, sphere[a] covers the sphere exactly once when 0 < u < 2π and −π/2 < v < π/2, we can compute the total area of a sphere by computer. As expected, the result is 4aπ. 12.5. EXAMPLES OF METRICS 375 Paraboloids The two types of paraboloid are captured by the single definition paraboloid[a, b](u, v) = (u, v, au2 + bv 2 ), copied from page 303, where both a and b are nonzero. If these two parameters have the same sign, we obtain the elliptical paraboloid, and otherwise the hyperbolic paraboloid; both types are visible in Figure 12.10. Figure 12.10: Elliptical and hyperbolic paraboloids The metric can be computed as a function of a, b, and the result is ds2 = (1 + 4a2 u2 )du2 + 8ab uv dudv + (1 + 4b2 v 2 )dv 2 . Only the sign of F = 4ab uv changes in passing from ‘elliptical’ to ‘hyperbolic’, as E and G are invariant. The element of area p dA = 1 + 4a2 u2 + 4b2 v 2 dudv is identical in the two cases. Cylinders Perhaps an even simpler example is provided by the two cylinders illustrated in Figure 12.11. Whilst the first is a conventional circular cylinder, the leaning one on the right is formed by continuously translating an ellipse in a constant direction. Indeed, by a ‘cylinder’, we understand the surface parametrized by cylinder[d, γ](u, v) = γ(u) + v d, where d is some fixed nonzero vector (representing the direction of the axis), and γ is a fixed curve. Our two cylinders are determined by the choices d = (0, 0, 1), γ(u) = (cos u, sin u, 0), d = (0, 1, 1), γ(u) = (sin u, 1 4 cos u, 0). 376 CHAPTER 12. METRICS ON SURFACES Figure 12.11: Cylinders Whilst the circular cylinder has the Pythagorean metric ds2 = du2 + dv 2 (as in (12.2) apart from the change of variables), the elliptical cylinder has ds2 = (1 − 15 16 sin2 u)du2 − 1 2 sin u dudv + 2 dv 2 . As we shall see later, this reflects the fact that no piece of the second cylinder can be deformed into the plane without distorting distance. The Helicoid The geometric definition of the helicoid is the surface generated by a line ℓ attached orthogonally to an axis m such that ℓ moves along m and also rotates, both at constant speed. The effect depends whether or not ℓ extends to both sides of m (see Figure 12.12); any point of ℓ not on the axis describes a circular helix. Figure 12.12: Helicoids The helicoid surface is (in Exercise 3) explicitly parametrized by (12.27) helicoid[a, b](u, v) = av cos u, av sin u, b u . 12.6. EXERCISES 377 The Monkey Saddle We can use Notebook 12 to find the metric and infinitesimal area of the monkey saddle defined on page 304. The Riemannian metric of the monkey saddle is given by ds2 = 1 + (3u2 − 3v 2 )2 du2 − 36u v(u2 − v 2 )dudv + 1 + 36u2 v 2 dv 2 , and a computation gives dA = p 1 + 9u4 + 18u2 v 2 + 9v 4 dudv. Numerical integration can be used to find the area of (for instance) the squarelike portion S = {monkeysaddle(p, q) | −1 < p, q < 1}, which is approximately 2.33. Twisted Surfaces Recall Definition 11.15 from the previous chapter. Lemma 12.18. Suppose x is a twisted surface in R3 whose profile curve is α = (ϕ, ψ). Then E = a2 + 2aϕ(v) cos bu + ϕ(v)2 cos2 bu + ψ(v)2 sin2 bu +b2 (ϕ(v)2 + ψ(v)2 ) − 2aψ(v) sin bu − ϕ(v)ψ(v) sin(2bu), F = b(−ψϕ′ + ϕψ ′ ), G = ϕ′2 + ψ ′2 . This can be proved computationally (see Exercise 4). 12.6 Exercises 1. Find the metric, infinitesimal area, and total area of a circular torus defined by torus[a, b](u, v) = (a + b cos v) cos u, (a + b cos v) sin u, b sin v . 2. Fill in the details of the proof of Lemma 12.12. M 3. Find the metric and infinitesimal area of the helicoid parametrized by (12.27). Compute the area of the region helicoid[a, b](p, q) | 0 < p < 2π, c < q < d , and plot the helicoid with a = b = 1. 378 CHAPTER 12. METRICS ON SURFACES M 4. Prove Lemma 12.18 using the program metric from Notebook 12. 5. Describe an explicit isometry between an open rectangle in the plane and a circular cylinder, in analogy to (12.13) and the analysis carried out in Section 12.2 for the circular cone. M 6. Determine the metric and infinitesimal area of Enneper’s minimal surface2 , defined by v3 u3 2 2 2 2 + u v , −v + − vu , u − v . enneper(u, v) = u − 3 3 Figure 12.13: Two views of Enneper’s surface Plot Enneper’s surface with viewpoints different from the ones shown in Figure 12.13. Finally, compute the area of the image under enneper of the set [−1, 1] × [−1, 1]. (See also page 509 and [Enn1].) M 7. Find formulas for the components E, F, G of the metrics of the following surfaces parametrized in Chapter 10: the circular paraboloid, the eight surface and the Whitney umbrella. M 8. Find E, F, G for the following surfaces parametrized in Chapter 11: the Möbius strip, the Klein bottle, Steiner’s Roman surface, the cross cap and the pseudo cross cap. 2 Alfred Enneper (1830–1885). Professor at the University of Göttingen. Enneper also studied minimal surfaces and surfaces of constant negative curvature. Chapter 13 Shape and Curvature In this chapter, we study the relationship between the geometry of a regular surface M in 3-dimensional space R3 and the geometry of R3 itself. The basic tool is the shape operator defined in Section 13.1. The shape operator at a point p of M is a linear transformation S of the tangent space Mp that measures how M bends in different directions. The shape operator can also be considered to be (minus) the differential of the Gauss map of M (Proposition 13.5), and its effect is illustrated using tangent vectors to coordinate curves on both the original surface and the sphere (Figure 13.1). In Section 13.2, we define a variant of the shape operator called normal curvature. Given a tangent vector vp to a surface M, the normal curvature k(vp ) is a real number that measures how M bends in the direction vp . This number is easy to understand geometrically because it is the curvature of the plane curve formed by the intersection of M with the plane passing through vp meeting M perpendicularly. Techniques for computing the shape operator and normal curvature are given in Section 13.3. The eigenvalues of the shape operator at p ∈ M are studied at the end of that section. They turn out to be the maximum and minimum of the normal curvature at p, the so-called principal curvatures, and the corresponding eigenvectors are orthogonal. The principal curvatures are graphed for the monkey saddle in Figure 13.6. The most important curvature functions of a surface in R3 are the Gaussian curvature and the mean curvature, both defined in Section 13.4. This leads to a unified discussion of the the first, second and third fundamental forms. Two separate techniques for computing curvature from the parametric representation of a surface are described in Section 13.5, which includes calculations from Notebook 13 with reference to Monge patches and other examples. Section 13.6 is devoted to a global curvature theorem, while we discuss how to compute the curvatures of nonparametric surfaces in Section 13.7. 385 386 CHAPTER 13. SHAPE AND CURVATURE 13.1 The Shape Operator We want to measure how a regular surface M bends in R3 . A good way to do this is to estimate how the surface normal U changes from point to point. We use a linear operator called the shape operator to calculate the bending of M. The shape operator came into wide use after its introduction in O’Neill’s book [ON1]; however, it occurs much earlier, for example, implicitly in Blashke’s classical book [Blas2]1 , and explicitly in [BuBu]2 . The shape operator applied to a tangent vector vp is the negative of the derivative of U in the direction vp : Definition 13.1. Let M ⊂ R3 be a regular surface, and let U be a surface normal to M defined in a neighborhood of a point p ∈ M. For a tangent vector vp to M at p we put S(vp ) = −Dv U. Then S is called the shape operator. Derivatives of vector fields were discussed in Section 9.5, and regular surfaces in Section 10.3. The precise definition of D v U relies on Lemma 9.40, page 281, since U is not defined away from the surface M. It is easy to see that the shape operator of a plane is identically zero at all points of the plane. For a nonplanar surface the surface normal U will twist and turn from point to point, and S will be nonzero. At any point of an orientable regular surface there are two choices for the unit normal: U and −U. The shape operator corresponding to −U is the negative of the shape operator corresponding to U. If M is nonorientable, we have seen that a surface normal U cannot be defined continuously on all of M. This does not matter in the present chapter, because all calculations involving U are local, so it suffices to perform the local calculations on an open subset U of M where U is defined continuously. 1 Wilhelm Johann Eugen Blaschke (1885–1962). Austrian-German mathematician. In 1919 he was appointed to a chair in Hamburg, where he built an important school of differential geometry. 2 Around 1900 the Gibbs–Heaviside vector analysis notation (which one can read about in the interesting book [Crowe]) spread to differential geometry, although its use was controversial for another 30 years. Blashke’s [Blas2] was one of the first differential geometry books to use vector analysis. The book coauthored by Burali-Forti and Burgatti [BuBu] contains an amusing attack on those resisting the new vector notation. In the multivolume works of Darboux [Darb1] and Bianchi [Bian] most formulas are written component by component. Compact vector notation, of course, is indispensable nowadays both for humans and for computers. 13.1. SHAPE OPERATOR 387 Recall (Lemma 10.26, page 299) that any point q in the domain of definition of a regular patch x: U → R3 has a neighborhood Uq of q such that x(Uq ) is a regular surface. Therefore, the shape operator of a regular patch is also defined. Conversely, we can use patches on a regular surface M ⊂ R3 to calculate the shape operator of M. The next lemmas establish some elementary properties. Lemma 13.2. Let x: U → R3 be a regular patch. Then S(xu ) = −Uu and S(xv ) = −Uv . Proof. Fix v and define a curve α by α(u) = x(u, v). Then by Lemma 9.40, page 281, we have S xu (u, v) = S α′ (u) = −Dα′ (u) U = −(U ◦ α)′ (u). But (U ◦ α)′ is just Uu . Similarly, S(xv ) = −Uv . Lemma 13.3. At each point p of a regular surface M ⊂ R3 , the shape operator is a linear map S : Mp −→ Mp . Proof. That S is linear follows from the fact that D av+bw = aDv + bDw . To prove that S maps Mp into Mp (instead of merely into R3p ), we differentiate the equation U · U = 1 and use (9.12) on page 276: 0 = vp [U · U] = 2(Dv U) · U(p) = −2S(vp ) · U(p), for any tangent vector vp . Since S(vp ) is perpendicular to U(p), it must be tangent to M; that is, S(vp ) ∈ Mp . Next, we find an important relation between the shape operator of a surface and the acceleration of a curve on the surface. Lemma 13.4. If α is a curve on a regular surface M ⊂ R3 , then α′′ · U = S(α′ ) · α′ . Proof. We restrict the vector field U to the curve α and use Lemma 9.40. Since α(t) ∈ M for all t, the velocity α′ is always tangent to M, so α′ · U = 0. When we differentiate this equation and use Lemmas 13.2 and 13.3, we obtain α′′ · U = −U′ · α′ = S(α′ ) · α′ . Observe that α′′ · U can be interpreted geometrically as the component of the acceleration of α that is perpendicular to M. 388 CHAPTER 13. SHAPE AND CURVATURE Figure 13.1: Coordinate curves and tangent vectors, together with (on the right) their Gauss images Lemma 13.2 allows us to illustrate −S by its effect on tangent vectors to coordinate curves to M. These are mapped to the tangent vectors to the corresponding curves on the unit sphere defined by U. This is expressed more invariantly by the next result that asserts that −S is none other than the tangent map of the Gauss map. Proposition 13.5. Let M be a regular surface in R3 oriented by a unit normal vector field U. View U as the Gauss map U: M → S 2 (1), where S 2 (1) denotes the unit sphere in R3 . If vp is a tangent vector to M at p ∈ M, then U∗ (vp ) is parallel to −S(vp ) ∈ Mp . Proof. By Lemma 9.10, page 269, we have U∗ (vp ) = vp [u1 ], vp [u2 ], vp [u3 ] U(p) . On the other hand, Lemma 9.28, page 275, implies that −S(vp ) = Dv U = vp [u1 ], vp [u2 ], vp [u3 ] p . Since the vectors (vp [u1 ], vp [u2 ], vp [u3 ])U(p) and (vp [u1 ], vp [u2 ], vp [u3 ])p are parallel, the lemma follows. We conclude this introductory section by noting a fundamental relationship between shape operators of surfaces and Euclidean motions of R3 . Theorem 13.6. Let F : R3 → R3 be an orientation-preserving Euclidean mo- tion, and let M1 and M2 be oriented regular surfaces such that F (M1 ) = M2 . Then 13.2. NORMAL CURVATURE 389 (i) the unit normals U1 and U2 of M1 and M2 can be chosen so that F∗ (U1 ) = U2 ; (ii) the shape operators S1 and S2 of the two surfaces (with the choice of unit normals given by (i)) are related by S2 ◦ F∗ = F∗ ◦ S1 . Proof. Since F∗ preserves lengths and inner products, it follows that F∗ (U1 ) is perpendicular to M2 and has unit length. Hence F∗ (U1 ) = ±U2 ; we choose the plus sign at all points of M2 . Let p ∈ M1 and vp ∈ M1p , and let wF (p) = F∗ (vp ). A Euclidean motion is an affine transformation, so by Lemma 9.35, page 278, we have (S2 ◦ F∗ )(vp ) = S2 (wF (p) ) = −Dw U2 = −Dw F∗ (U1 ) = −F∗ (D v U1 ) = (F∗ ◦ S1 )(vp ). Because vp is arbitrary, we have S2 ◦ F∗ = F∗ ◦ S1 . 13.2 Normal Curvature Although the shape operator does the job of measuring the bending of a surface in different directions, frequently it is useful to have a real-valued function, called the normal curvature, which does the same thing. We shall define it in terms of the shape operator, though it is worth bearing in mind that normal curvature is explicitly a much older concept (see [Euler3], [Meu] and Corollary 13.20). First, we need to make precise the notion of direction on a surface. Definition 13.7. A direction ℓ on a regular surface M is a 1-dimensional subspace of (that is, a line through the origin in) a tangent space to M. A nonzero vector vp in a tangent space Mp determines a unique 1-dimensional subspace ℓ, so we can use the terminology ‘the direction vp ’ to mean ℓ, provided the sign of vp is irrelevant. Definition 13.8. Let up be a tangent vector to a regular surface M ⊂ R3 with kup k = 1. Then the normal curvature of M in the direction up is k(up ) = S(up ) · up . More generally, if vp is any nonzero tangent vector to M at p, we put (13.1) k(vp ) = S(vp ) · vp . kvp k2 390 CHAPTER 13. SHAPE AND CURVATURE If ℓ is a direction in a tangent space Mp to a regular surface M ⊂ R3 , then k(vp ) is easily seen to be the same for all nonzero tangent vectors vp in ℓ (this is Exercise 9). Therefore, we call the common value of the normal curvature the normal curvature of the direction ℓ. Let us single out two kinds of directions. Definition 13.9. Let ℓ be a direction in a tangent space Mp , where M ⊂ R3 is a regular surface. If the normal curvature of ℓ is zero, we say that ℓ is an asymptotic direction. Similarly, if the normal curvature vanishes on a tangent vector vp to M, we say that vp is an asymptotic vector. An asymptotic curve on M is a curve whose trace lies on M and whose tangent vector is everywhere asymptotic. Asymptotic curves will be studied in detail in Chapter 18. Definition 13.10. Let M be a regular surface in R3 and let p ∈ M. The maximum and minimum values of the normal curvature k of M at p are called the principal curvatures of M at p and are denoted by k1 and k2 . Unit vectors e1 , e2 ∈ Mp at which these extreme values occur are called principal vectors. The corresponding directions are called principal directions. A principal curve on M is a curve whose trace lies on M and whose tangent vector is everywhere principal. The principal curvatures measure the maximum and minimum bending of a regular surface M at each point p ∈ M. Principal curves will be studied again in Section 15.2, and in more detail in Chapter 19. There is an important interpretation of normal curvature of a regular surface as the curvature of a space curve. Lemma 13.11. (Meusnier) Let up be a unit tangent vector to M at p, and let β be a unit-speed curve in M with β(0) = p and β ′ (0) = up . Then (13.2) k(up ) = κ[β](0) cos θ, where κ[β](0) is the curvature of β at 0, and θ is the angle between the normal N(0) of β and the surface normal U(p). Thus all curves lying on a surface M and having the same tangent line at a given point p ∈ M have the same normal curvature at p. Proof. Suppose that κ[β](0) 6= 0. By Lemma 13.4 and Theorem 7.10, page 197, we have (13.3) k(up ) = S(up ) · up = β ′′ (0) · U(p) = κ[β](0)N(0) · U(p) = κ[β](0) cos θ. In the exceptional case that κ[β](0) = 0, the normal N(0) is not defined, but we still have k(up ) = 0. 13.2. NORMAL CURVATURE 391 To understand the meaning of normal curvature geometrically, we need to find curves on a surface to which we can apply Lemma 13.11. Definition 13.12. Let M ⊂ R3 be a regular surface and up a unit tangent vec- tor to M. Denote by Π up , U(p) the plane determined by up and the surface normal U(p). The normal section of M in the up direction is the intersection of Π up , U(p) and M. Corollary 13.13. Let β be a unit-speed curve which lies in the intersection of a regular surface M ⊂ R3 and one of its normal sections Π through p ∈ M. Assume that β(0) = p and put up = β′ (0). Then the normal curvature k(up ) of M and the curvature of β are related by (13.4) k(up ) = ±κ[β](0). Proof. We may assume that κ[β](0) 6= 0, for otherwise (13.4) is an obvious consequence of (13.2). Since β has unit speed, κ[β](0)N(0) = β′′ (0) is perpendicular to β′ (0). On the other hand, both U(p) and N(0) lie in Π , so the only possibility is N(0) = ±U(p). Hence cos θ = ±1 in (13.2), and so we obtain (13.4). Figure 13.2: Normal sections to a paraboloid through asymptotic curves Corollary 13.13 gives an excellent method for estimating normal curvature visually. For a regular surface M ⊂ R3 , suppose we want to know the normal curvature in various directions at p ∈ M. Each unit vector up ∈ Mp , together with the surface normal U(p), determines a plane Π up , U(p) . Then the normal section in the direction up is the intersection of Π up , U(p) and M. This is a plane curve in Π up , U(p) whose curvature is given by (13.4). There are three cases: 392 CHAPTER 13. SHAPE AND CURVATURE • If k(up ) > 0, then the normal section is bending in the same direction as U(p). Hence in the up direction M is bending toward U(p). • If k(up ) < 0, then the normal section is bending in the opposite direction from U(p). Hence in the up direction M is bending away from U(p). • If k(up ) = 0, then the curvature of the normal section vanishes at p so the normal to a curve in the normal section is undefined. It is impossible to conclude that there is no bending of M in the up direction since κ[β] might vanish only at p. But in some sense the bending is small. As the unit tangent vector up turns, the surface may bend in different ways. A good example of this occurs at the center point of a hyperbolic paraboloid. Figure 13.3: Normal sections to a paraboloid through principal curves In Figure 13.2, both normal sections intersect the hyperbolic paraboloid in straight lines, and the normal curvature determined by each of these sections vanishes. These straight lines are in fact asymptotic curves, whereas the sections shown in Figure 13.3 intersect the surface in curves tangent to principal directions (recall Definitions 13.9 and 13.10). In the second case, the normal curvature determined by one section is positive, and that determined by the other is negative. The normal sections at the center of a monkey saddle are similar to those of the hyperbolic paraboloid, but more complicated. In this case, there are three asymptotic directions passing through the center point o of the monkey saddle. It is this fact that forces S to vanish as a linear transformation of the tangent space To M at the point o itself. 13.3. CALCULATION OF THE SHAPE OPERATOR 393 Figure 13.4: Normal sections to a monkey saddle 13.3 Calculation of the Shape Operator Symmetric linear transformations are much easier to work with than general linear transformations. We shall exploit this in developing the theory of the shape operator, which fortunately falls into this category. Lemma 13.14. The shape operator of a regular surface M is symmetric or self-adjoint, meaning that S(vp ) · wp = vp · S(wp ) for all tangent vectors vp , wp to M. Proof. Let x be an injective regular patch on M. We differentiate the formula U · xu = 0 with respect to v and obtain (13.5) 0= ∂ (U · xu ) = Uv · xu + U · xuv , ∂v where Uv is the derivative of the vector field v 7→ U(u, v) along any v-parameter curve. Since Uv = −S(xv ), equation (13.5) becomes (13.6) S(xv ) · xu = U · xuv . Similarly, (13.7) S(xu ) · xv = U · xvu . 394 CHAPTER 13. SHAPE AND CURVATURE From (13.6), (13.7) and the fact that xuv = xvu we get (13.8) S(xu ) · xv = U · xvu = U · xuv = S(xv ) · xu . The proof is completed by expressing vp , wp in terms of xu , xv and using linearity. Definition 13.15. Let x: U → R3 be a regular patch. Then (13.9)   e = −Uu · xu = U · xuu ,    f = −Uv · xu = U · xuv = U · xvu = −Uu · xv ,     g = −U · x = U · x . v v vv Classically, e, f, g are called the coefﬁcients of the second fundamental form of x. In Section 12.1 we wrote the metric as ds2 = E du2 + 2F dudv + Gdv 2 , and E, F, G are called the coefﬁcients of the ﬁrst fundamental form of x. The quantity e du2 + 2f dudv + g dv 2 has a more indirect interpretation, given on page 402. The notation e, f, g is that used in most classical differential geometry books, though many authors ′ ′′ use L, M, N in their place.√Incidentally, √ Gauss used the √notation D, D , D for 2 2 2 the respective quantities e EG − F , f EG − F , g EG − F . Theorem 13.16. (The Weingarten3 equations) Let x: U → R3 be a regular patch. Then the shape operator S of x is given in terms of the basis {xu , xv } by  eF − f E f F − eG    −S(xu ) = Uu = EG − F 2 xu + EG − F 2 xv , (13.10)    −S(xv ) = Uv = g F − f G xu + f F − g E xv . EG − F 2 EG − F 2 Proof. Since x is regular, and xu and xv are linearly independent, we can write   −S(xu ) = Uu = a11 xu + a21 xv , (13.11)  −S(x ) = U = a x + a x , v v 12 u 22 v 3 Julius Weingarten (1836–1910). Professor at the Technische Universität in Berlin. A surface for which there is a definite functional relation between the principal curvatures is called a Weingarten surface. 13.3. CALCULATION OF THE SHAPE OPERATOR 395 for some functions a11 , a21 , a12 , a22 , which we need to compute. We take the scalar product of each of the equations in (13.11) with xu and xv , and obtain  −e = Ea11 + F a21 ,      −f = F a + Ga , 11 21 (13.12)  −f = Ea12 + F a22 ,     −g = F a12 + Ga22 . Equations (13.12) can be written more concisely in terms of matrices: ! ! ! e f E F a11 a12 ; − = f g F G a21 a22 hence a11 a21 a12 a22 ! = − E F F G !−1 e f ! f g ! −1 = EG − F 2 G −F −F E −1 = EG − F 2 Ge − F f −F e + Ef e f f g ! Gf − F g −F f + Eg ! . The result follows from (13.11). Although S is a symmetric linear operator, its matrix (aij ) relative to {xu , xv } need not be symmetric, because xu and xv are not in general perpendicular to one another. There is also a way to express the normal curvature in terms of E, F, G and e, f, g: Lemma 13.17. Let M ⊂ R3 be a regular surface and let p ∈ M. Let x be an injective regular patch on M with p = x(u0 , v0 ). Let vp ∈ Mp and write vp = a xu (u0 , v0 ) + b xv (u0 , v0 ). Then the normal curvature of M in the direction vp is k(vp ) = e a2 + 2f ab + g b2 . E a2 + 2F ab + Gb2 Proof. We have kvp k2 = kaxu + b xv k2 = a2 E + 2ab F + b2 G, and S(vp ) · vp = aS(xu ) + b S(xv ) · (axu + b xv ) = a2 e + 2abf + b2 g. The result follows from (13.1). 396 CHAPTER 13. SHAPE AND CURVATURE Eigenvalues of the Shape Operator We first recall an elementary version of the spectral theorem in linear algebra. Lemma 13.18. Let V be a real n-dimensional vector space with an inner product and let A: V → V be a linear transformation that is self-adjoint with respect to the inner product. Then the eigenvalues of A are real and A is diagonalizable: there is an orthonormal basis {e1 , . . . , en } of V such that Aej = λj ej for > j = 1, . . . , n. Lemma 13.14 tells us that the shape operator S is a self-adjoint linear operator on each tangent space to a regular surface in R3 , so the eigenvalues of S must be real. These eigenvalues are important geometric quantities associated with each regular surface in R3 . Instead of proving Lemma 13.18 in its full generality, we prove it for the special case of the shape operator. Lemma 13.19. The eigenvalues of the shape operator S of a regular surface M ⊂ R3 at p ∈ M are precisely the principal curvatures k1 and k2 of M at p. The corresponding unit eigenvectors are unit principal vectors, and vice versa. If k1 = k2 , then S is scalar multiplication by their common value. Otherwise, the eigenvectors e1 , e2 of S are perpendicular, and S is given by (13.13) Se1 = k1 e1 , Se2 = k2 e2 . Proof. Consider the normal curvature as a function k : Sp1 → R, where Sp1 is the set of unit tangent vectors in the tangent space Mp . Since Sp1 is a circle, it is compact, and so k achieves its maximum at some unit vector, call it e1 ∈ Sp1 . Choose e2 to be any vector in Sp1 perpendicular to e1 , so {e1 , e2 } is an orthonormal basis of Mp and   Se1 = (Se1 · e1 )e1 + (Se1 · e2 )e2 , (13.14)  Se = (Se · e )e + (Se · e )e . 2 2 1 1 2 2 2 Define a function u = u(θ) by setting u(θ) = e1 cos θ + e2 sin θ, and write k(θ) = k u(θ) . Then (13.15) k(θ) = (Se1 · e1 ) cos2 θ + 2(Se1 · e2 ) sin θ cos θ + (Se2 · e2 ) sin2 θ, so that d k(θ) = 2(Se2 · e2 − Se1 · e1 ) sin θ cos θ + 2(Se1 · e2 )(cos2 θ − sin2 θ). dθ In particular, (13.16) 0= dk (0) = 2Se1 · e2 , dθ 13.4. GAUSSIAN AND MEAN CURVATURE 397 because k(θ) has a maximum at θ = 0. Then (13.14) and (13.16) imply (13.13). From (13.13) it follows that both e1 and e2 are eigenvectors of S, and from (13.15) it follows that the principal curvatures of M at p are the eigenvalues of S. Hence the lemma follows. The principal curvatures determine the normal curvature completely: Corollary 13.20. (Euler) Let k1 (p), k2 (p) be the principal curvatures of a regular surface M ⊂ R3 at p ∈ M, and let e1 , e2 be the corresponding unit principal vectors. Let θ denote the oriented angle from e1 to up , so that up = e1 cos θ + e2 sin θ. Then the normal curvature k(up ) is given by (13.17) k(up ) = k1 (p) cos2 θ + k2 (p) sin2 θ. Proof. Since S(e1 ) · e2 = 0, (13.15) reduces to (13.17). 13.4 Gaussian and Mean Curvature The notion of the curvature of a surface is a great deal more complicated than the notion of curvature of a curve. Let α be a curve in R3 , and let p be a point on the trace of α. The curvature of α at p measures the rate at which α leaves the tangent line to α at p. By analogy, the curvature of a surface M ⊂ R3 at p ∈ M should measure the rate at which M leaves the tangent plane to M at p. But a difficulty arises for surfaces that was not present for curves: although a curve can separate from one of its tangent lines in only two directions, a surface separates from one of its tangent planes in infinitely many directions. In general, the rate of departure of a surface from one of its tangent planes depends on the direction. There are several competing notions for the curvature of a surface in R3 : • the normal curvature k; • the principal curvatures k1 , k2 ; • the mean curvature H; • the Gaussian curvature K. We defined normal curvature and the principal curvatures of a surface M ⊂ R3 in Section 13.2. In the present section, we give the definitions of the Gaussian and mean curvatures; these are the most important functions in surface theory. First, we recall some useful facts from linear algebra. If S : V → V is a linear transformation on a vector space V , we may define the determinant and trace of S, written det S and tr S, merely as the determinant and trace of the matrix 398 CHAPTER 13. SHAPE AND CURVATURE A representing S with respect to any chosen basis. If P is an invertible matrix representing a change in basis then S is represented by P −1AP with respect to the new basis, but standard properties of the determinant and trace functions ensure that det(P −1AP ) = det A, (13.18) tr(P −1AP ) = tr A, so that our definitions are independent of the choice of basis. Definition 13.21. Let M be a regular surface in R3 . The Gaussian curvature K and mean curvature H of M are the functions K, H : M → R defined by (13.19) K(p) = det S(p) and H(p) = 12 tr S(p) . Note that although the shape operator S and the mean curvature H depend on the choice of unit normal U, the Gaussian curvature K is independent of that choice. The name ‘mean curvature’ is due to Germain4 . Definition 13.22. A minimal surface in R3 is a regular surface for which the mean curvature vanishes identically. A regular surface is ﬂat if and only if its Gaussian curvature vanishes identically. We shall see in Chapter 16 that surfaces of minimal area are indeed minimal in the sense of Definition 13.22. The Gaussian curvature permits us to distinguish four kinds of points on a surface. Definition 13.23. Let p be a point on a regular surface M ⊂ R3 . We say that • p is elliptic if K(p) > 0 (equivalently, k1 and k2 have the same sign); • p is hyperbolic if K(p) < 0 (equivalently, k1 and k2 have opposite signs); • p is parabolic if K(p) = 0, but S(p) 6= 0 (equivalently, exactly one of k1 and k2 is zero); • p is planar if K(p) = 0 and S(p) = 0 (equivalently, k1 = k2 = 0). It is usually possible to glance at almost any surface and recognize which points are elliptic, hyperbolic, parabolic or planar. Consider, for example, the paraboloids shown in Figure 12.10 on page 375. Not surprisingly, all the points 4 Sophie Germain (1776–1831). French mathematician, best known for her work on elasticity and Fermat’s last theorem. Germain (under the pseudonym ‘M. Blanc’) corresponded with Gauss regarding her results in geometry and number theory. 13.4. GAUSSIAN AND MEAN CURVATURE 399 on the left-hand elliptical paraboloid are elliptic, and all those on the righthand hyperbolic paraboloid are hyperbolic. Calculations from the next section show that the monkey surface (13.25) has all its points hyperbolic except for its central point, o = (0, 0, 0), which is planar. This corresponds to the fact, illustrated in Figure 13.5, that its Gaussian curvature K both vanishes and achieves an absolute maximum at o. 2 1 0 -1 -2 0 -1 -2 -3 Figure 13.5: Gaussian curvature of the monkey saddle There are two especially useful ways of choosing a basis of a tangent space to a surface in R3 . Each gives rise to important formulas for the Gaussian and mean curvatures, which are presented in turn by the following proposition and subsequent theorem. Proposition 13.24. The Gaussian curvature and mean curvature of a regular surface M ⊂ R3 are related to the principal curvatures by K = k1 k2 and H = 21 (k1 + k2 ). Proof. If we choose an orthonormal basis of eigenvectors of S for Mp , the matrix of S with respect to this basis is diagonal so that ! k1 0 K = det = k1 k2 0 k2 ! k1 0 1 H = 2 tr = 21 (k1 + k2 ). 0 k2 400 CHAPTER 13. SHAPE AND CURVATURE Let M ⊂ R3 be a regular surface. The Gaussian and mean curvatures are functions K, H : M → R; we have written K(p) and H(p) for their values at p ∈ M. We need a slightly different notation for a regular patch x: U → R3 . Strictly speaking, K and H are functions defined on x(U) → R. However, we follow conventional notation and abbreviate K ◦ x to K and H ◦ x to H. Thus K(u, v) and H(u, v) will denote the values of the Gaussian and mean curvatures at x(u, v). Theorem 13.25. Let x: U → R3 be a regular patch. The Gaussian curvature and mean curvature of x are given by the formulas (13.20) K = (13.21) H = eg − f 2 , EG − F 2 eG − 2f F + gE , 2(EG − F 2 ) where e, f, g are the coefficients of the second fundamental form relative to x, and E, F, G are the coefficients of the first fundamental form. Proof. This time we compute K and H using the basis {xu , xv }, and the matrix ! Ge − F f Gf − F g −1 EG − F 2 −F e + Ef −F f + Eg of Theorem 13.16. Taking the determinant and half the trace yields K= (f F − eG)(f F − gE) − (eF − f E)(gF − f G) eg − f 2 = , (EG − F 2 )2 EG − F 2 and H=− (f F − eG) + (f F − gE) eG − 2f F + gE = . 2 2(EG − F ) 2(EG − F 2 ) The importance of Proposition 13.24 is theoretical, that of Theorem 13.25 more practical. Usually, one uses Theorem 13.25 to compute K and H, and afterwards Proposition 13.24 to find the principal curvatures. More explicitly, Corollary 13.26. The principal curvatures k1 , k2 are the roots of the quadratic equation x2 − 2Hx + K = 0. Thus we can choose k1 , k2 so that p (13.22) k1 = H + H 2 − K and k2 = H − p H 2 − K. 13.4. GAUSSIAN AND MEAN CURVATURE 401 Corollary 13.27. Suppose that M ⊂ R3 has negative Gaussian curvature K at p. Then: (i) there are exactly two asymptotic directions at p, and they are bisected by the principal directions; (ii) the two asymptotic directions at p are perpendicular if and only if the mean curvature H of M vanishes at p. Proof. Let e1 and e2 be unit principal vectors corresponding to k1 (p) and k2 (p). Then K(p) = k1 (p)k2 (p) < 0 implies that k1 (p) and k2 (p) have opposite signs. Thus, there exists θ with 0 < θ < π/2 such that tan2 θ = − k1 (p) . k2 (p) Put up (θ) = e1 cos θ + e2 sin θ. Then (13.17) implies that up (θ) and up (−θ) are linearly independent asymptotic vectors at p. The angle between up (θ) and up (−θ) is 2θ, and it is clear that e1 bisects the angle between up (θ) and up (−θ). This proves (i). For (ii) we observe that H(p) = 0 if and only if θ equals ±π/4, up to integral multiples of π. The Three Fundamental Forms In classical differential geometry, there are frequent references to the ‘second fundamental form’ of a surface in R3 , a notion that is essentially equivalent to the shape operator S. Such references can for example be found in the influential textbook [Eisen1] of Eisenhart5 . Definition 13.28. Let M be a regular surface in R3 . The second fundamental form is the symmetric bilinear form II on a tangent space Mp given by II(vp , wp ) = S(vp ) · wp for vp , wp ∈ Mp . Since there is a second fundamental form, there must be a ﬁrst fundamental form. It is nothing but the inner product between tangent vectors: I(vp , wp ) = vp · wp . 5 Luther Pfahler Eisenhart (1876–1965). American differential geometer and dean at Princeton University. 402 CHAPTER 13. SHAPE AND CURVATURE Note that the first fundamental form I can in a sense defined whether or not the surface is in R3 ; this is the basis of theory to be discussed in Chapter 26. The following lemma is an immediate consequence of the definitions. Lemma 13.29. Let x: U → R3 be a regular patch. Then I(axu + b xv , axu + b xv ) = E a2 + 2F ab + Gb2 , II(axu + b xv , axu + b xv ) = e a2 + 2f ab + g b2 . The normal curvature is therefore given by k(vp ) = II(vp , vp ) I(vp , vp ) for any nonzero tangent vector vp . This lemma explains why we call E, F, G the coefficients of the first fundamental form, and e, f, g the coefficients of the second fundamental form. Finally, there is also a third fundamental form III for a surface in R3 given by III(vp , wp ) = S(vp ) · S(wp ) for vp , wp ∈ Mp . Note that III, in contrast to II, does not depend on the choice of surface normal U. The third fundamental form III contains no new information, since it is expressible in terms of I and II. Lemma 13.30. Let M ⊂ R3 be a regular surface. Then the following relation holds between the first, second and third fundamental forms of M: (13.23) III − 2H II + K I = 0, where H and K denote the mean curvature and Gaussian curvature of M. Proof. Although (13.23) follows from Corollary 13.26 and the Cayley–Hamilton6 Theorem (which states that a matrix satisfies its own characteristic polynomial), we prefer to give a direct proof. First, note that the product H II is independent of the choice of surface normal U. Hence (13.23) makes sense whether or not M is orientable. To 6 Sir William Rowan Hamilton (1805–1865). Irish mathematician, best known for having been struck with the concept of quaternions as he crossed Brougham Bridge in Dublin (see Chapter 23), and for his work in dynamics. 13.5. MORE CURVATURE CALCULATIONS 403 prove it, we observe that since its left-hand side is a symmetric bilinear form, it suffices to show that for each p ∈ M and some basis {e1 , e2 } of Mp we have (13.24) (III − 2H II + K I)(ei , ej ) = 0, for i, j = 1, 2. We choose e1 and e2 to be linearly independent principal vectors at p. Then (III − 2H II + K I)(e1 , e2 ) = 0 because each term vanishes separately. Furthermore, (III − 2H II + K I)(ei , ei ) = ki2 − (k1 + k2 )ki + k1 k2 = 0 for i = 1 and 2, as required. 13.5 More Curvature Calculations In this section, we show how to compute by hand the Gaussian curvature K and the mean curvature H for a monkey saddle and a torus. Along the way we compute the coefficients of their first and second fundamental forms. The Monkey Saddle For the surface parametrized by x(u, v) = monkeysaddle(u, v) = (u, v, u3 − 3u v 2 ), (13.25) and described on page 304, we easily compute xu (u, v) = (1, 0, 3u2 − 3v 2 ), xuu (u, v) = (0, 0, 6u), xv (u, v) = (0, 1, −6uv), xuv (u, v) = (0, 0, −6v), xvv (u, v) = (0, 0, −6u). Therefore, E = 1 + 9(u2 − v 2 )2 , F = −18uv(u2 − v 2 ), G = 1 + 36u2v 2 , and by inspection a unit surface normal is so that (−3u2 + 3v 2 , 6uv, 1) , U= p 1 + 9u4 + 18u2 v 2 + 9v 4 6u e = U · xuu = p , 1 + 9u4 + 18u2 v 2 + 9v 4 −6v , f = U · xuv = p 4 1 + 9u + 18u2 v 2 + 9v 4 −6u . g = U · xvv = p 4 1 + 9u + 18u2 v 2 + 9v 4 404 CHAPTER 13. SHAPE AND CURVATURE Theorem 13.25 yields K= −36(u2 + v 2 ) , (1 + 9u4 + 18u2 v 2 + 9v 4 )2 H= −27u5 + 54u3 v 2 + 81uv 4 . (1 + 9u4 + 18u2 v 2 + 9v 4 )3/2 A glance at these expressions shows that o = (0, 0, 0) is a planar point of the monkey saddle and that every other point is hyperbolic. Furthermore, the Gaussian curvature of the monkey saddle is invariant under all rotations about the z-axis, even though the monkey saddle itself does not have this property. The principal curvatures are determined by Corollary 13.26, and are easy to plot. Figure 13.6 shows graphically their singular nature at the point o, which contrasts with the surface itself sandwiched in the middle. Figure 13.6: Principal curvatures of the monkey saddle Alternative Formulas Classically, the standard formulas for computing K and H for a patch x are (13.20) and (13.21). It is usually too tedious to compute K and H by hand in one step. Therefore, the functions E, F, G and e, f, g need to be computed before any of the curvature functions are calculated. The computation of E, F , G is straightforward: first one computes the first derivatives xu and xv and then the dot products E = xu · xu , F = xu · xv and G = xv · xv . There are two methods for computing e, f, g. The direct approach using the definitions necessitates computing the surface normal via equation (10.11) on page 295, and then using the definition (13.9). The other method, explained in 13.5. MORE CURVATURE CALCULATIONS 405 the next lemma, avoids computation of the surface normal; it uses instead the vector triple product which can be computed as a determinant. Lemma 13.31. Let x: U → R3 be a regular patch. Then [xuu xu xv ] , e= √ EG − F 2 [xuv xu xv ] f= √ , EG − F 2 [xvv xu xv ] g= √ . EG − F 2 Proof. From (7.4) and (13.9) it follows that e = xuu · U = xuu · xu × xv [xuu xu xv ] . = √ kxu × xv k EG − F 2 The other formulas are proved similarly. On the other hand, at least theoretically, we can compute K and H for a regular patch x directly in terms of the first and second derivatives of x. Here are the relevant formulas. Corollary 13.32. Let x: U → R3 be a regular patch. The Gaussian and mean curvatures of M are given by the formulas (13.26) K = (13.27) H = [xuu xu xv ][xvv xu xv ] − [xuv xu xv ]2 , 2 kxu k2 kxv k2 − (xu · xv )2 [xuu xu xv ]kxv k2 − 2[xuv xu xv ](xu · xv ) + [xvv xu xv ]kxu k2 . 3/2 2 kxu k2 kxv k2 − (xu · xv )2 Proof. The equations follow from (13.20), (13.21) and Lemma 13.31 when we write out e, f, g and E, F, G explicitly in terms of dot products. Equations 13.26 and 13.27 are used effectively in Notebook 13. They are usually too complicated for hand calculation, but we do use them below in the case of the torus. It is neither enlightening nor useful to write out the formulas for the principal curvatures in terms of E, F, G, e, f, g in general. However, there is one special case when such formulas for k1 , k2 are worth noting. Corollary 13.33. Let x: U → R3 be a regular patch for which f = F = 0. With respect to this patch, the principal curvatures are e/E and g/G. Proof. When F = f = 0, the Weingarten equations (13.10) reduce to S(xu ) = e xu E and S(xv ) = g xv . G By definition, e/E and g/G are the eigenvalues of the shape operator S. 406 CHAPTER 13. SHAPE AND CURVATURE The Sphere We compute the principal curvatures of the patch (13.28) x(u, v) = sphere[a](u, v) = a cos v cos u, a cos v sin u, a sin v of the sphere with center o = (0, 0, 0) and radius a. We find that xu (u, v) = xv (u, v) = and so − a cos v sin u, a cos v cos u, 0 , − a sin v cos u, −a sin v sin u, a cos v , E = a2 cos2 v, F = 0, G = a2 . Furthermore, xuu (u, v) = xuv (u, v) = xvv (u, v) = − a cos v cos u, −a cos v sin u, 0 , a sin v sin u, −a sin v cos u, 0 , − a cos v cos u, −a cos v sin u, −a sin v = −x(u, v), and Lemma 13.31 yields   −a cos v cos u −a cos v sin u 0   det −a cos v sin u a cos v cos u 0  −a sin v cos u −a sin v sin u a cos v = −a cos2 v. e= a2 cos v Changing just the first row of the determinant gives f = 0 and g = −a. Therefore, K = a−2 , H = −a−1 , k1 = −a−1 = k2 , and the corresponding Weingarten matrix of the shape operator is ! a−1 0 . 0 a−1 The Circular Torus We compute the Gaussian and mean curvatures of the patch x(u, v) = torus[a, b, b](u, v) = (a + b cos v) cos u, (a + b cos v) sin u, b sin v , representing a torus with circular sections rather than the more general case on pages 210 and 305. 13.5. MORE CURVATURE CALCULATIONS 407 Setting a = 0 and then b = a reduces the parametrization of the torus to that of the sphere (13.28). For this reason, the calculations are only slightly more involved than those above, though we now assume that a > b > 0 and jump to the results e = − cos v(a + b cos v), and K= cos v , b(a + b cos v) k1 = − f = 0, H=− cos v , a + b cos v g = −b, a + 2b cos v , 2b(a + b cos v) 1 k2 = − . b It follows that the Gaussian curvature K of the torus vanishes along the curves given by v = ±π/2. These are the two circles of contact, when the torus is held between two planes of glass. These circles consist exclusively of parabolic points, since the angle featuring in (13.2) on page 390 is π/2, and the normal curvature cannot change sign. The set of hyperbolic points is {x(u, v) 12 π < v < 32 π}, and the set of elliptic points is {x(u, v) − 21 π < v < 21 π}. This situation can be illustrated using commands in Notebook 13 that produce different colors according to the sign of K. The Astroidal Ellipsoid If we modify the standard parametrization of the ellipsoid given on page 313 by replacing each coordinate by its cube, we obtain the astroidal ellipsoid astell[a, b, c](u, v) = (a cos u cos v)3 , (b sin u cos v)3 , (c sin v)3 . Therefore, astell[a, a, a] has the nonparametric equation x2/3 + y 2/3 + z 2/3 = a2 , and is called the astroidal sphere. Figure 13.7 depicts it touching an ordinary sphere. Notebook 13 computes the Gaussian curvature of the astroidal sphere, which is given by K= 1024 sec4 v 2. 9a6 (−18 + 2 cos 4u + cos(4u−2v) + 14 cos 2v + cos(4u+2v)) Surprisingly, this function is continuous on the edges of the astroid, and is singular only at the vertices. This is confirmed by Figure 13.8. 408 CHAPTER 13. SHAPE AND CURVATURE Figure 13.7: The astroidal sphere 4 3 0.5 2 1 0 2 2.2 2.4 -0.5 2.6 Figure 13.8: Gaussian curvature of the astroidal sphere Monge Patches It is not hard to compute directly the Gaussian and mean curvatures of graphs of functions of two variables; see, for example, [dC1, pages 162–163]. However, we can use computations from Notebook 13 to verify the following results: 13.5. MORE CURVATURE CALCULATIONS 409 Lemma 13.34. For a Monge patch (u, v) 7→ u, v, h(u, v) we have E = 1 + h2u , e= huu , (1 + + h2v )1/2 K= G = 1 + h2v , F = hu hv , h2u huu hvv − h2uv , (1 + h2u + h2v )2 f= huv , (1 + + h2v )1/2 h2u H= hvv , (1 + + h2v )1/2 g= h2u (1 + h2v )huu − 2hu hv huv + (1 + h2u )hvv . 2(1 + h2u + h2v )3/2 See Exercise 18 for the detailed general computations. Once the formulsa of Lemma (13.34) have been stored in Notebook 13, they can be applied to specific functions. For example, suppose we wish to determine the curvature of the graph of the function pm,n (u, v) = um v n . (13.29) One quickly discovers that K= m(1 − m − n)nu2m−2 v 2n−2 2. (1 + m2 u2m−2 v 2n−2 + n2 u2m v 2n−2 ) We see from this formula that the graph has nonpositive Gaussian curvature at all points, provided n + m > 1. For definiteness, let us illustrate what happens when m = 2 and n = 4. 0 1 -0.5 -1 0.5 -1.5 -1 0 -0.5 -0.5 0 0.5 1 -1 Figure 13.9: Gaussian curvature of the graph of (u, v) 7→ u2 v 4 Figure 13.9 shows that the Gaussian curvature of p2,4 is everywhere nonpositive. It is easier to see that all points are either hyperbolic or planar by plotting p in polar coordinates; this we do in Figure 13.10. 410 CHAPTER 13. SHAPE AND CURVATURE Figure 13.10: The graph of p2,4 in polar coordinates 13.6 A Global Curvature Theorem We recall the following fundamental fact about compact subsets of Rn (see page 373): Lemma 13.35. Let R be a compact subset of Rn , and let f : R → R be a continuous function. Then f assumes its maximum value at some point p ∈ R. For a proof of this fundamental lemma, see [Buck, page 74]. Intuitively, it is reasonable that for each compact surface M ⊂ R3 , there is a point p0 ∈ M that is furthest from the origin, and at p0 the surface bends towards the origin. Thus it appears that the Gaussian curvature K of M is positive at p0 . We now prove that this is indeed the case. The proof uses standard facts from calculus concerning a maximum of a differentiable function of one variable. Theorem 13.36. If M is a compact regular surface in R3 , there is a point p ∈ R3 at which the Gaussian curvature K is strictly positive. Proof. Let f : R3 → R be defined by f (p) = kpk2 . Then f is continuous (in fact, differentiable), since it can be expressed in terms of the natural coordinate functions of R3 as f = u21 + u22 + u23 . By Lemma 13.35, f assumes its maximum value at some point p0 ∈ M. Let v ∈ Mp be a unit tangent vector, and choose a unit-speed curve α: (a, b) → M such that a < 0 < b, α(0) = p0 and α′ (0) = v. 13.7. NONPARAMETRICALLY DEFINED SURFACES 411 Since the function g : (a, b) → R defined by g = f ◦ α has a maximum at 0, it follows that g ′ (0) = 0 and g ′′ (0) 6 0. But g(t) = α(t) · α(t), so that (13.30) 0 = g ′ (0) = 2α′ (0) · α(0) = 2v · p0 . In (13.30), v can be an arbitrary unit tangent vector, and so p0 must be normal to M at p0 . Clearly, p0 6= 0, so that (13.30) implies that p0 /kp0 k is a unit normal vector to M at p0 . Furthermore, 0 > g ′′ (0) = 2α′′ (0) · α(0) + 2α′ (0) · α′ (0) = 2 α′′ (0) · p0 + 1 , so that α′′ (0) · p0 6 −1, or (13.31) k(v) = α′′ (0) · 1 p0 6− , kp0 k kp0 k where k(v) is the normal curvature determined by the tangent vector v and the unit normal vector p0 /kp0 k. In particular, the principal curvatures of M at p0 (with respect to p0 /kp0 k) satisfy k1 (p0 ), k2 (p0 ) 6 − 1 . kp0 k This implies that the Gaussian curvature of M at p0 satisfies K(p0 ) = k1 (p0 )k2 (p0 ) > 1 > 0. kp0 k2 Noncompact surfaces of positive Gaussian curvature exist (see Exercise 15). On the other hand, for surfaces of negative curvature, we have the following result (see Exercise 16): Corollary 13.37. Any surface in R3 whose Gaussian curvature is everywhere nonpositive must be noncompact. 13.7 Nonparametrically Defined Surfaces So far we have discussed computing the curvature of a surface from its parametric representation. In this section we show how in some cases the curvature can be computed from the nonparametric form of a surface. Lemma 13.38. Let p be a point on a regular surface M ⊂ R3 , and let vp and wp be tangent vectors to M at p. Then the Gaussian and mean curvatures of M at p are related to the shape operator by the formulas (13.32) (13.33) S(vp ) × S(wp ) = K(p)vp × wp , S(vp ) × wp + vp × S(wp ) = 2H(p)vp × wp . 412 CHAPTER 13. SHAPE AND CURVATURE Proof. First, assume that vp and wp are linearly independent. Then we can write S(vp ) = avp + bwp and S(wp ) = cvp + dwp , so that a b c d ! is the matrix of S with respect to vp and wp . It follows from (13.19), page 398, that S(vp ) × wp + vp × S(wp ) = (avp + bwp ) × wp + vp × (cvp + dwp ) = (a + d)vp × wp = (tr S(p))vp × wp = 2H(p)vp × wp , proving (13.33) in the case that vp and wp are linearly independent. If vp and wp are linearly dependent, they are still the limits of linearly independent tangent vectors. Since both sides of (13.33) are continuous in vp and wp , we get (13.33) in the general case. equation (13.32) is proved by the same method (see Exercise 22). Theorem 13.39. Let Z be a nonvanishing vector field on a regular surface M ⊂ R3 which is everywhere perpendicular to M. Let V and W be vector fields tangent to M such that V × W = Z. Then (13.34) K = [Z DV Z DW Z] , kZk4 (13.35) H = [Z W DV Z] − [Z V DW Z] . 2kZk3 Proof. Let U = Z/kZk; then (9.5), page 267, implies that DV Z 1 Z. DV U = +V kZk kZk Therefore, S(V) = −D V U = −DV Z + NV , kZk where NV is a vector field normal to M. By Lemma 13.38 we have (13.36) K V × W = S(V) × S(W) −D V Z −DW Z = + NV × + NW . kZk kZk 13.7. NONPARAMETRICALLY DEFINED SURFACES 413 Since NV and NW are linearly dependent, it follows from (13.36) that KZ = D V Z × DW Z + some vector field tangent to M. kZk2 Taking the scalar product of both sides with Z yields (13.34). Equation (13.35) is proved in a similar fashion. In order to make use of Theorem 13.39, we need an important function that measures the distance from the origin of each tangent plane to a surface. Definition 13.40. Let M be an oriented regular surface in R3 with surface normal U. Then the support function of M is the function h: M → R given by h(p) = p · U(p). Geometrically, h(p) is the distance from the origin to the tangent space Mp . Corollary 13.41. Let M be the surface { (u1 , u2 , u3 ) ∈ R3 | f1 uk1 + f2 uk2 + f3 uk3 = 1 }, where f1 , f2 , f3 are constants, not all zero, and k is a nonzero real number. Then the support function, Gaussian curvature and mean curvature of M are given by (13.37) (13.38) 1 , h = q 2 u2k−2 + f 2 u2k−2 f12 u2k−2 + f 1 2 2 3 3 K = (k − 1)2 f1 f2 f3 (u1 u2 u3 )k−2 = h4 (k − 1)2 f1 f2 f3 (u1 u2 u3 )k−2 , 2 X 3 fi2 u2k−2 i i=1 (13.39) H = −k + 1 k−2 (f1 uk1 + f2 uk2 ) 23 f1 f2 (u1 u2 ) X 3 fi2 u2k−2 2 i i=1 +f2 f3 (u2 u3 )k−2 (f2 uk2 + f3 uk3 ) + f3 f1 (u3 u1 )k−2 (f3 uk3 + f1 uk1 ) . Proof. Let g(u1 , u2 , u3 ) = f1 uk1 + f2 uk2 + f3 uk3 − 1, so that M = { p ∈ R3 | g(p) = 0 }. Then Z = grad g is a nonvanishing vector field that is everywhere perpendicular to M. Explicitly, 3 X fi uik−1 Ui , Z=k i=1 414 CHAPTER 13. SHAPE AND CURVATURE u1 , u2 , u3 being the natural coordinate functions of R3 , and Ui = ∂/∂ui (see P Definition 9.20). The vector field X = ui Ui satisfies X (13.40) X·Z = k fi uki , in which the summations continue to be over i = 1, 2, 3. The support function of M is given by X fi uki Z = qX . h = X· kZk f 2 u2k−2 i i Since fi uki equals 1 on M, we get (13.37). Next, let X X V= vi Ui and W = wi Ui P be vector fields on R3 . Since f1 , f2 , f3 are constants, we have X X ]Ui = k(k − 1) fi vi uik−2 Ui , DV Z = k V[fi uk−1 i and similarly for W. Therefore, the triple product [Z D V Z D W Z] equals   k f1 u1k−1 k f2 u2k−1 k f3 u3k−1   det k(k − 1)f1 v1 u1k−2 k(k − 1)f2 v2 u2k−2 k(k − 1)f3 v3 u3k−2  k(k − 1)f1 w1 u1k−2 k(k − 1)f2 w2 u2k−2 k(k − 1)f3 w3 u3k−2 = k 3 (k − 1)2 f1 f2 f3 u1k−2 u2k−2 u3k−2 [X V W]. Now we choose V and W so that they are tangent to M and V × W = Z. Using (13.34), we obtain K= k 3 (k − 1)2 f1 f2 f3 u1k−2 u2k−2 u3k−2 X · Z (k − 1)2 f1 f2 f3 u1k−2 u2k−2 u3k−2 = . 2 X 4 3 kZk 2 2k−2 fi u i i=1 This proves (13.38). The proof of (13.39) is similar. Computations in Notebook 13 yield three special cases of Corollary 13.41. Corollary 13.42. The support function and Gaussian curvature of (i) the ellipsoid y2 z2 x2 + + = 1, a2 b2 c2 (ii) the hyperboloid of one sheet (iii) the hyperboloid of two sheets y2 z2 x2 + − = 1, a2 b2 c2 x2 y2 z2 − 2 − 2 =1 2 a b c 13.8. EXERCISES 415 are given in each case by h= y2 z2 x2 + 4 + 4 4 a b c −1/2 and K=± h4 a 2 b 2 c2 , where the minus sign only applies in (ii). We shall return to these quadrics in Section 19.6, where we describe geometrically useful parametrizations of them. We conclude the present section by considering a superquadric, which is a surface of the form f1 xk + f2 y k + f3 z k = 1, where k is different from 2. We do the special case k = 2/3, which is an astroidal ellipsoid (see page 407). Corollary 13.43. The Gaussian curvature of the superquadric f1 x2/3 + f2 y 2/3 + f3 z 2/3 = 1 is given by K= f1 f2 f3 2 . 2 2/3 9 f3 (xy) + f22 (xz)2/3 + f12 (yz)2/3 13.8 Exercises M 1. Plot the graph of p2,3 , defined by (13.29), and its Gaussian curvature. M 2. Plot the following surfaces and describe in each case (without additional calculation) the set of elliptic, hyperbolic, parabolic and planar points: (a) a sphere, (b) an ellipsoid, (c) an elliptic paraboloid, (d) a hyperbolic paraboloid, (e) a hyperboloid of one sheet, (f) a hyperboloid of two sheets. This exercise continues on page 450. 3. By studying the plots of the surfaces listed in Exercise 2, describe the general shape of the image of their Gauss maps. 416 CHAPTER 13. SHAPE AND CURVATURE 4. Compute by hand the coefficients of the first fundamental form, those of the second fundamental form, the unit normal, the mean curvature and the principal curvatures of the following surfaces: (a) the elliptical torus, (b) the helicoid, (c) Enneper’s minimal surface. Refer to the exercises on page 377. 5. Compute the first fundamental form, the second fundamental form, the unit normal, the Gaussian curvature, the mean curvature and the principal curvatures of the patch x(u, v) = u2 + v, v 2 + u, u v . 6. The translation surface determined by curves α, γ : (a, b) → R3 is the patch (u, v) 7→ α(u) + γ(v). It is the surface formed by moving α parallel to itself in such a way that a point of the curve moves along γ. Show that f = 0 for a translation surface. 7. Explain the difference between the translation surface formed by a circle and a lemniscate lying in perpendicular planes, and the twisted surface formed from a lemniscate according to Section 11.6. Figure 13.11: Translation and twisted surface formed by a lemniscate 8. Compute by hand the first fundamental form, the second fundamental form, the unit normal, the Gaussian curvature, the mean curvature and the principal curvatures of the patch xn (u, v) = un , v n , u v). For n = 3, see the picture on page 291. 13.8. EXERCISES 417 9. Prove the statement immediately following Definition 13.8. 10. Show that the first fundamental form of the Gauss map of a patch x coincides with the third fundamental form of x. 11. Show that an orientation-preserving Euclidean motion F : R3 → R3 leaves unchanged both principal curvatures and principal vectors. 12. Show that the Bohemian dome defined by bohdom[a, b, c, d](u, v) = (a cos u, b sin u + c cos v, d sin v) is the translation surface of two ellipses (see Exercise 6). Compute by hand the Gaussian curvature, the mean curvature and the principal curvatures of bohdom[a, b, b, a]. Figure 13.12: The Bohemian dome bohdom[1, 2, 2, 3] 13. Show that the mean curvature H(p) of a surface M ⊂ R3 at p ∈ M is given by Z 1 π k(θ)dθ, H(p) = π 0 where k(θ) denotes the normal curvature, as in the proof of Lemma 13.19. 14. (Continuation) Let n be an integer larger than 2 and for 0 6 i 6 n − 1, put θi = ψ + 2π i/n, where ψ is some angle. Show that H(p) = n−1 1X k(θi ). n i=0 418 CHAPTER 13. SHAPE AND CURVATURE 15. Give examples of a noncompact surface (a) whose Gaussian curvature is negative, (b) whose Gaussian curvature is identically zero, (c) whose Gaussian curvature is positive, (d) containing elliptic, hyperbolic, parabolic and planar points. 16. Prove Corollary 13.37 under the assumption that K is everywhere negative. 17. Show that there are no compact minimal surfaces in R3 . M 18. Find the coefficients the first, second and third fundamental forms to the following surfaces: (a) the elliptical torus defined on page 377, (b) the helicoid defined on page 377, (c) Enneper’s minimal surface defined on page 378, (d) a Monge patch defined on page 302. 19. Determine the principal curvatures of the Whitney umbrella. Its mean curvature is shown in Figure 13.13. 1000 500 0.015 0 0.014 -500 0.013 -1000 -0.1 0.012 -0.05 0 0.011 0.05 0.1 0.01 Figure 13.13: Gaussian curvature of the Whitney umbrella 13.8. EXERCISES 419 M 20. Find the formulas for the coefficients e, f, g of the second fundamental forms of the following surfaces parametrized in Chapter 11: the Möbius strip, the Klein bottle, Steiner’s Roman surface, the cross cap. M 21. Find the formulas for the Gaussian and mean curvatures of the surfaces of the preceding exercise. 22. Prove equation (13.32). 23. Prove Lemma 13.44. Let M ⊂ R3 be a surface, and suppose that x: U → M and y : V → M are coherently oriented patches on M with x(U) ∩ y(V) nonempty. Let x−1 ◦ y = (ū, v̄): U ∩ V → U ∩ V be the associated change of coordinates, so that y(u, v) = x ū(u, v), v̄(u, v) . Let ex , fx , gx denote the coefficients of the second fundamental form of x, and let ey , fy , gy denote the coefficients of the second fundamental form of y. Then  2 2 ∂ ū ∂v̄ ∂v̄ ∂ ū    ey = ex + 2fx , + gx   ∂u ∂u ∂u ∂u     ∂v̄ ∂v̄ ∂ ū ∂v̄ ∂ ū ∂v̄ ∂ ū ∂ ū (13.41) + gx + fx + , fy = ex  ∂u ∂v ∂u ∂v ∂v ∂u ∂u ∂v    2 2   ∂ ū ∂v̄ ∂v̄ ∂ ū    gy = ex + 2fx . + gx ∂v ∂v ∂v ∂v 24. Prove equation (13.39) of Corollary 13.41. Chapter 14 Ruled Surfaces We describe in this chapter the important class of surfaces, consistng of those which contain infinitely many straight lines. The most obvious examples of ruled surfaces are cones and cylinders (see pages 314 and 375). Ruled surfaces are arguably the easiest of all surfaces to parametrize. A chart can be defined by choosing a curve in R3 and a vector field along that curve, and the resulting parametrization is linear in one coordinate. This is the content of Definition 14.1, that provides a model for the definition of both the helicoid and the Möbius strip (described on pages 376 and 339). Several quadric surfaces, including the hyperbolic paraboloid and the hyperboloid of one sheet are also ruled, though this fact does not follow so readily from their definition. These particular quadric surfaces are in fact ‘doubly ruled’ in the sense that they admit two one-parameter families of lines. In Section 14.1, we explain how to parametrize them in such a way as to visualize the straight line rulings. We also define the Plücker conoid and its generalizations. It is a straightforward matter to compute the Gaussian and mean curvature of a ruled surface. This we do in Section 14.2, quickly turning attention to flat ruled surfaces, meaning ruled surfaces with zero Gaussian curvature. There are three classes of such surfaces, the least obvious but most interesting being the class of tangent developables. Tangent developables are the surfaces swept out by the tangent lines to a space curve, and the two halves of each tangent line effectively divide the surface into two sheets which meet along the curve in a singular fashion described by Theorem 14.7. This behaviour is examined in detail for Viviani’s curve, and the linearity inherent in the definition of a ruled surface enables us to highlight the useful role played by certain plane curves in the description of tangent developables. In the final Section 14.4, we shall study a class of surfaces that will help us to better understand how the Gaussian curvature varies on a ruled surface. 431 432 CHAPTER 14. RULED SURFACES 14.1 Definitions and Examples A ruled surface is a surface generated by a straight line moving along a curve. Definition 14.1. A ruled surface M in R3 is a surface which contains at least one 1-parameter family of straight lines. Thus a ruled surface has a parametrization x: U → M of the form (14.1) x(u, v) = α(u) + v γ(u), where α and γ are curves in R3 . We call x a ruled patch. The curve α is called the directrix or base curve of the ruled surface, and γ is called the director curve. The rulings are the straight lines v 7→ α(u) + v γ(u). In using (14.1), we shall assume that α′ is never zero, and that γ is not identically zero. We shall see later that any straight line in a surface is necessarily an asymptotic curve, pointing as it does in direction for which the normal curvature vanishes (see Definition 13.9 on page 390 and Corollary 18.6 on page 560). It follows that the rulings are asymptotic curves. Sometimes a ruled surface M has two distinct ruled patches on it, so that a ruling of one patch does not belong to the other patch. In this case, we say that x is doubly ruled. We now investigate a number of examples. The Helicoid and Möbius Strip Revisited We can rewrite the definition (12.27) as helicoid[a, b](u, v) = α(u) + v γ(u), where (14.2) ( α(u) = (0, 0, b u), γ(u) = a(cos u, sin u, 0). In this way, helicoid[a, b] is a ruled surface whose base curve has the z-axis as its trace, and director curve γ that describes a circle. In a similar fashion, our definition (11.3), page 339, of the Möbius strip becomes moebiusstrip(u, v) = α(u) + v γ(u), where   α(u) = (cos u, sin u, 0), (14.3)  γ(u) = cos u cos u, cos u sin u, sin u . 2 2 2 This time, it is the circle that is the base curve of the ruled surface. The Möbius strip has a director curve γ which lies on a unit sphere, shown in two equivalent ways in Figure 14.1. 14.1. DEFINITIONS AND EXAMPLES 433 Figure 14.1: Director curve for the Möbius strip The curve γ is not one that we encountered in the study of curves on the sphere in Section 8.4. It does however have the interesting property that whenever p belongs to its trace, so does the antipodal point −p. The Hyperboloid of One Sheet We have seen that the elliptical hyperboloid of one sheet is defined nonparametrically by x2 y2 z2 (14.4) + 2 − 2 = 1. 2 a b c Planes perpendicular to the z-axis intersect the surface in ellipses, while planes parallel to the z-axis intersect it in hyperbolas. We gave the standard parametrization of the hyperboloid of one sheet on page 313, but this has the disadvantage of not showing the rulings. Let us show that the hyperboloid of one sheet is a doubly-ruled surface by finding two ruled patches on it. This can be done by fixing a, b, c > 0 and defining x± (u, v) = α(u) ± v α′ (u) + v (0, 0, c), where (14.5) α(u) = ellipse[a, b](u) = (a cos u, b sin u, 0). is the standard parametrization of the ellipse y2 x2 + =1 a2 b2 in the xy-plane. It is readily checked that (14.6) x± (u, v) = a(cos u ∓ v sin u), b(sin u ± v cos u), cv , 434 CHAPTER 14. RULED SURFACES and that both x+ and x− are indeed parametrizations of the hyperboloid (14.4). Hence the elliptic hyperboloid is doubly ruled; in both cases, the base curve can be taken to be the ellipse (14.5). We also remark that x+ can be obtained from x− by simultaneously changing the signs of the parameter c and the variable v. Figure 14.2: Rulings on a hyperboloid of one sheet The Hyperbolic Paraboloid The hyperbolic paraboloid is defined nonparametrically by z= (14.7) x2 y2 − a2 b2 (though the constants a, b here are different from those in (10.16)). It is doubly ruled, since it can be parametrized in the two ways x± (u, v) = = (au, 0, u2 ) + v(a, ±b, 2u) (a(u + v), ±b v, u2 + 2uv). Although we have tacitly assumed that b > 0, both parametrizations can obviously be obtained from the same formula by changing the sign of b, a fact exploited in Notebook 14. A special case corresponds to taking a = b and carrying out a rotation by π/2 about the z-axis, so as to define new coordinates  1  x̄ = √ (x − y)   2   1 ȳ = √ (x + y),   2    z̄ = z. This transforms (14.7) into the equation a2 z̄ = 2 x̄ȳ. 14.1. DEFINITIONS AND EXAMPLES 435 Figure 14.3: Rulings on a hyperbolic paraboloid Plücker’s Conoid The surface defined nonparametrically by z= 2xy x2 + y 2 is called Plücker’s conoid1 [BeGo, pages 352,363]. Its Monge parametrization is obviously 2uv (14.8) pluecker(u, v) = u, v, 2 u + v2 A computer plot using (14.8) does not reveal any rulings (Figure 14.4, left). To see that this conoid is in fact ruled, one needs to convert (u, v) to polar coordinates, as explained in Section 10.4. Let us write pluecker(r cos θ, r sin θ) = (r cos θ, r sin θ, 2 cos θ sin θ) = (0, 0, sin 2θ) + r(cos θ, sin θ, 0). Thus, the z-axis acts as base curve and the circle θ 7→ (cos θ, sin θ) as director curve for the parametrization in terms of (r, θ). Using this parametrization, the rulings are clearly visible passing through the z-axis (Figure 14.4, right). 1 Julius Plücker (1801–1868). German mathematician. Until 1846 Plücker’s original research was in analytic geometry, but starting in 1846 as professor of physics in Bonn, he devoted his energies to experimental physics for nearly twenty years. At the end of his life he returned to mathematics, inventing line geometry. 436 CHAPTER 14. RULED SURFACES Figure 14.4: Parametrizations of the Plücker conoid It is now an easy matter to define a generalization of Plücker’s conoid that has n folds instead of 2: (14.9) plueckerpolar[n](r, θ) = (r cos θ, r sin θ, sin nθ). Each surface plueckerpolar[n] is a ruled surface with the rulings passing through the z-axis. For a generalization of (14.9) that includes a variant of the monkey saddle as a special case, see Exercise 5. Even more general than (14.9) is the right conoid, which is a ruled surface with rulings parallel to a plane and passing through a line that is perpendicular to the plane. For example, if we take the plane to be the xy-plane and the line to be the z-axis, a right conoid will have the form (14.10) rightconoid[ϑ, h](u, v) = v cos ϑ(u), v sin ϑ(u), h(u) . This is investigated in Notebook 14 and illustrated in Figure 14.12 on page 449. Figure 14.5: The conoids plueckerpolar[n] with n = 3 and 7 14.2. CURVATURE OF A RULED SURFACE 437 14.2 Curvature of a Ruled Surface Lemma 14.2. The Gaussian curvature of a ruled surface M ⊂ R3 is every- where nonpositive. Proof. If x is a ruled patch on M, then xvv = 0; consequently g = 0. Hence it follows from Theorem 13.25, page 400, that (14.11) K= −f 2 6 0. EG − F2 As an example, we consider the parametrization (14.3) of the Möbius strip. The coefficients of its first and second fundamental forms, its Gaussian and mean curvatures are all computed in Notebook 14. For example, with a = 2 as in Figure 14.6, we obtain K(u, v) = − 16 16 + 3v 2 + 16v cos u2 + 2v 2 cos u 2 8 + 2v 2 + 8v cos u2 + v 2 cos u sin u2 H(u, v) = −2 3/2 . 16 + 3v 2 + 16v cos u2 + 2v 2 cos u It is clear that, for this particular parametrization, K is never zero. The righthand plot of K in Figure 14.6 exhibits its minima, which correspond to definite regions of the strip which are especially distorted. Figure 14.6: Curvature of a Möbius strip Recall that a flat surface is a surface whose Gaussian curvature is everywhere zero. Such a surface is classically called a developable surface. An immediate consequence of (14.11) is the following criterion for flatness. Corollary 14.3. M is flat if and only if f = 0. 438 CHAPTER 14. RULED SURFACES The most obvious examples of flat surfaces, other than the plane, are circular cylinders and cones (see the discussion after Figure 14.7). Paper models of both can easily be constructed by bending a sheet of paper and, as we shall see in Section 17.2, this operation leaves the Gaussian curvature unchanged and identically zero. Circular cylinders and cones are subsumed into parts (ii) and (iii) of the following list of flat ruled surfaces. Definition 14.4. Let M ⊂ R3 be a surface. Then: (i) M is said to be the tangent developable of a curve α: (a, b) → R3 if M can be parametrized as (14.12) x(u, v) = α(u) + v α′ (u); (ii) M is a generalized cylinder over a curve α: (a, b) 7→ R3 if M can be parametrized as y(u, v) = α(u) + v q, where q ∈ R3 is a fixed vector; (iii) M is a generalized cone over a curve α: (a, b) → R3 , provided M can be parametrized as z(u, v) = p + v α(u), where p ∈ R3 is fixed (it can be interpreted as the vertex of the cone). Our next result gives criteria for the regularity of these three classes of surfaces. We base it on Lemma 10.18. (i) Let α : (a, b) → R3 be a regular curve whose curvature κ[α] is everywhere nonzero. The tangent developable x of α is regular everywhere except along α. Lemma 14.5. (ii) A generalized cylinder y(u, v) = α(u) + v q is regular wherever α′ × q does not vanish. (iii) A generalized cone z(u, v) = p + v α(u) is regular wherever v α × α′ is nonzero, and is never regular at its vertex. Proof. For a tangent developable x, we have (14.13) (xu × xv )(u, v) = (α′ + v α′′ ) × α′ = v α′′ × α′ . If κ[α] 6= 0, then α′′ × α′ is everywhere nonzero by (7.26). Thus (14.13) implies that x is regular whenever v 6= 0. The other statements have similar proofs. 14.2. CURVATURE OF A RULED SURFACE 439 Figure 14.7: Cylinder and cone over a figure eight Figure 14.7 illustrates cases (ii) and (iii) of Definition 14.4. Proposition 14.6. If M is a tangent developable, a generalized cylinder or a generalized cone, then M is flat. Proof. By Corollary 14.3, it suffices to show that f = 0 in each of the three cases. For a tangent developable x, we have xu = α′ + v α′′ , xv = α′ and xuv = α′′ ; hence the triple product f= [α′′ (α′ +v α′′ ) α′ ] kxu × xv k is zero. It is obvious that f = 0 for a generalized cylinder y, since yuv = 0. Finally, for a generalized cone, we compute f= [α′ vα′ α] = 0. kxu × xv k The general developable surface is in some sense the union of tangent developables, generalized cylinders and generalized cones. This remark is explained in the paragraph directly after the proof of Theorem 14.15. The normal surface and the binormal surface to a space curve α can be defined by mimicking the construction of the tangent developable. It suffices to replace the tangent vector α′ (u) in (14.12) by the unit normal or binormal vector N(u) or B(u). However, unlike a tangent developable, the normal and binormal surfaces are not in general flat. Consider the normal and binormal surfaces to Viviani’s curve, defined in Section 7.5. Output from Notebook 14 gives the following expression for the binormal surface of viviani[1]: 440 CHAPTER 14. RULED SURFACES √ −2 2 v cos3 u2 v 3 sin u + sin 3u 2 , √ + sin u, 1 + cos u + √ 2 26 + 6 cos u 13 + 3 cos u √ 2 2v u √ + 2 sin 2 13 + 3 cos u Figure 14.8 shows parts of the normal and binormal surfaces together, with the small gap representing Viviani’s curve itself. The curvature of these surfaces is computed in Notebook 14, and shown to be nonzero. Figure 14.8: Normal and binormal surfaces to Viviani’s curve 14.3 Tangent Developables Consider the surface tandev[α] = α(u) + v α′ (u); this was first defined in (14.12), though we now use notation from Notebook 14. We know from Lemma 14.5 that tandev[α] is singular along the curve α. We prove next that it is made up of two sheets which meet along the trace of α in a sharp edge, called the edge of regression. Theorem 14.7. Let α : (a, b) → R3 be a unit-speed curve with a < 0 < b, and let x be the tangent developable of α. Suppose that α is differentiable at 0 and that the curvature and torsion of α are nonzero at 0. Then the intersection of the trace of x with the plane perpendicular to α at α(0) is approximated by a semicubical parabola with a cusp at α(0). 14.3. TANGENT DEVELOPABLES 441 Figure 14.9: Tangent developable to a circular helix Proof. Since α has unit speed, we have α′ (s) = T(s). The Frenet formulas (Theorem 7.10 on page 197) tell us that α′′ (s) = κ(s)N(s) and moreover α′′′ (s) = −κ(s)2 T(s) + κ′ (s)N(s) + κ(s)τ (s)B(s), α′′′′ (s) = −3κ(s)κ′ (s)T(s) + − κ(s)3 + κ′′ (s) − κ(s)τ (s)2 N(s) + 2κ′ (s)τ (s) + κ(s)τ ′ (s) B(s). We next substitute these formulas into the power series expansion α(s) = α0 + sα′0 + where the subscript (14.14) s2 ′′ s3 ′′′ s4 ′′′′ α0 + α0 + α0 + O s5 , 2 6 24 denotes evaluation at s = 0. We get s4 s3 α(s) = α0 + T0 s − κ02 − κ0 κ′0 6 8 2 s s3 s4 + N0 κ0 + κ′0 + − κ03 + κ′′0 − κ0 τ 02 2 6 24 3 4 s s + B0 2κ′0 τ 0 + κ0 τ ′0 + O s5 . κ0 τ 0 + 6 24 0 This is a significant formula that gives a local expression for a space curve relative to a fixed Frenet frame; see [dC1, §1-6]. Taking the derivative of (14.14) 442 CHAPTER 14. RULED SURFACES gives s2 s3 α′ (s) = T0 1 − κ02 − κ0 κ′0 2 2 s3 s2 − κ03 + κ′′0 − κ0 τ 02 + N0 sκ0 + κ′0 + 2 6 2 3 s s + B0 κ0 τ 0 + 2κ′0 τ 0 + κ0 τ ′0 + O s4 . 2 6 The tangent developable of α is therefore given by x(u, v) = α(u) + v α′ (u) u3 u2 v 2 u3 v κ0 − κ0 κ′0 = α0 + T0 u − κ02 + v − 6 2 2 2 u u3 u2 v ′ u3 v + N0 − κ03 + κ′′0 − κ0 τ 02 κ0 + κ′0 + u v κ0 + κ0 + 2 6 2 6 3 u u3 v u2 v ′ ′ + B0 κ0 τ 0 + 2κ0 τ 0 + κ0 τ 0 + O u4 . + 6 2 6 We want to determine the intersection of x with the plane perpendicular to α at α0 . Therefore, we set the above coefficient of T0 equal to zero and solve for v to give v = − u − 61 u3 κ02 + O u4 u3 + O u4 . = −u − κ02 ′ 4 1 2 2 1 3 3 1 − 2 u κ0 − 2 u κ0 κ0 + O u Substituting back this value yields the following power series expansions: (14.15)  u2    x = coefficient of N0 = − κ0 − · · · 2 3    y = coefficient of B = − u κ τ + · · · 0 0 0 3 By hypothesis, κ0 6= 0 6= τ 0 , so that we can legitimately ignore higher order terms. Doing this, the plane curve described by (14.15) is approximated by the implicit equation 8τ 02 x3 + 9κ0 y 2 = 0, which is a semicubical parabola. 14.3. TANGENT DEVELOPABLES 443 Figure 14.10: Tangent developable to Viviani’s curve Figures 14.10 illustrates a case where the hypotheses of Theorem 14.7 are not valid. It shows the plane generated by the normal N and binormal B unit vectors to Viviani’s curve t (14.16) α(t) = 1 − cos t, − sin t, 2 cos , −3π 6 t 6 π 2 at α(0) = (0, 0, 2). We have reparametrized the curve (7.35) on page 207 so that the point at which the torsion vanishes has parameter value 0 rather than π (see Figure 7.7). We can find the intersection of this plane with the tangent developable explicitly, using the following general considerations. We seek points x(u, v) = α(u) + vα′ (u) on the tangent developable surface satisfying the orthogonality condition (x(u, v) − α(0)) · α′ (0) = 0. The resulting equation is linear in v, and has solution v = v(u) = (α(0) − α(u)) · α′ (0) . α′ (u) · α′ (0) The cuspidal section is now represented by the plane curve (x, y) where ( x(u) = (x(u, v(u)) − α(0)) · N, y(u) = (x(u, v(u)) − α(0)) · B. 444 CHAPTER 14. RULED SURFACES 2.2 2.1 -4 -3 -2 -1 1.9 1.8 Figure 14.11: Viviani cusps for t = 0.3, 0.2, 0.1 and (in center) t = 0 Carrying out the calculations for (14.16) in Notebook 14 at t = 0 shows that (x, y) equals 1 u 3u u 3u √ − 4 + 4 cos u − 3 cos − cos , −2 + 2 cos u + 6 cos + 2 cos . 2 2 2 2 2 5 cos u It follows that (x(−u), y(−u)) = (x(u), y(u)), and so the two branches of the cusp coincide. The situation is illustrated by Figure 14.11 for neighboring values of t. For a fuller discussion of the two sheets that form the tangent developable, see [Spivak, vol 3, pages 207–213] and [Stru2, pages 66–73]. The following lemma is easy to prove. Lemma 14.8. Let β : (c, d) → R3 be a unit-speed curve. The metric of the tangent developable tandev[β] depends only on the curvature of β. Explicitly: E = 1 + v 2 κ[β]2 , F = 1, G = 1. The helix on page 200 has the same constant curvature as a circle of radius (a2 + b2 )/a. Lemma 14.8 can then be used to provide a local isometry between a portion of the plane (the tangent developable to the circle) and the tangent developable to the helix. 14.4 Noncylindrical Ruled Surfaces We shall now impose the following hypothesis on a class of ruled surfaces to be studied further. Definition 14.9. A ruled surface parametrized by x(u, v) = β(u)+v γ(u) is said to be noncylindrical provided γ × γ ′ never vanishes. 14.4. NONCYLINDRICAL RULED SURFACES 445 The rulings are always changing directions on a noncylindrical ruled surface. We show how to find a useful reference curve on a noncylindrical ruled surface. This curve, called a striction curve, is a generalization of the edge of regression of a tangent developable. e be a parametrization of a noncylindrical ruled surface of Lemma 14.10. Let x e(u, v) = β(u) + v γ(u). Then x e has a reparametrization of the form the form x (14.17) x(u, v) = σ(u) + v δ(u), e. where kδk = 1 and σ ′ · δ ′ = 0. The curve σ is called the striction curve of x Proof. Since γ×γ ′ is never zero, γ is never zero. We define a reparametrization e e of x e by x v γ(u) v e e(u, v) = x e u, = β(u) + . x kγ(u)k kγ(u)k e e has the same trace as x e. If we put δ(u) = γ(u)/kγ(u)k, then Clearly, x e e(u, v) = β(u) + v δ(u). x Furthermore, kδ(u)k = 1 and so δ(u) · δ ′ (u) = 0. Next, we need to find a curve σ such that σ ′ (u) · δ ′ (u) = 0. To this end, we write (14.18) σ(u) = β(u) + t(u)δ(u) for some function t = t(u) to be determined. We differentiate (14.18), obtaining σ ′ (u) = β′ (u) + t′ (u)δ(u) + t(u)δ ′ (u). Since δ(u) · δ ′ (u) = 0, it follows that σ ′ (u) · δ ′ (u) = β′ (u) · δ ′ (u) + t(u)δ ′ (u) · δ ′ (u). Since γ × γ ′ never vanishes, γ and γ ′ are always linearly independent, and consequently δ ′ never vanishes. Thus if we define t by (14.19) t(u) = − β′ (u) · δ ′ (u) , kδ ′ (u)k2 we get σ ′ (u) · δ ′ (u) = 0. Now define e e(u, t(u) + v). x(u, v) = x e e and x e all Then x(u, v) = β(u) + (t(u) + v)δ(u) = σ(u) + v δ(u), so that x, x have the same trace, and x satisfies (14.17). 446 CHAPTER 14. RULED SURFACES Lemma 14.11. The striction curve of a noncylindrical ruled surface x does not depend on the choice of base curve. e be two base curves for x. In the notation of the previous Proof. Let β and β proof, we may write (14.20) e β(u) + v δ(u) = β(u) + w(v)δ(u) e be the corresponding striction curves. for some function w = w(v). Let σ and σ Then β′ (u) · δ ′ (u) δ(u) σ(u) = β(u) − kδ ′ (u)k2 and e ′ (u) · δ ′ (u) β e e (u) = β(u) σ − δ(u), kδ ′ (u)k2 so that ′ ′ e′ e − (β − β ) · δ δ. e =β−β (14.21) σ−σ ′ 2 kδ k On the other hand, it follows from (14.20) that (14.22) e = (w(v) − v)δ. β−β The result follows by substituting (14.22) and its derivative into (14.21). There is a nice geometric interpretation of the striction curve σ of a ruled surface x, which we mention without proof. Let ε > 0 be small. Since nearby rulings are not parallel to each other, there is a unique point P (ε) on the straight line v 7→ x(u, v) that is closest to the line v 7→ x(u+ε, v). Then P (ε) → σ(u) as ε → 0. This follows because (14.19) is the equation that results from minimizing kσ ′ k to first order. Definition 14.12. Let x be a noncylindrical ruled surface given by (14.17). Then the distribution parameter of x is the function p = p(u) defined by (14.23) p= [σ ′ δ δ ′ ] . δ ′ · δ′ Whilst the definition of σ requires σ ′ to be perpendicular to δ ′ , the function p measures the component of σ ′ perpendicular to δ × δ ′ . Lemma 14.13. Let M be a noncylindrical ruled surface, parametrized by a patch x of the form (14.17). Then x is regular whenever v 6= 0, or when v = 0 and p(u) 6= 0. Furthermore, the Gaussian curvature of x is given in terms of its distribution parameter by (14.24) K= −p(u)2 p(u)2 + v 2 2 . 14.4. NONCYLINDRICAL RULED SURFACES 447 Also, E = kσ ′ k2 + v 2 kδ ′ k2 , F = σ ′ · δ, G = 1, ′ 2 E G − F 2 = (p2 + v 2 )kδ k , and g = 0, pkδ ′ k f= p . p2 + v 2 Proof. First, we observe that both σ ′ × δ and δ ′ are perpendicular to both δ and σ ′ . Therefore, σ ′ × δ must be a multiple of δ ′ , and σ ′ × δ = pδ ′ where p= [σ ′ δ δ ′ ] . δ′ · δ′ Since xu = σ ′ + v δ ′ and xv = δ, we have xu × xv = pδ ′ + v δ ′ × δ, so that kxu × xv k2 = kpδ ′ k2 + kv δ ′ × δk2 = (p2 + v 2 )kδ ′ k2 . It is now clear that the regularity of x is as stated. Next, xuv = δ ′ and xvv = 0, so that g = 0 and f= Therefore, [xuv xu xv ] pkδ ′ k δ ′ · (pδ ′ + v δ ′ × δ) p p = . = kxu × xv k p2 + v 2 kδ ′ k p2 + v 2 K= pkδ ′ k − p p2 + v 2 !2 −f 2 , = kxu × xv k2 (p2 + v 2 )kδ ′ k2 which simplifies into (14.24). Equation (14.24) tells us that the Gaussian curvature of a noncylindrical ruled surface is generally negative. However, more can be said. Corollary 14.14. Let M be a noncylindrical ruled surface given by (14.17) with distribution parameter p, and Gaussian curvature K(u, v). (i) Along a ruling (so u is fixed), K(u, v) → 0 as v → ∞. (ii) K(u, v) = 0 if and only if p(u) = 0. (iii) If p never vanishes, then K(u, v) is continuous and |K(u, v)| assumes its maximum value 1/p2 at v = 0. Proof. All of these statements follow from (14.24). 448 CHAPTER 14. RULED SURFACES Next, we prove a partial converse of Proposition 14.6. Theorem 14.15. Let x(u, v) = β(u) + v δ(u) with kδ(u)k = 1 parametrize a flat ruled surface M. (i) If β ′ (u) ≡ 0, then M is a cone. (ii) If δ ′ (u) ≡ 0, then M is a cylinder. (iii) If both β ′ and δ ′ never vanish, then M is the tangent developable of its striction curve. Proof. Parts (i) and (ii) are immediate from the definitions, so it suffices to prove (iii). We can assume that β is a unit-speed striction curve, so that β ′ · δ ′ ≡ 0. (14.25) Since K ≡ 0, it follows from (14.24) and (14.23) that [β ′ δ δ ′ ] ≡ 0. (14.26) Then (14.25) and (14.26) imply that β′ and δ are collinear. Of course, cases (i), (ii) and (iii) of Theorem 14.15 do not exhaust all of the possibilities. If there is a clustering of the zeros β or δ, the surface can be complicated. In any case, away from the cluster points a developable surface is the union of pieces of cylinders, cones and tangent developables. Indeed, the following result is proved in [Krey1, page 185]. Every flat ruled patch (u, v) 7→ x(u, v) can be subdivided into sufficiently small u-intervals so that the portion of the surface corresponding to each interval is a portion of one of the following: a plane, a cylinder, a cone, a tangent developable. Examples of Striction Curves The parametrization (12.27) of the circular helicoid can be rewritten as x(u, v) = (0, 0, b u) + av(cos u, sin u, 0), which shows that it is a ruled surface. The striction curve σ and director curve δ are given by σ(u) = (0, 0, b u) Then and δ(u) = (cos u, sin u, 0). v x u, = σ(u) + v δ(u) = (v cos u, v sin u, b u), a and the distribution parameter assumes the constant value b. 14.5. EXERCISES 449 The hyperbolic paraboloid (10.12), page 296, when parametrized as x(u, v) = (u, 0, 0) + v(0, 1, u), has σ(u) = (u, 0, 0) as its striction curve and (0, 1, u) δ(u) = √ 1 + u2 as its director curve. Thus x(u, v p 1 + u2 ) = σ(u) + v δ(u), and the distribution parameter is given by p(u) = 1 + u2 . Figure 14.12: Right conoid 14.5 Exercises 1. Compute the Gaussian and mean curvatures of the generalized hyperbolic paraboloid defined and plotted on page 434. M 2. Compute the Gaussian and mean curvatures of the right conoid defined by equation (14.10) and illustrated in Figure 14.12. 3. Explain why Lemma 14.2 is also a consequence of the fact (mentioned on page 432) that a ruled surface has asymptotic curves, namely, the rulings. 450 CHAPTER 14. RULED SURFACES M 4. Further to Exercise 2 of the previous chapter, describe the sets of elliptic, hyperbolic, parabolic and planar points for the following surfaces: (a) a cylinder over an ellipse, (b) a cylinder over a parabola, (c) a cylinder over a hyperbola, (d) a cylinder over y = x3 , (e) a cone over a circle. M 5. Compute the Gaussian and mean curvature of the patch plueckerpolar[m, n, a](r, θ) = a r cos θ, r sin θ, rm sin nθ that generalizes plueckerpolar on page 436. Relate the case m = n = 3 to the monkey saddle on page 304. 6. Complete the proof of Lemma 14.5 and prove Lemma 14.8. M 7. Wallis’ conical edge2 is defined by p wallis[a, b, c](u, v) = v cos u, v sin u, c a2 − b2 cos2 u . Show that wallis[a, b, c] is a right conoid, as in (14.10). Figure 14.13 illus√ trates the case a = 1 − c and b = 3. Compute and plot its Gaussian and mean curvatures. Figure 14.13: Back and front of Wallis’s conical edge 2 John Wallis (1616–1703). English mathematician. Although ordained as a minister, Wallis was appointed Savilian professor of geometry at Oxford in 1649. He was one of the first to use Cartesian methods to study conic sections instead of employing the traditional synthetic approach. The sign ∞ for infinity (probably adapted from the late Roman symbol for 1000) was first introduced by Wallis. 14.5. EXERCISES 451 8. Show that the parametrization exptwist[a, c] of the expondentially twisted helicoid described on page 563 of the next chapter is a ruled patch. Find the rulings. M 9. Plot the tangent developable to the twisted cubic defined on page 202. M 10. Plot the tangent developable to the Viviani curve defined on page 207 for the complete range −2π 6 t 6 2π (see Figure 14.10). 11. Plot the tangent developable to the bicylinder defined on page 214. M 12. Carry out the calculations of Theorem 14.7 by computer. M 13. Show that the normal surface to a circular helix is a helicoid. Draw the binormal surface to a circular helix and compute its Gaussian curvature. M 14. For any space curve α, there are other surfaces which lie between the normal surface and the binormal surface. Consider perpsurf[φ, α](u, v) = α(u) + v cos φN(u) + sin φB(u), where N, B are the normal and binormal vector fields to α. Clearly, perpsurf[0, α] is the normal surface of α, whilst perpsurf[ π2 , α] is its binormal surface. Use perpsurf to construct several of these intermediate surfaces for a helix. Chapter 15 Surfaces of Revolution and Constant Curvature Surfaces of revolution form the most easily recognized class of surfaces. We know that ellipsoids and hyperboloids are surfaces of revolution provided that two of their axes are equal; this is evident from the figures on pages 312 and 314. Similarly for tori and elliptical paraboloids. Indeed, many objects from everyday life such as cans, table glasses, and furniture legs are surfaces of revolution. The process of lathing wood produces surfaces of revolution by its very nature. We begin an analytic study of surfaces of revolution by defining standard parametrizations whose coordinate curves are parallels and meridians. Then, in order to provide a fresh example, we define the catenoid, an important minimal surface. We compare it with a hyperboloid of revolution, which (only) at first sight it resembles. We also display a surface obtained by rotating a plane curve with assigned curvature (Figure 15.3). We point out in Section 15.2 that parallels and meridians are principal curves, being tangent to the directions determined by the principal curvatures. We pursue this topic analytically in Section 15.3, in which formulas are given for the Gaussian and mean curvatures of a surface of revolution. All the accompanying computations are carried out in Notebook 15. More curvature formulas are given in Section 15.4 for the generalized helicoids, which constitute a class of surfaces encompassing both helicoids and surfaces of revolution. While the concept of a surface of revolution, like that of a ruled surface, may be understood in terms of elementary geometry, some of the most interesting surfaces of revolution are those of constant Gaussian curvature. We develop this theme in the second half of the chapter by exhibiting surfaces of revolution with constant positive and negative Gaussian curvature in Sections 15.5 and 15.6 respectively. Understanding their equations requires at least cursory knowledge 461 462 CHAPTER 15. SURFACES OF REVOLUTION of the elliptic integrals of Legendre1 , which we mention. Other than the sphere, there are two types of surfaces of revolution with constant positive curvature, one shaped like an American football, and the other like a barrel. There are also three types in the negative case, the most famous of which is the pseudosphere, the surface of revolution obtained by rotating a tractrix. Their equations will be studied further in Chapter 21. In Section 15.7, we take a glimpse at more exotic surfaces with constant Gaussian curvature. We define a flat generalized helicoid, thereby extending a discussion of flat ruled surfaces in Section 14.2. Then we illustrate the surfaces of Kuen and Dini, whose definitions will be justified mathematically in Sections 21.7 and 21.8. 15.1 Surfaces of Revolution A surface of revolution is formed by revolving a plane curve about a line in R3 . More precisely: Definition 15.1. Let Π be a plane in R3 , let ℓ be a line in Π , and let C be a point set in Π . When C is rotated in R3 about ℓ, the resulting point set M is called the surface of revolution generated by C , which is called the profile curve. The line ℓ is called the axis of revolution of M. For convenience, we shall choose Π to be the xz-plane and ℓ to be the z-axis. We shall assume that the point set C has a parametrization α: (a, b) → C that is differentiable. Write α = (ϕ, ψ). Definition 15.2. The patch surfrev[α]: (0, 2π) × (a, b) → R3 defined by (15.1) surfrev[α](u, v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) . is called the standard parametrization of the surface of revolution M. The patch is of necessity defined with the angle u lying in the open interval (0, 2π), which means that the standard parametrization must be combined with an analogous patch in order to cover the surface. One usually assumes ϕ(v) > 0 in (15.1) to ensure that the profile curve does not cross the axis of revolution; 1 Adrien Marie Legendre (1752–1833). French mathematician, who made numerous contributions to number theory and the theory of elliptic functions. In 1782, Legendre determined the attractive force for certain solids of revolution by introducing an infinite series of polynomials that are now called Legendre polynomials. In his 3-volume work Traité des fonctions elliptiques (1825,1826,1830) Legendre founded the theory of elliptic integrals. 15.1. SURFACES OF REVOLUTION 463 however, we do not make this assumption initially. In particular, the curvature formulas that we derive hold in the general case. A meridian on the earth is a great circle that passes through the north and south poles. A parallel is a circle on the earth parallel to the equator. These notions extend to an arbitrary surface of revolution. Definition 15.3. Let C be a point set in a plane Π ⊂ R3 , and let M[C ] be the surface of revolution in R3 generated by revolving C about a line ℓ ⊂ Π . A meridian on M[C ] is the intersection of M[C ] with a plane containing the axis of the surface of revolution ℓ. A parallel on M[C ] is the intersection of M[C ] with a plane orthogonal to the axis of the surface of revolution. For a surface parametrized by (15.1), the parallels (15.2) u 7→ surfref[α](u, v0 ) = ϕ(v0 ) cos u, ϕ(v0 ) sin u, ψ(v0 ) , and the meridians (15.3) v 7→ surfref[α](u0 , v) = ϕ(v) cos u0 , ϕ(v) sin u0 , ψ(v) are coordinate curves. Many of the computer-generated plots of surfaces of revolution display (polygonal approximations of) the parallels and meridians by default. This phenomenon is evident in Figure 15.1, which shows a surface of revolution flanked by its parallels (on the left) and meridians (on the right). Figure 15.1: Parallels and meridians on a surface of revolution generated by the curve t 7→ 2 + 1 2 sin 2t, t The quantities ϕ(v0 ) and ψ(v0 ) in (15.2) have geometric interpretations: |ϕ(v0 )| represents the radius of the parallel, whereas ψ(v0 ) can be interpreted as the distance (measured positively or negatively) of the center of the same parallel from the origin. 464 CHAPTER 15. SURFACES OF REVOLUTION The Catenoid We have already discussed various basic surfaces of revolution. Next, we consider the catenoid, the surface of revolution generated by a catenary. The catenoid has a standard parametrization v v catenoid[c](u, v) = c cos u cosh , c sin u cosh , v . c c It is easy to compute the principal curvatures of the catenoid, with the result that 1 . k1 = −k2 = v 2 c cosh c The Gaussian curvature is therefore K= −1 c2 cosh v 4 c , whereas the mean curvature H vanishes. It follows that the catenoid is a minimal surface, a concept defined on page 398, and we shall prove in Section 16.3 that any surface of revolution which is also a minimal surface is contained in a catenoid or a plane. Figure 15.2: Hyperboloid of revolution and catenoid The shape of a small portion of the catenoid around the equator is similar to that of the hyperboloid of revolution around the equator, but large portions are quite different, as are their Gauss maps. We know that the image of the Gauss map of the whole hyperboloid omits disks around the north and south 15.2. PRINCIPAL CURVES 465 poles (see Figure 11.2 on page 336 and the discussion there). By contrast, the image of the Gauss map of the whole catenoid omits only the north and south poles. This is demonstrated in Notebook 15. There is a well-known relationship between the catenoid and the helicoid, discussed in Section 16.4. Revolving Curves with Prescribed Curvature One can also consider the surfaces formed by revolving curves with specified curvature. This provides a means of rendering ‘solid’ many of the examples of curves studied in Chapter 6. An example is illustrated in Figure 15.3; others are given in Notebook 15. Figure 15.3: Surface of revolution whose meridian has κ2(s) = sin s 15.2 Principal Curves In this section, we take the first step in the study of curves with particular properties that lie on surfaces. Later, in Chapter 18, we shall study other classes, namely asymptotic curves and geodesics. For simplicity, we shall deal only with orientable surfaces. For such a surface M, we choose a globally-defined surface unit normal U. Recall Definition 15.4. A curve α on a regular surface M ⊂ R3 is called a principal curve if and only if the velocity α′ always points in a principal direction. Thus, S(α′ ) = ki α′ , where S denotes the shape operator of M with respect to U, and ki (i = 1 or 2) is a principal curvature of M. A useful characterization of a principal vector is provided by 466 CHAPTER 15. SURFACES OF REVOLUTION Lemma 15.5. A nonzero tangent vector vp to a regular surface M ⊂ R3 is principal if and only if S(vp ) × vp = 0. Hence a curve α on M is a principal curve if and only if S(α′ ) × α′ = 0. Proof. If S(vp ) = ki vp , then S(vp ) × vp = ki vp × vp = 0. Conversely, if S(vp ) × vp = 0, then S(vp ) and vp are linearly dependent. More often than not, principal curves can be found by geometrical considerations. For example, we shall construct them in Chapter 19 using triply orthogonal families of surfaces. Another example is the next theorem, due to Terquem2 and Joachimsthal3 [Joach1], which provides a simple but useful criterion for the intersection of two surfaces to be a principal curve on both. Theorem 15.6. Let α be a curve whose trace lies in the intersection of regular surfaces M1 , M2 ⊂ R3 . Denote by Ui the unit surface normal to Mi , i = 1, 2. Suppose that along α the surfaces M1 , M2 meet at a constant angle; that is, U1 · U2 is constant along α. Then α is a principal curve in M1 if and only if it is a principal curve in M2 . Proof. Along the curve of intersection we have d d d (15.4) U1 · U2 + U1 · U2 . 0 = (U1 · U2 ) = dt dt dt Suppose that the curve of intersection α is a principal curve in M1 . Then (15.5) d U1 = −k1 α′ , dt where k1 is a principal curvature on M1 . But α′ is also orthogonal to U2 , so we conclude from (15.4) and (15.5) that d U1 · (15.6) U2 = 0. dt Since dU2 /dt is also perpendicular to U2 , it follows from (15.6) that d U2 = −k2 α′ dt for some k2 . In other words, α is a principal curve in M2 . 2 Olry Terquem (1782–1862). French mathematician and religious correspondent. Known for his study about the nine point circle of a given triangle, and also for suggesting some radical reforms aimed at impoving the standing of the Jewish community in France. 3 Ferdinand Joachimsthal (1818–1861). German mathematician, student of Kummer and professor at Halle and Breslau. Joachimsthal was a great teacher; his book [Joach2] was one of the first to explain the results of the Monge school and Gauss. 15.2. PRINCIPAL CURVES 467 As an important application of Theorem 15.6, we find principal curves on a surface of revolution. Theorem 15.7. Suppose the surface of revolution M[α] generated by a plane curve α is a regular surface. Then the meridians and parallels on M[α] are principal curves. Proof. Each meridian is sliced from M[α] by a plane Πm containing the axis of rotation of M[α]. For p ∈ M[α] ∩ Πm it is clear that the surface normal U(p) of M[α] lies in Πm . Hence U(p) and the unit surface normal of Πm are orthogonal. Therefore, Theorem 15.6 implies that the meridians are principal curves of M[α]. Next, let Πp be a plane orthogonal to the axis of M[α]. By rotational symmetry, the unit surface normal U of M[α] makes a constant angle with the unit surface normal of Πp . Again, Theorem 15.6 implies that the parallels are principal curves. Principal curves give rise to an important class of patches for which curvature computations are especially simple. Definition 15.8. A principal patch is a patch x: U → R3 for which the curves u 7→ x(u, v) and v 7→ x(u, v) are principal curves. In now follows that the standard parametrization (15.1) of a surface of revolution is a principal patch. This fact will be derived independently in the next section, using Lemma 15.9. Let x: U → R3 be a patch. (i) If F = 0 = f at all points of U, then x is a principal patch. (ii) If x is a principal patch with distinct principal curvatures, then F = 0 = f on U. Proof. If F = 0 = f , the Weingarten equations (Theorem 13.16, page 394) imply that xu and xv are eigenvectors of the shape operator. Conversely, if x is principal, then both xu and xv are eigenvectors of the shape operator. If in addition the principal curvatures are distinct, Lemma 13.19 (page 396) implies that F = 0 = f. In the case of the plane or sphere, the principal curvatures are equal and every point of the surface is said to be umbilic. Moreover, there are infinitely many principal curves through each point of the plane or sphere. This situation will be discussed further in Chapter 19. 468 CHAPTER 15. SURFACES OF REVOLUTION 15.3 Curvature of a Surface of Revolution First, we compute the coefficients of the first and second fundamental forms, and also the unit surface normal for a general surface of revolution. Lemma 15.10. Let M be a surface of revolution with profile curve α = (ϕ, ψ). Let x = surfref[α] be the standard parametrization (15.1) of M. Then (15.7) E = ϕ2 , F = 0, G = ϕ′2 + ψ ′2 . Thus x is regular wherever ϕ and ϕ′2 + ψ ′2 are nonzero. When this is the case, (15.8) −|ϕ|ψ ′ , e= p ϕ′2 + ψ ′2 and the unit surface normal is (15.9) f = 0, g= (sign ϕ)(ϕ′′ ψ ′ − ϕ′ ψ ′′ ) p , ϕ′2 + ψ ′2 sign ϕ U(u, v) = p (ψ ′ cos u, ψ ′ sin u, −ϕ′ ). ϕ′2 + ψ ′2 Proof. From (15.1) it follows that the first partial derivatives of x are given by   xu = −ϕ(v) sin u, ϕ(v) cos u, 0 , (15.10)  x = ϕ′ (v) cos u, ϕ′ (v) sin u, ψ ′ (v). v Then (15.7) is immediate from (15.10) and the definitions of E, F, G. The unit normal xu × xv U= kxu × xv k is easily computed. Next, it is necessary to write down the second partial derivatives of x, and use Lemma 13.31 on page 405 to find e, f, g. We omit the calculations, which are easily checked. Recalling Lemma 15.9, we obtain another proof of the following result. Corollary 15.11. The standard parametrization (15.1) of a surface of revolution is a principal patch. At this point, one could compute the Gaussian curvature K by means of Theorem 13.25. However, we prefer to compute the principal curvatures first. We know that the coordinate curves (15.2) and (15.3) are principal curves. Hence the principal curvatures for a surface of revolution have special meaning, so we denote them by kp , km instead of the usual k1 , k2 . To be specific, kp is the curvature of the parallel (15.2) and km is the curvature of the meridian (15.3) (the reverse of alphabetical order!). 15.3. CURVATURE OF A SURFACE OF REVOLUTION 469 Theorem 15.12. The principal curvatures of a surface of revolution parametrized by (15.1) are given by  −ψ ′ e   p , = k =  p  E |ϕ| ϕ′2 + ψ ′2 (15.11)  g (sign ϕ)(ϕ′′ ψ ′ − ϕ′ ψ ′′ )   = .  km = G (ϕ′2 + ψ ′2 )3/2 The Gaussian curvature is given by (15.12) K= −ψ ′2 ϕ′′ + ϕ′ ψ ′ ψ ′′ , ϕ(ϕ′2 + ψ ′2 )2 and the mean curvature by (15.13) H= ϕ(ϕ′′ ψ ′ − ϕ′ ψ ′′ ) − ψ ′ (ϕ′2 + ψ ′2 ) . 2|ϕ|(ϕ′2 + ψ ′2 )3/2 Proof. Since F = f = 0, it follows that xu xv , kxu k kxv k forms an orthonormal basis which diagonalizes the shape operator S wherever x is regular. Hence by Corollary 13.33 on page 405, (15.14) S(xu ) = e xu E and S(xv ) = g xv . G Thus, km = g/G, kp = e/E, and (15.11) follows from (15.7) and (15.8). Finally, (15.12) and (15.13) follow from the relations K = kp km , H = 21 (kp + km ). We know from the geometric description of the normal curvature given in Section 13.2 that km at p ∈ M coincides, up to sign, with the curvature κ of the meridian through p. The second equation of (15.11) confirms this fact. Corollary 15.13. For a surface of revolution, the functions K, H, kp , km , E, F, G, e, f, g are all constant along parallels. Proof. All these functions are expressible in terms of ϕ and ψ and their derivatives. But ϕ and ψ do not depend on the angle u. We may in theory choose a profile curve to have unit speed. In this case, the formulas we have derived so far for a surface of revolution simplify considerably. 470 CHAPTER 15. SURFACES OF REVOLUTION Corollary 15.14. Let x be the standard parametrization (15.1) of a surface of revolution in R3 whose profile curve α = (ϕ, ψ) has unit speed. Then E = ϕ2 , e = −|ϕ|ψ ′ , kp = F = 0, f = 0, −ψ ′ , |ϕ| G = 1, g = (sign ϕ)(ϕ′′ ψ ′ − ϕ′ ψ ′′ ), km = (sign ϕ)(ϕ′′ ψ ′ − ϕ′ ψ ′′ ), 2H = (sign ϕ)(ϕ′′ ψ ′ − ϕ′ ψ ′′ ) − ψ′ , |ϕ| K= −ϕ′′ . ϕ In particular, the signed curvature κ2[α] equals ±km . p Proof. Since 1 = kα′ k = ϕ′2 + ψ ′2 , all formulas, except for the last, are immediate from (15.7), (15.8), (15.11). That for K follows from (15.12) and the equality ϕ′ ϕ′′ + ψ ′ ψ ′′ = 0. The last statement follows from Newton’s equation κ2[α] = −ψ ′ ϕ′′ + ϕ′ ψ ′′ (ϕ′2 + ψ ′2 )3/2 on page 15. 15.4 Generalized Helicoids Both helicoids and surfaces of revolution are examples of the class of surfaces known as generalized helicoids, which were first studied by Minding4 in 1839 (see [Mind]). Definition 15.15. Let Π be a plane in R3 , ℓ be a line in Π , and C be a point set in Π . Suppose that C is rotated in R3 about ℓ and simultaneously displaced parallel to ℓ so that the speed of displacement is proportional to the speed of rotation. Then the resulting point set M is called the generalized helicoid generated by C , which is called the profile curve of M. The line ℓ is called the axis of M. The ratio of the speed of displacement to the speed of rotation is called the slant of the generalized helicoid M. The Euclidean motion (consisting of a simultaneous translation and rotation) used in this definition is called a screw motion. Clearly, a surface of revolution 4 Ernst Ferdinand Adolf Minding (1806–1885). German professor, later dean of the faculty, at the University of Dorpat (now Tartu) in Estonia. Minding was a self-taught mathematician. While a school teacher, he studied for his doctorate, which was awarded by Halle for a thesis on approximating the values of double integrals. In 1864, Minding became a Russian citizen and in the same year was elected to the St. Petersburg Academy. In the 1830s Minding was one of the first mathematicians to use Gauss’s approach to the differential geometry of surfaces. 15.4. GENERALIZED HELICOIDS 471 is a generalized helicoid of slant 0, and a generalized helicoid reduces to an ordinary helicoid when the profile curve is t 7→ (bt, 0). Darboux ([Darb2, Volume 1, page 128]) put it very well when he observed that a surface of revolution has an important kinematic property: when it rotates about its axis, it glides over itself. In more modern language, we would say that a surface of revolution is the orbit of a curve in a plane Π by a 1-parameter group of rotations of R3 about a line in Π . Generalized helicoids enjoy a similar property: when a generalized helicoid is subjected to a screw motion, it glides over itself. Thus, a generalized helicoid is the orbit of a curve in a plane Π by a 1-parameter group of screw motions of R3 about a line in Π . Just as we did for surfaces of revolution on page 462, we choose Π to be the xz-plane and ℓ to be the z-axis. We shall assume that the point set C has a parametrization α: (a, b) → C which is differentiable, and we write α = (ϕ, ψ). This allows us to give the alternative Definition 15.16. Let α be a plane curve and write α(t) = (ϕ(t), ψ(t)). Then the generalized helicoid with profile curve α and slant c is the surface in R3 parametrized by genhel[c, α](u, v) = ϕ(v) cos u, ϕ(v) sin u, cu + ψ(v) . To summarize, a generalized helicoid is generated by a plane curve, which is rotated about a fixed axis and at the same time translated in the direction of the axis with speed a constant multiple of the speed of rotation. It is also possible to define the notion of meridian for a generalized helicoid. Definition 15.17. Let C be a point set in a plane Π ⊂ R3 , and let M[C , c] be the generalized helicoid in R3 of slant c generated by revolving C about a line ℓ ⊂ Π and simultaneously displacing it parallel to ℓ. A meridian on M[C , c] is the intersection of M[C , c] with a plane containing the axis ℓ. The generalization to generalized helicoids of the notion of parallel given on page 463 is more complicated. For a surface of revolution a parallel is a circle; for a generalized helicoid the corresponding curve is a helical curve, which we call a parallel helical curve. The v-parameter curves are the meridians and the u-parameter curves are the parallel helical curves. In contrast to the situation with a surface of revolution, the meridians and parallel helical curves will not be perpendicular to one another for a generalized helicoid of nonzero slope. In Theorem 15.25, page 483, we shall show that a generalized helicoid with the property that its meridians are principal curves must either be a surface of revolution or Dini’s surface illustrated in Figure 15.13. Generalized helicoids can be considered to be ‘twisted’ surfaces of revolution, though this is unrelated to the construction on page 348. For example, Figure 15.4 is evidently a twisted version of Figure 15.1; it has slant 1/2. 472 CHAPTER 15. SURFACES OF REVOLUTION Figure 15.4: A generalized helicoid together with its parallel helical curves and meridians The proof of the following lemma is an easy extension of the proof of Lemma 15.10, and the calculations are carried out in Notrebook 15. Lemma 15.18. Let x be a generalized helicoid in R3 whose profile curve is α = (ϕ, ψ). Then E = ϕ2 + c2 , F = c ψ′ , G = ϕ′2 + ψ ′2 . Therefore, x is regular wherever EG − F 2 = ϕ2 (ϕ′2 + ψ ′2 ) + c2 ϕ′2 is nonzero. In this case, denoting its square root by D, we have e= −ϕ2 ψ ′ , D Furthermore (15.15) K= f= c ϕ′2 , D g= ϕ(ϕ′′ ψ ′ − ϕ′ ψ ′′ ) . D −c2 ϕ′4 + ϕ3 ψ ′ (−ψ ′ ϕ′′ + ϕ′ ψ ′′ ) , D4 and H= −2c2 ϕ′2 ψ ′ − ϕ2 ψ ′ (ϕ′2 + ψ ′2 ) + ϕ(c2 + ϕ2 )(ψ ′ ϕ′′ − ϕ′ ψ ′′ ) . 2 D3 We know that the sphere S 2 (a) is a surface of revolution parametrized by (15.16) sphere[a](u, v) = a cos v cos u, a cos v sin u, a sin v ; it has constant positive Gaussian curvature K = 1/a2 . We call the twisted version of the sphere the corkscrew surface or twisted sphere. Explicitly, it is given by 15.5. CONSTANT POSITIVE CURVATURE 473 twisphere[a, b](u, v) = a cos u cos v, a sin u cos v, a sin v + b u . In Notebook 15, the Gaussian curvature of the twisted sphere is found to be 4a2 cos4 v − 4b2 sin4 v a2 + b2 + (a2 − b2 ) cos 2v 2 , and simplifies to (cos 2v)/a2 if b = a. This formula is used to color the left-hand side of Figure 15.5. Figure 15.5: Corkscrew surfaces colored by Gaussian and mean curvature 15.5 Surfaces of Constant Positive Curvature The sphere is not the only surface in R3 that has constant curvature. We can take a spherical cap made up of some thin inelastic material, for example, half of a ping-pong ball. Then the spherical cap can be bent without stretching into many different shapes. Since there is no stretching involved and the Gaussian curvature is an isometric invariant (see Theorem 17.5 on page 536), a bent cap also has constant positive curvature. On the other hand, it is clear intuitively that a whole ping-pong ball cannot be bent. Indeed, Liebmann’s Theorem (Theorem 19.13, page 603) states that the sphere S 2 (a) is rigid in this sense. 474 CHAPTER 15. SURFACES OF REVOLUTION To find other surfaces of revolution in R3 that have constant positive curvature, we proceed backwards: we assume that we are given a surface of revolution M with constant positive curvature and seek restrictions on a parametrization x of M. First, we determine the profile curves of a surface of constant positive Gaussian curvature. Theorem 15.19. Let M be a surface of revolution whose Gaussian curvature is a positive constant 1/a2 , where a > 0. Then M is part of a surface parametrized by a patch x of the form x(u, v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) , where  v  ϕ(v) = b cos ,   a Z v/ap    ψ(v) = a2 − b2 sin2 t dt, (15.17) 0 for some constant b > 0. The parameter v has one of the following ranges: πa πa 6v6 ; 2 2 if b = a, then − if b < a, then −∞ 6 v 6 ∞; if b > a, then −a arcsin 6 v 6 a arcsin . a b a b The patch x is regular at (u, v) if and only if ϕ(v) 6= 0, that is, v 6= (n + 12 )πa. Proof. Assume without loss of generality that x = surfrev[α] is given by (15.1), and that the profile curve α = (ϕ, ψ) has unit speed, so that ϕ′2 + ψ ′2 = 1. If M has constant positive curvature 1/a2 , then Corollary 15.14, page 470, implies that ϕ satisfies the differential equation (15.18) ϕ′′ + 1 ϕ = 0, a2 whose general solution is ϕ(v) = b cos((v/a) + c). Without loss of generality, we can assume that c = 0; this amounts to translating the profile curve along the axis of revolution so that the profile curve is farthest from the axis of revolution when v = 0. Also, by taking a mirror image if necessary, we may assume that b > 0. Thus we get the first equation of (15.17). We can assume that ψ ′ (v) > 0 for all v; otherwise, replace v by −v. Then ′2 ϕ + ψ ′2 = 1 implies that r v ′ (15.19) aψ (v) = a2 − b2 sin2 , a 15.5. CONSTANT POSITIVE CURVATURE 475 and when we integrate this equation from 0 to v, and then rescale the variable, we get the second equation of (15.17). In order that ψ(v) be well defined, it is necessary that the quantity under the square root in (15.19) be nonnegative. For b < a this quantity is always positive, so ψ(v) is defined for all v. When b = a the profile curve is a part of a circle; in that case the requirement that −π/2 6 v/a 6 π/2 ensures that the profile curve does not overlap itself and is thus a semicircle. If b > a then (15.19) is defined if and only if −a/b 6 sin(v/a) 6 a/b. When b = a, the profile curve is v v , v 7→ a cos , a sin a a − v π π 6 6 , 2 2 a which when revolved about the z-axis yields a sphere S 2 (a) of radius a. If b < a, the profile curve (ϕ, ψ) makes a shallower arc and first crosses the z-axis when z=± Z 0 π/2p a2 − b2 sin2 t dt. Its length between these two points is πa, which is the same as the length of a semicircle of radius a. But the profile curve can be continued indefinitely, so as to weave its way up and down the z-axis, instead of closing up (a hint of this behavior can be seen in Figure 15.6). When b > a the profile curve is only defined on the interval −a arcsin a a 6 v 6 a arcsin , b b and the resulting surface of revolution resembles a barrel when b is moderately larger than a. The closer the ratio a/b is to zero, the larger the hole in the middle. The outside surface in Figure 15.7 illustrates the case a/b = 3/4. In conclusion, Corollary 15.20. Let S(a, b) be the surface of revolution whose profile curve is α = (ϕ, ψ) with ϕ, ψ given by (15.17). (i) S(a, a) is an ordinary sphere of radius a. (ii) (Spindle type) If 0 < b < a, then S(a, b) is a surface of revolution that resembles an infinite string of beads, each of which is shaped like a football with its vertices on the axis of revolution. (iii) (Bulge type) If 0 < a < b, then S(a, b) is barrel-shaped and does not meet the axis of revolution. 476 CHAPTER 15. SURFACES OF REVOLUTION 2 1 -0.4 0.4 -1 -2 Figure 15.6: Profile curves with b 6 a and b > a Figure 15.7: The surfaces S(a, b) with a/b = 2 and 3/4 Elliptic Integrals We shall make a slight detour into the complicated subject of elliptic functions and integrals, which will be useful when we study surfaces of revolution of constant negative curvature. 15.6. CONSTANT NEGATIVE CURVATURE 477 Definition 15.21. The elliptic integral of the second kind is defined by E φ m = Z φp 0 1 − m sin2 θ dθ, whereas the complete elliptic integral of the second kind is Z π/2p π E 1 − m sin2 θ dθ. m = 2 0 The motivation for this definition comes from the formula for the arc length of an ellipse, given in Exercise 9. But we have already made implicit use of it, for in (15.17) we may write v b2 , ψ(v) = aE a a2 after having brought a factor of a outside the square root. Notice that E φ 1 = sin φ; thus E φ m can be considered to be a generalization of the sine function. The corresponding generalization of the hyperbolic sine function turns out to be −i E iφ −m because −i E iφ −m = (15.20) Z φp 1 − m sinh2 θ dθ, 0 as can be checked by changing variables in the integral and using the identity sinh ix = i sin x. We shall need (15.20) in the next section. 15.6 Surfaces of Constant Negative Curvature The determination of the surfaces of revolution of constant negative curvature proceeds along the same lines as that in Section 15.5. However, the resulting surfaces are quite different in appearance. Theorem 15.22. Let M be a surface of revolution whose Gaussian curvature is a negative constant −1/a2. Then M is part of a surface parametrized by a patch x such that x(u, v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) , where the profile curve α = (ϕ, ψ) is one of the following types: (i) (Pseudosphere)  Z vp  −v/a −2t/a dt  ae , 1 − e   (15.21) 0 α(v) =     Z vp v/a 2t/a 1−e dt ae , 0 for 0 6 v < ∞, for −∞ < v 6 0. 478 CHAPTER 15. SURFACES OF REVOLUTION (ii) (Hyperboloid type) (15.22)  v  ϕ(v) = b cosh ,   a Z v/ap  iv −b2 2  2 2  ψ(v) = , a − b sinh t dt = −iaE a a2 0 for some constant b > 0, and v satisfies −a arcsinh a a 6 v 6 a arcsinh . b b (iii) (Conic type) (15.23)  v  ϕ(v) = b sinh ,   a    Z v/ap  ψ(v) = a2 − b2 cosh2 t dt   0   p  −b2 iv  2 2  , = −i a − b E a a2 − b 2 for some constant b with 0 < b 6 a, and v must satisfy √ √ a2 − b 2 a2 − b 2 −a arcsinh 6 v 6 a arcsinh . b b Proof. Without loss of generality, a > 0. The general solution of the negative analogue ϕ′′ − ϕ/a2 = 0 of (15.18) is (15.24) ϕ(v) = Aev/a + B e−v/a . Case 1. First, suppose that A is zero in (15.24). We can assume that B > 0 by reflecting the profile curve about the y-axis if necessary, so that (ϕ, ψ) becomes (−ϕ, ψ). Moreover, substituting v v + a log B − a log a, we can assume that B = a and ϕ(v) = ae−v/a . Since 0 6 ψ ′ (v)2 = 1 − ϕ′ (v)2 = 1 − e−2v/a , we must have v > 0. Thus we get the first alternative in (15.21). Similarly, B = 0 leads to the second alternative. Next, suppose that A and B are both different from zero in (15.24). Using the change of variables a B v v + log 2 A if necessary, we may assume that |A| = |B|. 15.6. CONSTANT NEGATIVE CURVATURE 479 2 2 1 1 1.2 -1 -0.5 -1 -1 -2 -2 0.5 1 Figure 15.8: Profile curves for constant negative curvature Figure 15.9: Surfaces of hyperboloid and conic type Case 2. When A = B, we can (using a mirror image of the profile curve if necessary) assume that A > 0. Then v ϕ(v) = A ev/a + e−v/a = 2A cosh . a We put b = 2A and obtain (15.22). Case 3. If A = −B, we can (changing v to −v if necessary) assume that A > 0. Thus (15.24) becomes v ϕ(v) = A ev/a − e−v/a = 2A sinh . a 480 CHAPTER 15. SURFACES OF REVOLUTION We put b = 2A and obtain (15.23). Corollary 15.23. The surface of revolution whose profile curve is given by (15.21) is a pseudosphere or tractoid; that is, the surface of revolution of a tractrix. Proof. This is an immediate consequence of Lemma 2.1, page 52, and part (i) of Theorem 15.22. In the light of the previous corollary, equation (2.16) gives rise to the parametrization v . pseudosphere[a](u, v) = a cos u sin v, sin u sin v, cos v + log tan 2 In Notebook 15, we verify that this surface has constant curvature −1/a2 . Figure 15.10: Pseudosphere 15.7 More Examples of Constant Curvature In this final section, we determine the flat generalized helicoids and introduce the surfaces of Dini and Kuen, which will be studied in greater detail by transform methods in Chapter 21. 15.7. MORE EXAMPLES OF CONSTANT CURVATURE 481 Flat Generalized Helicoids Recall the parametrization of a generalized helicoid given in Definition 15.16. Theorem 15.24. A generalized helicoid is flat if and only if its profile curve can be parametrized as α(t) = (t, ψ(t)), where r c2 c (15.25) ±ψ(t) = t a2 − 2 + c arcsin . t at 3 2.5 2 1.5 -3 -2 -1 1 2 3 Figure 15.11: Zero-curvature profile curve Figure 15.12: Zero-curvature generalized helicoid Proof. Equation (15.15), page 472, implies that a generalized helicoid is flat if and only if its profile curve α = (ϕ, ψ) satisfies the differential equation (15.26) ϕ3 (ϕ′ ψ ′ ψ ′′ − ψ ′2 ϕ′′ ) − c2 ϕ′4 = 0. The profile curve can be parametrized so that ϕ(t) = t; then (15.26) reduces to t3 ψ ′ ψ ′′ − c2 = 0. 482 CHAPTER 15. SURFACES OF REVOLUTION Thus, 2 c2 d ′ 2 (ψ ) = 3 , dt t and r c2 ψ ′ (t) = ± a2 − 2 , (15.27) t 2 where a is a constant of integration, and a > 0. Then (15.27) can be integrated by computer to give (15.25), apart from an irrelevant constant of integration. In Notebook 15, we verify that the generalized helicoid generated by t 7→ t, ψ(t) , with ψ defined in (15.25), does indeed have zero Gaussian curvature. Figure 15.13: Dini’s surface Dini’s Surface The twisted pseudosphere is associated with the name of Dini5 . The explicit parametrization of Dini’s surface is obtained using by applying the genhel construction on page 471 to a tractrix, and is therefore v + cu . dini[a, b](u, v) = a cos u sin v, a sin u sin v, a cos v + log tan 2 5 Ulisse Dini (1845–1918). Italian mathematician who worked mainly in Pisa. He made fundamental contributions to surface theory and real analysis. His statue can be found off the Piazza dei Cavalieri, not far from the leaning tower. 15.7. MORE EXAMPLES OF CONSTANT CURVATURE 483 In contrast to the corkscrew surface, Dini’s has constant curvature. Figure 15.13 shows clearly that it is a deformation by twisting of the pseudosphere. The following theorem is taken from volume 1 page 353 of Luigi Bianchi’s classical text6 , Lezioni di Geometria Differenziale [Bian]. Theorem 15.25. Let M be a generalized helicoid with the property that the meridians are principal curves. Then M is part of Dini’s surface. Proof. Without loss of generality, we can suppose that the profile curve of the generalized helicoid is of the form α(t) = (t, ψ(t)), so that genhel[c, α](u, v) = v cos u, v sin u, cu + ψ(v) . The unit normal U to genhel[c, α] is given by −c sin u + vψ ′ (v) cos u, c cos u + vψ ′ (v) sin u, −v p , U(u, v) = c2 + v 2 + v 2 ψ ′ (v)2 a formula extracted from Notebook 15. The meridian v 7→ genhel[c, α](u, v) is the intersection of the generalized helicoid with a plane Π through its axis. The unit normal to this plane is given by V(u, v) = (− sin u, cos u, 0). Now suppose that α is a principal curve on the generalized helicoid. Since α is automatically a principal curve on Π , Theorem 15.26 (which is Exercise 4) implies that the vector fields U and V meet at a constant angle σ along α. Thus c cos σ = U · V = p (15.28) . 2 2 c + v + v 2 ψ ′ (v)2 Then (15.28) implies that ψ satisfies the differential equation v 2 1 + ψ ′ (v)2 = c2 tan2 σ. But this is equivalent to the differential equation (2.17) on page 51, so α is a tractrix. Hence the generalized helicoid is a surface of Dini. 6 Luigi Bianchi (1856–1928). Italian mathematician who worked mainly in Pisa. Although he is most remembered for the ‘Bianchi identities’, he also made fundamental contributions to surface theory. 484 CHAPTER 15. SURFACES OF REVOLUTION Figure 15.14: Kuen’s surface Kuen’s Surface A more complicated surface of constant negative curvature is that of Kuen7 [Kuen]. It can be parametrized by cos v (cos u + u sin u) sin v (sin u − u cos u) sin v 1 v + . , , 2 log tan 2 1 + u2 sin2 v 1 + u2 sin2 v 1 + u2 sin2 v A computation in Notebook 15 shows that this surface has Gaussian curvature K = −4. It also turns out that, for this parametrization, the ‘mixed’ coefficients F and f of both the first and second fundamental forms vanish identically. As a consequence, the u and v parameter curves are principal; these are the curves visible in Figure 15.14. We shall explain this fact and see exactly how the equation for Kuen’s surface arises in Section 21.7. An extraordinary plaster model was made of this surface; plate 86 of [Fischer] is a photo of this plaster model. Reckziegel has given an excellent description of Kuen’s surface (see pages 30–41 of the Commentaries to [Fischer]). 7 Th. Kuen used Bianchi’s parametrization to make a plaster model of his surface (see [Kuen]) 15.8. EXERCISES 485 15.8 Exercises M 1. Show that the surface of revolution generated by the graph of a function h: R → R can be parametrized by x(u, v) = v cos u, v sin u, h(v) , 0 < u < 2π. Determine the associated functions E, F, G, e, f, g, kp , km , K, in terms of the function h. 2. Find the formulas for the Gaussian, mean and principal curvatures of a surface of revolution with profile curve α(t) = (t, ψ(t)). M 3. Plot the surfaces of revolution corresponding to each of the following curves: cycloid, cissoid, logarithmic spiral, lemniscate, cardioid, astroid, deltoid, nephroid, triangle. 4. Prove the following converse to Theorem 15.6 that was used to prove Theorem 15.25. Theorem 15.26. Let α be a curve which lies on the intersection of regular surfaces M1 , M2 ⊂ R3 . Suppose that α is a principal curve in M1 and also in M2 . Then the normals to M1 and M2 meet at a constant angle along α. 5. With reference to Lemma 15.9, show that the mapping x: R2 → R3 defined by x(u, v) = (u + v, u, 0) is a principal patch for which F 6= 0. M 6. Compute the Gaussian curvature and mean curvature of the generalized helicoid of a catenary. Plot the generalized helicoid of catenary[1] with slant 1/2. 7. Show that a surface of revolution whose Gaussian curvature is zero is a part of a plane, circular cone or circular cylinder. These are therefore the only flat surfaces of revolution. M 8. Plot the elliptic functions n φ 7→ E φ 5 for n = 0, 1, 2, 3, 4, 5. and n φ 7→ −i E iφ − 5 486 CHAPTER 15. SURFACES OF REVOLUTION 9. Find a formula for the length of the ellipse (x/a)2 + (y/b)2 = 1. in terms of the elliptic integral E. 10. A surface of revolution is formed by moving a curve γ in a plane Π1 about a circle in a plane Π2 , where Π1 and Π2 are perpendicular. There is a more general construction in which the circle is replaced by an arbitrary curve α in Π2 : gensurfrev[α, γ](u, v) = α1 (u)γ 1 (v), α2 (u)γ 2 (v), α3 (u)γ 3 (v) . Determine when a generalized surface of revolution is a principal patch. M 11. Compute the Gaussian curvature K and the mean curvatures H of the generalized surface of revolution formed by moving an eight curve along an eight curve. This is represented in Figure 15.15 left), while the function K is plotted on the right. Figure 15.15: A generalized surface of revolution and its Gaussian curvature M 12. Find the principal curvatures and the Gauss map of a pseudosphere. M 13. Sievert’s surface is defined by x = (x1 , x2 , x3 ), where p  √  2 2 + 2 sin2 u sin v u   √ cos − x = 2 tan u) + arctan( 1   2 2 − sin2 v cos2 u     p  √ 2 2 + 2 sin2 u sin v u √ + arctan( 2 tan u) , sin − x =  2 2  2  2 − sin v cos2 u      4 cos v v   x3 = log(tan ) + . 2 2 − sin2 v cos2 u 15.8. EXERCISES 487 Show that it has constant positive Gaussian curvature (see [Siev] and [Fischer, Commentary, page 38]). Investigate the more general patch, sievert[a], depending upon a parameter a (equal to 1 above) defined in Notebook 15. Figure 15.16 illustrates the case a = 4/5. Figure 15.16: The surface sievert[0.8] 14. Suppose that x is a generalized helicoid in R3 with slant c whose profile curve α = (ϕ, ψ) has unit speed. Simplify the equations of Lemma 15.18 that apply in this case. 15. Fill in the details of the proof of Theorem 15.25. Chapter 16 A Selection of Minimal Surfaces The origins of minimal surface theory can be traced back to 1744 with Euler’s paper [Euler2], and to 1760 with Lagrange’s paper [Lag]. Euler showed that the catenoid is a minimal surface, and Lagrange wrote down the partial differential equation that must be satisfied for a surface of the form z = f (x, y) to be minimal. In 1776, Meusnier1 rediscovered the catenoid and also showed that the helicoid is a minimal surface [Meu]. The mathematical world had to wait over 50 years until other examples were found by Scherk. These include the surfaces now called ‘Scherk’s minimal surface’, ‘Scherk’s fifth minimal surface’, and a family of surfaces that includes both the catenoid and the helicoid. The importance of minimal surfaces as those of least potential surface energy was illustrated by the experiments of Plateau2 , who dipped wires in the form of space curves into a solution of soapy water and glycerin, thus realizing minimal surfaces experimentally. Plateau’s problem is that of determining the minimal surfaces through a given curve. It has been formulated and studied in great generality by many mathematicians. In this chapter, we begin a study of minimal surfaces by showing that a minimal surface is a critical point of the area function in an appropriate sense (Theorem 16.4). In Section 16.2, we provide the first of a series of new examples 1 Jean Babtiste Meusnier de la Place (1754–1793). French mathematician, a student of Monge. He was a general in the revolutionary army and died of battle wounds. 2 Joseph Antoine Ferdinand Plateau (1801–1883). Belgian physicist. His thesis concerned the impressions that light can have on the eye. Unfortunately, this led him to stare into the bright sun, which had an adverse effect on his sight. In 1840, he began a series of experiments with surfaces for which he became famous. 501 502 CHAPTER 16. A SELECTION OF MINIMAL SURFACES of minimal surfaces by investigating the celebrated deformation from the helicoid to the catenoid. This construction also provides examples of local isometries of surfaces, a concept defined in Chapter 12. In Section 16.3, we prove that any surface of revolution which is also a minimal surface is contained in a catenoid or a plane. The minimal surfaces of Enneper, Catalan and Henneberg are described in Section 16.4. The second is intimately related to a cycloid curve that it contains, and this example anticipates the definition of geodesic torsion in Section 17.4 and the theory of geodesics in Chapter 18. A discussion of Scherk’s surface in Section 16.5 begins with the minimal surface equation for a Monge patch. In Section 16.6 we show that the Gauss map of a minimal surface is conformal, an important result that lies at the heart of much modern research into minimal surfaces [Os1]. Isothermal coordinates for surfaces are introduced in Section 16.7, and provide the foundation for our further study of minimal surfaces using a complex variable in Chapter 22. 16.1 Normal Variation On page 398, we defined a minimal surface in R3 as a surface whose mean curvature vanishes. This is Lagrange’s 1760 definition. A more intuitive meaning of minimal surface is the surface of least area among a family of surfaces having the same boundary. In order to show how these two definitions coincide, we define the normal variation of a surface M in R3 to be a family of surfaces t 7→ M(t) representing how M changes when pulled in a normal direction. Let A(t) denote the area of M(t). We show that the mean curvature of M vanishes if and only if the first derivative of t 7→ A(t) vanishes at M. First, we make precise the notion of normal variation. The notion of a bounded subset was discussed on page 373, and the general formula for the area of a bounded region of a surface in Rn was given by (12.24). In the case n = 3, (12.22) on the foot of page 372 yields Lemma 16.1. Let M be a regular surface in R3 , and x: U → M a regular patch. Given a bounded region Q in U, the area of x(Q) is given by ZZ kxu × xv k dudv. area(x(Q)) = Q We wish to study how area changes with a small perturbation of a surface. The simplest perturbation is one that is normal to the surface. Definition 16.2. Let x: U → R3 be a regular patch, and choose a bounded region Q ⊂ U. Suppose that h: Q → R is differentiable and ε > 0. Let U denote a 16.1. NORMAL VARIATION 503 unit vector field such that U(u, v) is perpendicular to x(u, v) for all (u, v) ∈ U. Then the normal variation of x and Q, determined by h, is the map X: (−ε, ε) × Q −→ R3 given by (16.1) Xt (u, v) = x(u, v) + th(u, v)U(u, v) for (u, v) ∈ Q and −ε < t < ε. It follows from this definition that Xt is a patch for each t with −ε < t < ε for sufficiently small ε. Let   E(t) = (Xt )u · (Xt )u ,    (16.2) F (t) = (Xt )u · (Xt )v ,     G(t) = (X ) · (X ) . t v t v Then E = E(0), F = F (0), G = G(0), and by Lemma 16.1 the area of Xt (Q) is given by ZZ p (16.3) E(t)G(t) − F (t)2 dudv. A(t) = Q We compute the derivative at zero of the function A(t). Lemma 16.3. We have (16.4) ′ A (0) = −2 ZZ hH Q p E G − F 2 dudv, where H denotes the mean curvature of M. Proof. Differentiating (16.1) with respect to u and v gives   (Xt )u = xu + thu U + thUu , (16.5)  (X ) = x + th U + thU . t v v v v From (13.9), page 394, (16.5) and the definitions (16.2), it follows that (16.6) Similarly, (16.7) E(t) = (Xt )u · (Xt )u = (xu + thu U + thUu ) · (xu + thu U + thUu ) = E + 2thxu · Uu + O t2 = E − 2the + O t2 . F (t) = G(t) = F − 2thf + O t2 , G − 2thg + O t2 . 504 CHAPTER 16. A SELECTION OF MINIMAL SURFACES From (16.6),(16.7) and Theorem 13.25, page 400, we get G − 2thg + O t2 E(t)G(t) − F (t)2 = E − 2the + O t2 − F − 2thf + O t2 = E G − F 2 − 2th(E g − 2F f + Ge) + O t2 = E G − F 2 (1 − 4thH) + O t2 , and hence 2 p p E(t)G(t) − F (t)2 = (E G − F 2 ) (1 − 4thH) + O(t2 ) p p = E G − F 2 1 − 4thH + O(t2 ) p = E G − F 2 (1 − 2thH) + O t2 . Combined with (16.3), we obtain ZZ p j E G − F 2 (1 − 2thH) + O t2 dudv A(t) = Q ZZ ZZ p p hH E G − F 2 dudv + O t2 . E G − F 2 dudv − 2t = Q Q When we differentiate with respect to t and evaluate the resulting expression at t = 0, we obtain (16.4). Theorem 16.4. Let x: U → R3 be a regular patch, and choose a bounded region Q ⊂ U. Then x is minimal on Q if and only if A′ (0) = 0 for a normal variation of x and Q with respect to any h: Q → R. Proof. If H is identically zero for x, then (16.4) implies that A′ (0) = 0 for any h. Conversely, suppose that A′ (0) = 0 for any differentiable function h: Q → R but that there is q ∈ Q for which H(q) 6= 0. Choose h such that h(q) = H(q), with h identically zero outside of a small neighborhood of q on which hH > 0. But now (16.4) implies that A′ (0) < 0. This contradiction shows that H(q) = 0. Since q is arbitrary, x is minimal. We have said nothing about the second derivative of A at 0, so that a minimal surface, although a critical point of A, may not actually be a minimum. More details of this approach, as well as extensions of the results of this section, can be found in [Nits, pages 90–116]. 16.2 Deformation from the Helicoid to the Catenoid We have already considered the helicoid on page 376 and the catenoid on page 464. In fact, these two surfaces constitute the initial and final points of a 16.2. DEFORMATION FROM HELICOID TO CATENOID 505 deformation through isometric minimal surfaces. For each t with 0 6 t 6 π/2, define z[t](u, v) = cos t sinh v sin u, − sinh v cos u, u (16.8) + sin t cosh v cos u, cosh v sin u, v . Figure 16.1: z h nπ 16 i with 0 6 n 6 8 506 CHAPTER 16. A SELECTION OF MINIMAL SURFACES The exact relation between this family of surfaces and the helicoid and catenoid is provided by the equations π π z[0](u, v) = helicoid[1, 1] u − , sinh v + 0, 0, , 2 π z[ ](u, v) 2 = 2 catenoid[1](u, v), verified in Notebook 16. Theorem 16.5. The 1-parameter family of surfaces (16.8) is a deformation from the helicoid to the catenoid such that z[0] is (a reparametrization of) a helicoid and z[π/2] is a catenoid. Furthermore, each z[t] is a minimal surface which is locally isometric to z[0]. In particular, the helicoid is locally isometric to the catenoid. Proof. Let E(t), F (t), G(t) denote the coefficients of the first fundamental form of z[t]. An easy calculation (by hand or with the help of Notebook 16) shows that E(t) = cosh2 v = G(t), F (t) = 0. In particular, E(t), F (t), G(t) are constant functions of t. The result follows from Lemma 12.7 on page 366. It is now an easy matter to plot accurate pictures of the deformation t 7→ z[t] like those of Figure 16.1, that are colored by a function of Gaussian curvature. The immersion z[t] obviously does depend on t; mathematically, this is actually a special case of a subsequent result on page 724. Neither the catenoid nor the helicoid intersects itself; however, every intermediate surface z[t] has self-intersections. It is also true that asymptotic curves on the helicoid are gradually transformed into principal curves on the catenoid (see page 390 for the definitions, and page 725 for the result). Figure 16.2 illustrates two particular regions of the helicoid and catenoid that correspond under the deformation. The boundaries of these regions include a helix and a circle, with respective parametrizations α(u) = sinh 1 sin u, − sinh 1 cos u, u , 0 6 u < 2π, γ(u) = cosh 1 cos u, cosh 1 sin u, 1 , 0 6 u < 2π. The length of the helix is Z 2π p |α′ (u)|du = 2π sinh2 1 + 1, 0 and that of the circle is Z 0 2π |γ ′ (u)|du = 2π cosh 1. The fact that these two lengths are equal is consistent with the statement that the helix maps isometrically onto the circle. 16.3. MINIMAL SURFACES OF REVOLUTION 507 Figure 16.2: Isometric regions of the helicoid and catenoid 16.3 Minimal Surfaces of Revolution The catenoid is the only member of the family z[t] that is a surface of revolution. We next prove that the catenoid and the plane are in fact essentially the only surfaces of revolution that are simultaneously minimal surfaces. The following result is therefore an analogue of Exercise 7 of Chapter 15, that dealt with the case of zero Gaussian curvature. Theorem 16.6. A surface of revolution M which is a minimal surface is con- tained in either a plane or a catenoid. Proof. Let x be a patch whose trace is contained in M, and let α = (ϕ, ψ) be the profile curve. Then x is given by (15.1). There are three cases: Case 1. ψ ′ is identically 0. Then ψ is constant, so that α is a horizontal line and M is part of a plane perpendicular to the axis of revolution. Case 2. ψ ′ is never 0. Then by the inverse function theorem, ψ has an inverse ψ −1 . Define e α(t) = α ψ −1 (t) = h(t), t , where h = ϕ ◦ ψ −1 , and a new patch y by y(u, v) = h(v) cos u, h(v) sin u, v . e is a reparametrization of α, it follows that x and y have the same trace. Since α Thus it suffices to show that the surface of revolution y is part of a catenoid. 508 CHAPTER 16. A SELECTION OF MINIMAL SURFACES In the notation of page 469, equations (15.11) reduce to  h′′ g   , = k = m  G (h′2 + 1)3/2 (16.9)  e 1   kp = = − √ . E h h′2 + 1 It follows from the assumption that H = 0 and (16.9) that h must satisfy the differential equation (16.10) h′′ h = 1 + h′2 . To solve (16.10), we first rewrite it as 2h′ 2h′ h′′ = . ′2 1+h h Integrating both sides yields log(1 + h′2 ) = log(h2 ) − log(c2 ) for some constant c 6= 0, and exponentiating we obtain 2 h 1 + h′2 = (16.11) . c The first-order differential equation (16.11) can be written as 1 h′ /c p = . c (h/c)2 − 1 (16.12) Both sides of (16.12) can be integrated to yield arccosh v h = + b. c c Thus the solution of (16.10) is v h(v) = c cosh +b , c and so M is part of a catenoid. Case 3. ψ ′ is zero at some points, but nonzero at others. In fact, this case cannot occur. Suppose, for example, that ψ ′ (v0 ) = 0, but ψ ′ (v) > 0 for v < v0 . By Case 2, the profile curve is a catenary for v < v0 , whose slope is given by ϕ′ /ψ ′ . Then ψ ′ (v0 ) = 0 implies that the slope becomes infinite at v0 . But this is impossible, since the profile curve is the graph of the function cosh . Nicely complementing Theorem 16.6 is a theorem due to Catalan which assets that the helicoid is the only minimal surface other than the plane which is also ruled (see [dC1]). 16.4. MORE EXAMPLES 509 16.4 More Examples of Minimal Surfaces Enneper’s Minimal Surface One of the simplest minimal surfaces is the one found by Enneper in 1864 (see [Enn1]). It is defined by u3 v3 2 2 2 2 enneper(u, v) = u − + u v , −v + − vu , u − v . 3 3 Determining its first and second fundamental forms was the object of Exercise 6 of Chapter 12 (see Figure 12.13 on page 378) and Exercise 4 of Chapter 13. Figure 16.3: Enneper’s surface with shadows It is easy to verify directly that H is identically zero. Firstly, xu xv whence (16.13) = (1 − u2 + v 2 , −2uv, 2u), = (2uv, −1 − u2 + v 2 , −2v), E = 1 + u2 + v 2 2 = G, F = 0. Setting ρ = 1 + u2 + v 2 helps further calculations, such as xu × xv = 2uρ, 2vρ, ρ2 − 2ρ . 510 CHAPTER 16. A SELECTION OF MINIMAL SURFACES However, we do not need this normal vector, since the vanishing of the mean curvature already follows from (16.13) and the equations xuu = −2(u, v, −1) = −xvv . In spite of the simplicity of its definition, Enneper’s surface is complicated because of self-intersections. These are best visualized by projecting the surface to the coordinate planes as shown in Figure 16.3. Catalan’s Minimal Surface The minimal surface of Catalan3 is parametrized by u v (16.14) x(u, v) = u − sin u cosh v, 1 − cos u cosh v, −4 sin sinh . 2 2 We defer to Notebook 16 for the verification that the mean curvature H is identically zero. Figure 16.4: Catalan’s minimal surface The intersection of the plane z = 0 with the trace of (16.14) contains the coordinate curve (16.15) α(u) = x(u, 0) = (u − sin u, 1 − cos u, 0), that defines a cycloid in the xy-plane. Much of the interest in Catalan’s surface derives from its properties in relation to this curve, that we now investigate. We denote by s an arc length function along the curve, whose speed is therefore √ ds u = |α′ (u)| = 2 − 2 cos u = 2 sin . (16.16) du 2 3 Eugène Charles Catalan (1814–1894). Belgian mathematician, who had difficulty obtaining a position in France because of his left-wing views. Catalan’s constant is ∞ n=0 (−1)n ≈ 0.915966. (2n + 1)2 16.4. MORE EXAMPLES 511 One can use this to write down s as a function of u, though we shall not need an explicit formula. Consider the vectors xu , xv tangent to the surface at points of the curve. The first is the tangent vector to the curve itself, since xu (u, 0) = α′ (u) = (1 − cos u, sin u, 0). By contrast, u xv (u, 0) = 0, 0, −2 sin , 2 and so this second vector is orthogonal to the plane containing the trace of α. The acceleration vector α′′ (s) is perpendicular to α′ (s) = α′ (u) du , ds since the latter has constant norm. But α′′ (s) must lie in the xy-plane, and is thus orthogonal to xv . We have therefore established the next result without any real calculation. Lemma 16.7. The acceleration α′′ (s) of the curve (16.15) is everywhere parallel to the normal vector xu × xv of the surface (16.14). In general, given a curve α whose trace lies on a surface, the tangential component of α′′ (s) defines the so-called geodesic curvature, which in the next chapter we prove depends only on the first fundamental form. The curve α is called pregeodesic if its acceleration vector α′′ (s) satisfies the conclusion of Lemma 16.7. The term geodesic is reserved for the case in which the curve’s parameter is a constant times arc length; for then the mapping s 7→ α(s) satisfies the so-called geodesic equations, that are further explained in Chapter 18. An interesting problem in minimal surface theory is the determination of a minimal surface that contains a given curve as a geodesic (see Section 22.6), or a curve tangent to asymptotic or principal directions. See [Nits, page 140]. Henneberg’s Minimal Surface The following elementary parametrization of the minimal surface of Henneberg4 is given on page 144 of [Nits]: henneberg(u, v) = 2 sinh u cos v − 23 sinh 3u cos 3v, 2 sinh u sin v + 32 sinh 3u sin 3v, 2 cosh 2u cos 2v . 4 Ernst Lebrecht Henneberg (1850–1922). German mathematician. Professor at the University of Darmstadt. 512 CHAPTER 16. A SELECTION OF MINIMAL SURFACES This patch fails to be regular precisely at the points (0, nπ/2), where n is an integer. Note that (16.4) is periodic in v with period 2π, and that (16.17) henneberg(u, v) = henneberg(−u, v + π). Hence for any region U in the right half-plane { (p, q) | p > 0 } there will be a region Ue in the left half-plane { (p, q) | p < 0 } whose image under henneberg is the same subset of R3 . But (16.17) implies that the unit normal U of henneberg satisfies U(u, v) = −U(−u, v + π). e will have opposite orientations. An inHence henneberg(U) and henneberg(U) stance of this is illustrated by the different shadings on the left and right of Figure 16.5. Figure 16.5: The henneberg images of {(p, q) | 1 3 6 |p| 6 56 , −π 6 q < π } In Chapter 22, we shall show how to associate to a minimal surface M a f and an isometric deformation z[t] from M to M f conjugate minimal surface M, through minimal surfaces, that generalizes equation (16.8). It turns out that the conjugate of Henneberg’s surface is given by hennebergconj(u, v) = 2 cosh u sin v − 23 cosh 3u sin 3v, 2 cosh u cos v + 23 cosh 3u cos 3v, 2 sinh 2u sin 2v , with a parametrization that is obtained from henneberg by merely interchanging sinh u ↔ cosh u and sin v ↔ cos v (see Exercise 12 on page 754). The resulting surface is apparently more complicated to visualize (Figure 16.6). Notebook 16 contains a verification that its mean curvature vanishes. 16.5. MONGE PATCHES AND SCHERK’S SURFACE 513 Figure 16.6: A realization of the conjugate Henneberg minimal surface 16.5 Monge Patches and Scherk’s Minimal Surface We have the following immediate consequence of Lemma 13.34, page 409: Lemma 16.8. A Monge patch (u, v) 7→ u, v, h(u, v) is a minimal surface if and only if (16.18) (1 + h2v )huu − 2hu hv huv + (1 + h2u )hvv = 0. One can hope to find interesting examples of minimal surfaces by assuming that h has a special form. In 1835 Scherk5 determined the minimal surfaces of the form (u, v) 7→ u, v, f (u) + g(v) (see [Scherk], [BaCo]). Theorem 16.9. If a Monge patch x: U → M with h(u, v) = f (u) + g(v) is a minimal surface, then either M is part of a plane or there are constants a, c1 , c2 , c3 , c4 , with a 6= 0, such that (16.19) 1 f (u) = − log cos(au + c1 ) + c2 , a 1 log cos(av + c3 ) + c4 . g(v) = a 5 Heinrich Ferdinand Scherk (1798–1885). German mathematician, who also worked in number theory. He studied with Bessel in Königsberg, and was elected rector of the University of Kiel three times, but was forced to leave in 1848 by the Danes. 514 CHAPTER 16. A SELECTION OF MINIMAL SURFACES Proof. When h(u, v) = f (u) + g(v), we have huu = f ′′ (u), hvv = g ′′ (v). huv = 0, Hence equation (16.18) reduces to (16.20) −g ′′ (v) f ′′ (u) = . 1 + f ′ (u)2 1 + g ′ (v)2 Here u and v are independent variables, so each side of (16.20) must equal a constant, call it a. If a = 0, then both f and g are linear, so that M is part of a plane. Otherwise, the two equations (16.21) f ′′ (u) −g ′′ (v) = a = 1 + f ′ (u)2 1 + g ′ (v)2 are easily solved by integrating twice. The result is (16.19). Figure 16.7: A piece of Scherk’s minimal surface As suggested by Theorem 16.9, we define Scherk’s minimal surface by cos av 1 scherk[a](u, v) = u, v, log . a cos au For simplicity, we also take a = 1. Figure 16.7 shows the image by this patch of (almost) the open rectangle −π/2 < u, v < π/2. In fact, scherk[1][u, v] is well-defined on the set R = (u, v) | cos u cos v > 0 that can be imagined as the union of the black squares on an infinite chess board. To see this, consider a single square π π π π Q(m, n) = (x, y) mπ − < x < mπ + , nπ − < y < nπ + 2 2 2 2 16.5. MONGE PATCHES AND SCHERK’S SURFACE 515 that we color black if m + n is even and white if m + n is odd. Then [ Q(m, n) | m and n are integers with m + n even . R= It is easy to see that scherk[1](u + 2mπ, v + 2nπ) = scherk[1](u, v) for all real u and v and all integers m and n. Hence the piece of Scherk’s surface over a black square Q(m, n), m + n even, is a translate of the portion over Q(0, 0). We can now plot identical pieces of the surface over the black squares, and these fit together to produce the entire surface. Figure 16.8: Chequered board Figure 16.9: Scherk’s minimal surface 516 CHAPTER 16. A SELECTION OF MINIMAL SURFACES Actually, Figure 16.9 was obtained in Notebook 16 from an implicit equation of Scherk’s surface. 16.6 The Gauss Map of a Minimal Surface Orientability was defined in Section 11.1 using the notion of a complex structure J , and this led to Definition 11.3 of the Gauss map on page 333. We next show that the Gauss map of a minimal surface has special properties that leads one to use complex analysis to advantage in the study of minimal surfaces. First, we need the following notion: Definition 16.10. Let M1 , M2 be oriented regular surfaces in Rn , and let J 1 , J 2 be the corresponding complex structures. A map Φ: M1 → M2 is called (i) a complex map if (16.22) Φ∗ ◦ J 1 = J 2 ◦ Φ∗ . (ii) an anticomplex map if (16.23) Φ∗ ◦ J 1 = −J 2 ◦ Φ∗ . An elementary but important property of complex and anticomplex maps (also called holomorphic and antiholomorphic maps) is Lemma 16.11. Let Φ: M1 → M2 be a complex or anticomplex map for which Φ∗ is nowhere zero. Then Φ is a conformal map. Proof. Let p ∈ M1 . Choose a nonzero tangent vector vp to M1 at p. We can assume that kΦ∗ (vp )k = λkvp k for some λ > 0. Since Φ is complex or anticomplex, abbreviating (J 1 )p to Jp , we have kΦ∗ (J p vp )k = λkvp k and Φ∗ (vp ) · Φ∗ (J p vp ) = 0. It now follows that kΦ∗ (avp + bJ p vp )k2 = λ2 kavp + bJ p vp k2 for all a, b ∈ R. Since {vp , Jp vp } is a basis of Mp , we can set λ = λ(p) in Definition 12.11 on page 370 to deduce that Φ is conformal. We have a ready-made example of an anticomplex map. Theorem 16.12. The Gauss map of an oriented minimal surface M ⊂ R3 is anticomplex, and (16.24) J p Sp = −Sp J p for all p ∈ M, where Sp is the shape operator of M at p. 16.6. GAUSS MAP OF A MINIMAL SURFACE 517 Proof. Let p 7→ U(p) be the Gauss map of M. Let p ∈ M, and let {e1 , e2 } be an orthonormal basis of Mp which diagonalizes Sp , and let k1 , k2 be the corresponding principal curvatures. The unit vector Up determines a complex structure J p on Mp , via (11.1) on page 332, that satisfies J p e1 = ±e2 and J p e2 = ∓e1 . We use the fact that M is a minimal surface to compute J p Sp e1 = J p k1 e1 = ±k1 e2 = ∓k2 e2 = ∓Sp e2 = −Sp J p e1 . Similarly, J p Sp e2 = −Sp J p e2 . In this way, we have established (16.24). But Lemma 13.5, on page 388, tells us that Sp is the negative of the tangent map of U at p. Hence (16.24) implies that the Gauss map is anticomplex. Figure 16.10: Enneper’s surface and its Gauss image The shape operator Sp at a point of M vanishes if and only if k1 = 0 = k2 , so that (by definition) p is a planar point. On a minimal surface this is equivalent to asserting that the Gaussian curvature K(p) is zero. Strictly speaking, we need to exclude such points when applying Lemma 16.11 to the conclusion of Theorem 16.12. We then obtain Corollary 16.13. The Gauss map of a minimal surface without planar points is conformal. Conformality means that the Gauss map preserves the proportions of infinitesimally small rectangles. When we plot a surface in the usual way with a sufficiently fine mesh, it is divided into lots of small, approximately rectangular shapes. The ratios of the lengths of the sides of these rectangles, and also the angles between the sides must be approximately preserved by the Gauss map. This phenomenon can be examined in Figure 16.10 which plots Enneper’s surface along with its image under the Gauss map. 518 CHAPTER 16. A SELECTION OF MINIMAL SURFACES 16.7 Isothermal Coordinates In this final section, we shall define an important type of patch that can be found on an arbitrary surface. It has a special significance for minimal surfaces, and will play an important role in the subsequent study in Chapter 22 in which we shall make more explicit the role of complex structures. Definition 16.14. Let U ⊆ R2 be an open subset. A patch x: U → Rn is called isothermal if there exists a differentiable function λ: U → R such that (16.25) xu · xu = xv · xv = λ2 and xu · xv = 0. We call λ the scaling function of the isothermal patch. Note that such a patch is regular at all points where λ is nonzero. The intuitive meaning of an isothermal patch x can be described as follows. Since xu and xv have the same length and are orthogonal, an isothermal patch maps an infinitesimal square in U into an infinitesimal square on its image. A more general patch would transform an infinitesimal square into an infinitesimal quadrilateral. The name ‘isothermal’ is due to Lamé 6 , and dates from 1833. The following lemma is obvious from the definition of conformal map on page 370. Lemma 16.15. A patch x: U → Rn is isothermal if and only if it is conformal, considered as a mapping x: U → x(U). Let us first deal with the case of the sphere. Its standard parametrization, given on page 288, is (16.26) x(u, v) = a cos u cos v, sin u cos v, sin v . The metric of this patch was computed to be (16.27) ds2 = a2 cos2 v du2 + dv 2 (this is equation (12.26) on page 374), so the patch is certainly not isothermal. As a hint of what is in store in Chapter 22, we use complex numbers to factor the right-hand side of (16.27) so as to obtain i dv i dv 2 2 2 ds = a cos v du + (16.28) du − . cos v cos v 6 Gabriel Lamé (1795–1870). French engineer, mathematician and physicist. He worked on a wide variety of topics, such as number theory, differential geometry (we shall mention his work on triply orthogonal systems in Chapter 19), elasticity (where two elastic constants are named after him) and difusion in crystalline material. 16.7. ISOTHERMAL COORDINATES 519 This has the effect of partially separating the variables, and if we set w = log(tan v + sec v), then dv . dw = cos v Moreover, sin v + 1 cos v 2 ew + e−w = + = , cos v sin v + 1 cos v so that cos v = sech w. Thus, (16.28) becomes ds2 = λ2 (du2 + dw2 ) with λ = a sech w. In summary, Lemma 16.16. The patch (16.29) y(u, w) = a cos u sech w, sin u sech w, tanh w is an isothermal patch on a sphere S 2 (a) of radius a. This patch is a special case of Mercator’s parametrization of the ellipsoid given in Exercise 6 of Chapter 10. But a computation in Notebook 16 confirms that it is only isothermal in the spherical case. It follows from Corollary 16.13 and Lemma 16.16 that the composition p 7→ U(p) 7→ y−1 (p), defined on a suitable open subset of M, is conformal. Its inverse will define an isothermal patch on the minimal surface M. The following well-known result actually guarantees the existence of isothermal patches on an arbitrary surface. Theorem 16.17. Let M be a surface, and suppose ds2 is a metric on M. Let p ∈ M. Then there exists an open set U ⊂ R2 and an isothermal patch x: U → M such that p ∈ x(U) and ds2 = λ2 (du2 + dv 2 ). For a proof, see [Bers, page 15], [Os1, page 31] or [FoTu, page 54]. It is also possible to argue in the way that led to Lemma 16.16. For we can factor an arbitrary metric ds2 = E dp2 + 2F dpdq + Gdq 2 as ! ! √ √ √ √ F + i EG − F2 F − i EG − F2 2 √ √ ds = dq dq . E dp + E dp + E E One can then establish the existence of a complex-valued integrating factor µ for which the differential system √ √ F + i EG − F2 √ µ(du + i dv) = E dp + dq E 520 CHAPTER 16. A SELECTION OF MINIMAL SURFACES can be resolved for real-valued functions u, v; see [AhSa, pages 125–126] for details. Setting λ = |µ| gives ds2 = |µ|2 (du + i dv)(du − i dv) = λ2 (du2 + dv 2 ), and the coordinates u, v provide an isothermal patch. We finish with a result of special significance for minimal surfaces. Lemma 16.18. Let x: U → R3 be a regular isothermal patch with scaling func- tion λ and mean curvature H. Then we have xuu + xvv = 2λ2 H U, (16.30) where U = (xu × xv )/kxu × xv k is the unit normal. Proof. Since x is isothermal, we can differentiate equations (16.25), obtaining xuu · xu = xuv · xv and xvv · xu = −xvu · xv . Therefore, (xuu + xvv ) · xu = xuv · xv − xvu · xv = 0. Similarly, (xuu + xvv ) · xv = 0. It follows that xuu + xvv is normal to M, and so xuu + xvv is a multiple of U. To find out which multiple, we use Theorem 13.25, page 400, and the assumption that x is isothermal to compute H= (xuu + xvv ) · U e+g e G − 2f F + g E = , = 2 2 2(E G − F ) 2λ 2λ2 so that we get (16.30). We denote by ∆= ∂2 ∂2 + ∂u2 ∂v 2 the Laplacian of R2 , as defined in Section 9.5. Extending this to vector-valued functions, we may write ∆x = xuu + xvv , and conclude the chapter with Corollary 16.19. A minimal isothermal patch satisfies the Laplace equation ∆x = 0. . 16.8. EXERCISES 521 16.8 Exercises 1. Prove that the unit normal of the patch z[t] defined at the top of page 505 does not depend upon the variable t. Show that the coefficients e(t), f (t) and g(t) of the second fundamental form of z[t] satisfy e(t)(u, v) = −g(t)(u, v) = − sin t, f (t)(u, v) = cos t. More general results can be found in Section 22.1. M 2. Check that the surfaces of Henneberg and Catalan are indeed minimal surfaces and compute the Gaussian curvature of each. 3. Prove that the acceleration α′′ (s) of the curve (16.15) is parallel to the u u vector (cos , sin ), where the parameters u, s are related by (16.16). 2 2 4. Verify that the image of the patch (16.29) is contained in the sphere S 2 (a). Compute xu and xv and prove directly that the patch is isothermal. M 5. Scherk’s fifth minimal surface is defined implicitly by (16.31) sin z = sinh x sinh y. Show that it indeed is a minimal surface. Figure 16.11: Scherk’s fifth minimal surface plotted implicitly 522 CHAPTER 16. A SELECTION OF MINIMAL SURFACES 6. Show that a surface of revolution is isothermal if and only if ψ ′2 = ϕ2−ϕ′2 , with notation from page 462. Hence the surface catenoid[a] is isothermal if and only if a = ±1, even though it is a minimal surface for all a. M 7. Portions of Scherk’s fifth minimal surface are parametrized by scherk5[a, b, c](u, v) = a arcsinh u, b arcsinh v, c arcsin(uv) , where a, b, c are each equal to ±1. Show that scherk5[a, b, c] has zero mean curvature H, and compute its Gaussian curvature K. Figure 16.12: Part of Scherk’s fifth minimal surface 8. Let x be an isothermal patch. Show that the Weingarten equations (13.10), page 394, simplify to  f e    S(xu ) = λ2 xu + λ2 xv , (16.32)    S(x ) = f x + g x . v u v λ2 λ2 M 9. A twisted generalization of Scherk’s minimal surface is given by scherk[a, θ](u, v) = sec 2θ(u cos θ + v sin θ), sec 2θ(u sin θ + v cos θ), cos sec 2θ(u sin θ + v cos θ) 1 . log a cos sec 2θ(u cos θ + v sin θ) Show that scherk[a, θ] is minimal, and compute its Gaussian curvature. Chapter 17 Intrinsic Surface Geometry The second fundamental form of a regular surface M ⊂ R3 helps to describe precisely how M sits inside the Euclidean space R3 . The first fundamental form of M, on the other hand, can be used to measure distance on M without reference to the ambient space. In other words, distance is intrinsic to M, and does not require knowledge of the second fundamental form. It is a remarkable fact that the Gaussian curvature K of a regular surface M ⊂ R3 is also intrinsic to M, in spite of the fact that K is the product of two principal curvatures, each of which is nonintrinsic. In order to prove this fact, we establish Brioschi’s formula in Section 17.1. It is an explicit (but complicated) formula for K in terms of E, F, G and their first and second partial derivatives. Brioschi’s formula is used in Section 17.2 to prove Gauss’s Theorema Egregium, which states that a surface isometry preserves the Gaussian curvature. The Theorema Egregium has enormous practical value as well as theoretical importance. It implies, for example, that no portion of the earth, which has nonzero Gaussian curvature, can be mapped isometrically onto a plane – maps of the earth cannot fail to distort distances in one way or another. It also makes it possible to study surfaces in the more abstract setting of Chapter 26. Christoffel symbols of surfaces are defined and discussed in Section 17.3. In Section 17.4, we study the geodesic curvature κg and geodesic torsion τg of a curve on a surface. The geodesic curvature is the generalization to surfaces of the signed curvature κ2 that we used in Chapters 1 – 5 to study plane curves. An analog of the Frenet formulas (7.12), called the Darboux formulas, is established for a curve on a surface. 17.1 Intrinsic Formulas for the Gaussian Curvature We begin by giving a formula for the Gaussian curvature of a patch x in terms of the dot products of the first and second partial derivatives of x. 531 532 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Theorem 17.1. Let x: U → R3 be a patch. Then the Gaussian curvature of x is given by   xuu · xvv        1 det K=  xu · xvv 2 2 (E G − F )       xv · xvv xuu · xu xu · xu xuu · xv    xu · xv    xv · xv xv · xu  xuv · xuv   − det  xu · xuv  xv · xuv xuv · xu xuv · xv xu · xu xu · xv xv · xu xv · xv        .      Proof. It follows from Lemma 13.31, page 405, and Theorem 13.25, page 400, that eg − f2 [xuu xu xv ][xvv xu xv ] − [xuv xu xv ]2 K= (17.1) = , EG − F2 (E G − F 2 )2 where [xuu xu xv ] denotes the vector triple product (page 193). Also, if we write x = (x1 , x2 , x3 ), then (17.2) Similarly, (17.3) [xuu xu xv ][xvv xu xv ]     x1uu x2uu x3uu x1vv x2vv x3vv         = det x1u x2u x3u  det x1u x2u x3u      x1v x2v x3v x1v x2v x3v    x1vv x1u x1v x1uu x2uu x3uu       = det x1u x2u x3u   x2vv x2u x2v     x3vv x3u x3v x1v x2v x3v   xuu · xvv xuu · xu xuu · xv     = det xu · xvv xu · xu xu · xv .   xv · xvv xv · xu xv · xv  xuv · xuv   [xuv xu xv ]2 = det xu · xuv  xv · xuv xuv · xu xu · xu xv · xu xuv · xv    xu · xv .  xv · xv The equation of the theorem follows from (17.1), (17.2) and (17.3). 17.1. INTRINSIC FORMULAS FOR GAUSSIAN CURVATURE 533 Except for xuu · xvv and xuv · xuv , we know how to express each of the dot products in the statement of Theorem 17.1 in terms of E, F, G and their derivatives. Next, we show that we can write at least the difference of the remaining two dot products in terms of the second derivatives of E, F and G. Lemma 17.2. Let x: U → R3 be a patch. Then (17.4) xuu · xvv − xuv · xuv = − 21 Evv + Fuv − 21 Guu . Proof. We have (17.5) xuu · xvv − xuv · xuv = (xu · xvv )u − xu · xvvu −(xu · xuv )v + xu · xuvv . Since xvvu = xuvv , equation (17.5) reduces to xuu · xvv − xuv · xuv = (xu · xvv )u − (xu · xuv )v = (xu · xv )v − xuv · xv − 12 (xu · xu )vv u = (xu · xv )vu − 1 2 (xv · xv )uu − 12 (xu · xu )vv = − 12 Evv + Fuv − 21 Guu . Now we are ready to derive the formula of Brioschi1 (see [Brio]). It is a great deal more complicated than (17.1), but it has the advantage that it expresses the Gaussian curvature of a patch x: U → R3 entirely in terms of the first fundamental form. Theorem 17.3. (Brioschi’s Formula) Let x: U → R3 be a patch. Then the Gaussian curvature of x is given by   1 − 2 Evv + Fuv − 21 Guu      1  K= det Fv − 21 Gu 2 2  (E G − F )     1 2 Gv 1 2 Eu Fu − 21 Ev E F F  G   − det  1      0 1 2 Ev 1 2 Gu 1 2 Ev E F 1 2 Gu F G        .     Francesco Brioschi (1824–1897). Italian mathematician. Professor first in Padua, then in Milan. Brioschi was one of the founders of modern mathematics in Italy. In addition to his work in differential geometry, Brioschi used elliptical modular functions to solve equations of the fifth and sixth degrees. 534 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Proof. We can (with the help of equations (17.9) below) rewrite the statement of Theorem 17.1 as    xuu · xvv 12 Eu Fu − 12 Ev       1   1 det K=  G E F F −  u v 2 2 2  (E G − F )      1 G F G 2 v   xuv · xuv 21 Ev 12 Gu        1 − det . E E F  2 v      1 F G 2 Gu It is possible to expand both of these 3 × 3 determinants by means of 2 × 2 determinants. In the resulting expansion each of the expressions xuu · xvv and xuv · xuv occurs with the same factor E G − F 2 . Thus we can manipulate the determinants using (17.4) so as to obtain the required formula for K. Brioschi’s formula in its full generality is tedious to use in hand calculations. However, in a frequently occurring case Brioschi’s formula simplifies considerably. Corollary 17.4. Let x: U → R3 be a patch for which F = 0. Then the Gaussian curvature of x is given by ( √ ! √ !) ∂ −1 ∂ 1 ∂ G 1 ∂ E √ √ K = √ (17.6) + ∂v E G ∂u E ∂u G ∂v G E −1 ∂ ∂ √ u √ v = √ + . ∂v 2 E G ∂u EG EG Proof. When F = 0, Brioschi’s Formula reduces to   1  − 2 Evv − 21 Guu 21 Eu − 21 Ev     1     1 det K = G E 0 −   2 u (E G)2       1 0 G 2 Gv   1 1 0  2 Ev 2 Gu      1  − det 2 Ev E 0       1 G 0 G u 2 1 n − 12 Evv − 21 Guu E G = 2 (E G) o + 41 Eu Gu G + 14 Ev Gv E + 14 Ev2 G + 41 G2u E , 17.2. GAUSS’S THEOREMA EGREGIUM 535 so that K=− Guu Eu Gu Ev2 Ev Gv G2u Evv − + + + + . 2 2 2 2E G 4E G 4E G 2E G 4E G 4 E2G Then (17.6) follows by combining this equation with the expression ( √ ! √ ) √ Gv ∂ E ∂ 1 1 1 ∂ E 1 ∂2 E √ √ − = √ +√ 3 E G ∂v G ∂v EG G ∂v 2 2G 2 ∂v Gv Ev 1 ∂ −1 1 2 E Ev − + √ = √ 3√ EG 2 G ∂v 4G 2 E Gv Ev Ev2 1 Evv 1 √ √ √ − − + + = √ 3 3 EG 2 G E 4G 2 E 2E 2 =− Gv Ev Ev2 Evv − + , 2 2 4G E 4E G 2E G together with 1 ∂ √ E G ∂u √ ! Eu Gu 1 ∂ G Guu G2u √ =− 2 − + . 2 4E G 4E G 2E G E ∂u Notebook 17 incorporates both the extrinsic formula of Theorem 17.1, and Brioschi’s implicit formula in Theorem 17.3. There we verify the consistency of these equations with those of Chapter 13. 17.2 Gauss’s Theorema Egregium We can now prove one of the most celebrated theorems of the 19th century. Mathematicians at the end of the 18th century such as Euler and Monge had used the Gaussian curvature, but only defined as the product of the principal curvatures. Since each principal curvature depends on the particular way the surface is embedded in R3 , there is no obvious reason to suppose that the product of the principal curvatures is intrinsic to M. Gauss’s discovery ([Gauss2], published in 1828) that the product of the principal curvatures depends only on the intrinsic geometry of the surface revolutionized differential geometry. This is the Theorema Egregium. Gauss’s proof of the Theorema Egregium is not simple. The difficulty of the proof can be compressed into Brioschi’s formula of Theorem 17.3. (A similar formula already occurs in [Gauss1].) As a matter of fact, Brioschi derived his formula in [Brio] in order to provide a new proof of the Theorema Egregium. Another proof of the Theorema Egregium using geodesic balls was given in the 1848 paper [BDP]. 536 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Theorem 17.5. (Gauss’s Theorema Egregium2) Let Φ: M1 → M2 be a local isometry between regular surfaces M1 , M2 ⊂ R3 . Denote the Gaussian curvatures of M1 and M2 by K1 and K2 . Then K1 = K2 ◦ Φ. Proof. Let x: U → R3 be a regular patch on M1 , and put y = Φ ◦ x. Then the restriction Φ|x(U): x(U) → y(U) is a local isometry. Lemma 12.7, page 366, tells us that Ex = Ey , Fx = Fy , Gx = Gy . But then (Ex )u = (Ey )u , (Fx )u = (Fy )u , (Gx )u = (Gy )u , and similarly for the other partial derivatives. In particular, Theorem 17.3 implies that Kx = Ky , where Kx and Ky denote the Gaussian curvatures of x and y. Hence K2 ◦ Φ = Ky ◦ y−1 ◦ Φ = Kx ◦ x−1 = K1 . For another proof of Theorem 17.5, see page 601. Examples Illustrating the Theorema Egregium (i) The patch x: R2 → R3 defined by x(u, v) = cos u, sin u, v can be considered to be a surface mapping between the plane and a circular cylinder. It is easy to check that E = G = 1 and F = 0 for both the plane and the parametrization x of the cylinder; hence x is a local isometry. On the other hand, x preserves neither the second fundamental form nor the mean curvature. (ii) We showed in Section 16.2 that the helicoid is locally isometric to the catenoid. The two surfaces therefore have the same Gaussian curvature, and this fact is graphically illustrated by Figure 16.1 on page 505. (iii) A sphere and plane are not locally isometric because the Gaussian curvature of a sphere is nonzero, unlike that of the plane. The earth is not exactly a sphere, but still has nonzero Gaussian curvature. As we remarked in the chapter’s introduction, this is the precise reason why any flat map of a portion of the earth must distort distances. 2 Si superficies curva in quamcumque aliam superficiem explicatur, mensura curvaturae in singulis punctis invariata manet. 17.3. CHRISTOFFEL SYMBOLS 537 (iv) Theorem 17.5 states that a local isometry preserves curvature. There are, however, curvature preserving diffeomorphisms that are not local isometries. We describe a classical example of such a map Φ from the funnel surface funnel[a, a, b] parametrized on page 564 of the next chapter to the helicoid parametrized on page 377. It is given by Φ(av cos u, av sin u, b log v) = (av cos u, av sin u, b u). The Gaussian curvatures of the two surfaces are computed in Notebook 17 using Bioschi’s formula, and they turn out to be the same, namely −b2 . (b2 + a2 v 2 )2 Hence Φ is a diffeomorphism which preserves curvature. The metric of the funnel surface is given by b2 ds2 = a2 v 2 du2 + a2 + 2 dv 2 , v while the metric of the helicoid is given by ds2 = (b2 + a2 v 2 )du2 + a2 dv 2 . There is no local isometry between the two surfaces, since the v-parameters would have to correspond to preserve K, but then E would change. (v) In a similar vein, we noted on page 404 that any rotation about the z-axis preserves the Gaussian curvature of the monkey saddle. It can be checked (see Exercise 9) that only the rotations by an angle which is an integer multiple of 2π/3 are isometries. Like (iv), this example also shows that the converse to Theorem 17.5 is false. 17.3 Christoffel Symbols Let x: U → R3 be a regular patch, and let U= xu × xv kxu × xv k be its associated unit normal vector. According to the definitions (13.9), page 394, of e, f, g, the normal components of xuu , xuv , xvv are given by (17.7) x⊥ uu = e U, In these formulas, the superscript tangent space. x⊥ uv = f U, ⊥ x⊥ vv = g U. denotes the component orthogonal to the 538 CHAPTER 17. INTRINSIC SURFACE GEOMETRY For many applications, one also needs the tangential components x⊤ uu , x⊤ uv , x⊤ vv of the patch’s second partial derivatives. Since x is regular, xu , xv and U form a basis for the tangent space R3p to R3 at each point p in the trace of x. It follows that there exist six functions Γ111 , such that (17.8) Γ211 , Γ112 , Γ212 , Γ211 , Γ222   xuu = Γ111 xu + Γ211 xv + e U,     xuv = Γ112 xu + Γ212 xv + f U,      x = Γ1 x + Γ2 x + g U. vv 22 u 22 v Given that U · U = 1, and U · xu = 0 = U · xv = 0, the coefficients e, f, g of U on the far right of (17.8) ensure that (17.7) is satisfied. Bearing in mind the symmetry of the mixed partial derivatives, one also defines Γ121 = Γ112 , Γ221 = Γ212 , since consistent use of the subscript notation also requires that the equation xvu = Γ121 xu + Γ221 xv + f U be valid. The functions Γijk are called the Christoffel3 symbols, and were originally defined in [Chris]. It turns out that the tangential components of xuu , xuv , xvv are ultimately expressible in terms of E, F, G and their first and second partial derivatives. In order to prove this, we first state Lemma 17.6. Let x: U → R3 be a regular patch. Then (17.9) 3  1 Γ11 E + Γ211 F = 12 Eu ,      Γ112 E + Γ212 F = 12 Ev ,      1 Γ22 E + Γ222 F = Fv − 21 Gu , Γ111 F + Γ211 G = Fu − 21 Ev , Γ112 F + Γ212 G = 12 Gu , Γ122 F + Γ222 G = 12 Gv . Elwin Bruno Christoffel (1829–1900). German mathematician, professor in Strasbourg. 17.3. CHRISTOFFEL SYMBOLS 539 Proof. To determine the tangential coefficients in (17.8), one takes the scalar product of each of the equations in (17.8) with xu and xv , thus giving Γ111 E + Γ211 F = xuu · xu = 1 ∂ kxu k2 = 2 ∂u 1 2 Eu , ∂ (xu · xv ) − xu · xuv = Fu − 12 Ev , ∂u 1 ∂ xuv · xu = kxu k2 = 12 Ev , 2 ∂v 1 ∂ kxv k2 = 12 Gu , xuv · xv = 2 ∂u ∂ xvv · xu = (xu · xv ) − xv · xuv = Fv − 21 Gu , ∂v 1 ∂ xvv · xv = kxv k2 = 21 Gv . 2 ∂v Γ111 F + Γ211 G = xuu · xv = Γ112 E + Γ212 F = Γ112 F + Γ212 G = Γ122 E + Γ222 F = Γ122 F + Γ222 G = This establishes (17.9). We can now state Theorem 17.7. Let x: U → R3 be a regular patch. Then  GEu − 2F Fu + F Ev   Γ111 = ,   2(E G − F 2 )       GEv − F Gu , Γ112 = 2(E G − F 2)        2GFv − GGu − F Gv    Γ122 = , 2(E G − F 2 ) Γ211 = 2EFu − EEv − F Eu , 2(E G − F 2 ) Γ212 = E Gu − F Ev , 2(E G − F 2 ) Γ222 = E Gv − 2F Fv + F Gu . 2(E G − F 2 ) Proof. These equations are easily obtained by solving the system (17.9) in pairs. Theorem 17.7 will be needed when we study geodesics in Chapter 18 and geodesic curvature in Sections 17.4 and 26.3. In one significant case, the equations simplify greatly: Corollary 17.8. Suppose that x is a patch for which F is identically zero. Then the Christoffel symbols are given by  Eu −Ev  Γ111 = , Γ211 = ,    2E 2G    Gu Ev , Γ212 = , Γ1 =  12 2E 2G      Gv  Γ1 = −Gu , Γ222 = . 22 2E 2G 540 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Corollary 17.9. Let x be a regular patch. Then (17.10)  p   Γ111 + Γ212 = log E G − F 2 , u p   Γ112 + Γ222 = log E G − F 2 . v Proof. We prove the first equation of (17.10). Using Theorem 17.7, we compute 2(E G − F 2 )(Γ111 + Γ212 ) = GEu − 2F Fu + E Gu = (E G − F 2 )u . Hence Γ111 + Γ212 = p (E G − F 2 )u 2 . E G − F = log 2(E G − F 2 ) u Here is an application to minimal surfaces. It refers to isothermal patches, introduced in Section 16.7 at the end of the previous chapter. Their relevance here is that such patches satisfy the hypothesis of Corollary 17.8. Theorem 17.10. The Gaussian curvature K of an isothermal patch x: U → R3 is given in terms of the scaling function λ by K= (17.11) −∆ log λ . λ2 Furthermore, the Christoffel symbols of x are given by (17.12) Γ111 = −Γ122 = Γ212 = λu λ and Γ112 = −Γ211 = Γ222 = λv . λ Proof. Formula (17.11) is a special case of (17.6). In detail, ∂ 1 ∂λ 1 ∂λ −1 ∂ + K = 2 λ ∂u λ ∂u ∂v λ ∂v 2 ∂2 −∆ log λ −1 ∂ log λ + log λ = . = 2 λ ∂u2 ∂v 2 λ2 Similarly, (17.12) is a special case of Corollary 17.8. 17.4 Geodesic Curvature of Curves on Surfaces Let M ⊂ Rn be a regular surface and β : (c, d) → M a unit-speed curve. We know that the acceleration of β in Rn has a component parallel to M and a component perpendicular to M. Therefore, we can write (17.13) β′′ (s) = β ′′ (s)⊤ + β ′′ (s)⊥ for c < s < d, where β ′′ (s)⊤ is by definition the component of β ′′ (s) tangent to M, and (it follows that) β′′ (s)⊥ is the component parallel to a normal vector to the surface. 17.4. GEODESIC CURVATURE OF CURVES 541 e g [β] and the normal curDefinition 17.11. The unsigned geodesic curvature κ vature κn [β] of a unit-speed curve β : (c, d) → M ⊂ Rn are given by (17.14) e g [β](s) = kβ′′ (s)⊤ k κ and κn [β](s) = kβ′′ (s)⊥ k. The normal and unsigned geodesic curvatures of an arbitrary curve α are defined to be the respective curvatures of a unit-speed reparametrization of α. The expression ‘normal curvature’ was introduced for surfaces in Chapter 13, and we shall see shortly (via Lemma 17.15) that there is a close link between these definitions. We first establish a simple relation between the quantities (17.14) and the curvature κ[β] of a space curve that was defined in Section 7.2. Lemma 17.12. Let M ⊂ Rn be a regular surface and β : (c, d) → M a unit- speed curve. Then the unsigned geodesic and normal curvatures of β are related to the curvature κ[β] of β as a curve in Rn by the formula Proof. We have e g [β]2 + κn [β]2 . κ[β]2 = κ κ[β](s)2 = kβ′′ (s)k2 = kβ ′′ (s)⊤ k2 + kβ ′′ (s)⊥ k2 e g [β](s)2 + κn [β](s)2 . = κ Next, we specialize to an oriented regular surface M ⊂ R3 with unit normal e g [β]. The associated complex U, so that we can define a refined version of κ structure of M is given by J vp = U × vp , where vp is any tangent vector to M (page 332). Let β : (c, d) → M be a unit-speed curve. Since β′′ (s) · β ′ (s) = 0, it follows that β′′ (s)⊤ is a multiple of J β ′ (s) = U × β ′ (s) for c < s < d. Definition 17.13. Let M be an oriented regular surface in R3 . The geodesic curvature κg [β] of a unit-speed curve β : (c, d) → M is given (for c < s < d) by (17.15) β ′′ (s)⊤ = κg [β](s)J β ′ (s) = κg [β](s)U × β ′ (s). The geodesic curvature of a curve α : (a, b) → R is again defined to be that of the unit-speed reparametrization of α. To make the general case precise, given α we may compute its arc length s = s(t) measured from any given point α(c), and the inverse function t = t(s). We may then define β(s) = α(t(s)), (17.16) κg [α](t) = κg [β](s(t)). We return to this situation in the proof of Lemma 17.16. 542 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Here is the refined version of Lemma 17.12: Corollary 17.14. Let M be an oriented regular surface in R3 and suppose that β : (c, d) → M is a unit-speed curve. Then e g [β] |κg [β]| = κ and κ[β]2 = κg [β]2 + κn [β]2 . Proof. We have e g [β](s) = kβ′′ (s)⊤ k = kκg [β](s)J β ′ (s)k κ = |κg [β](s)| kJ β′ (s)k = |κg [β](s)|. We collect some elementary facts about the curvatures κg and κn for the case of a curve in an oriented regular surface M ⊂ R3 . First, we show that the normal curvature of a unit-speed curve contains the same geometric information as the normal curvature that we defined in Section 13.2. Lemma 17.15. Let M ⊂ R3 be an oriented regular surface, and choose a unit surface normal U to M. Let up be a unit tangent vector to M at p. If β is any unit-speed curve in M with β(s) = p and β′ (s) = up , then κn [β](s) = k(up ), where k(up ) denotes the normal curvature of M in the direction up . Proof. By definition (see page 389), we have k(up ) = S(up ) · up = S β′ (s) · β′ (s). But according to Lemma 13.4, page 387, we have S β ′ (s) · β ′ (s) = β′′ (s) · U. Because (β ′′ (s) · U)U is the component of β′′ (s) perpendicular to M, the lemma follows from the definition of κn [β]. Next, we find simple formulas for the geodesic and normal curvatures when the curve in question does not necessarily have unit speed. Recall that [u v w] denotes the vector triple product u · (v × w), defined on page 193. Lemma 17.16. Let α : (a, b) → M ⊂ R3 be an arbitrary-speed curve in an oriented regular surface M in R3 . Let U be a surface normal to M. Then the normal and geodesic curvatures of α are given (for a < t < b) by κn [α](t) = (17.17) and (17.18) κg [α](t) = α′′ (t) · U kα′ (t)k2 α′′ (t) · J α′ (t) [α′ (t) α′′ (t) U] = . kα′ (t)k3 kα′ (t)k3 17.4. GEODESIC CURVATURE OF CURVES 543 Proof. In the notation of (17.16), α′ (t) = s′ (t)β ′ (s) and α′′ (t) = s′′ (t)β ′ (s) + s′ (t)2 β′′ (s), so that κn [α](t) = κn [β](s) = β ′′ (s) · U = α′′ (t) · U . s′ (t)2 Since s′ (t) = kα′ (t)k, we get (17.17). To prove (17.18) we first note that α′′ (t) × α′ (t) = s′′ (t)β ′ (s) + s′ (t)2 β ′′ (s) × s′ (t)β ′ (s) = s′ (t)3 β ′′ (s) × β ′ (s). Therefore, [α′′ (t) U α′ (t)] = −[U α′′ (t) α′ (t)] = −s′ (t)3 [U β ′′ (s) β ′ (s)]. Furthermore, κg [α](t) = κg [β](s) = β ′′ (s) · J β ′ (s) and so we get (17.18). [α′ (t) α′′ (t) U] = β ′′ (s) · U × β ′ (s) = , s′ (t)3 Corollary 17.17. In the case of a plane curve, the geodesic curvature κg coincides with the signed curvature κ2. Proof. Formulas (1.12) and (17.18) are the same. Thus the geodesic curvature is the generalization to an arbitrary surface of the signed curvature of a plane curve. In fact, we have the following geometric interpretaion of geodesic curvature: Theorem 17.18. Let α: (a, b) → M be an arbitrary-speed curve in an oriented regular surface M in R3 , and let p = α(t0 ), where a < t0 < b. Then the geodesic curvature of α at t0 coincides with the signed curvature of the projection of α in the tangent plane Mp to M at p. b : (a, b) → Mp denote the projection of α in the tangent plane Proof. Let α Mp . If U denotes the unit normal to M, then U(p) is also the unit normal to Mp at the origin of Mp (where Mp is considered as a surface in R3 in its own right). Clearly, for a < t < b we have b (t) = α(t) − α(t) · U(p) U(p) b ′ (t0 ) = α′ (t0 ). α and α 544 CHAPTER 17. INTRINSIC SURFACE GEOMETRY This implies that κg [α](t0 ) = = [α′ (t0 ) α′′ (t0 ) U(p)] kα′ (t0 )k3 b ′ (t0 ) α b ′′ (t0 ) U(p)] [α b 0 ). = κg [α](t b ′ (t0 )k3 kα Next, we show that the geodesic curvature κg is intrinsic to M ⊂ R3 by expressing κg in terms of the coefficients E, F, G of the first fundamental form with respect to a patch. Since the computation is local, there is no loss of generality by assuming that M is oriented. Theorem 17.19. Let α : (a, b) → M ⊂ R3 be a curve whose trace is contained in x(U), where x: U → M is a patch. Write α(t) = x(u(t), v(t)) for a < t < b. Then p κg [α] kα′ k3 = − E G − F 2 − Γ211 u′3 + Γ122 v ′3 − (2Γ212 − Γ111 )u′2 v ′ +(2Γ112 − Γ222 )u′ v ′2 + u′′ v ′ − v ′′ u′ . Proof. We have α′ = xu u′ + xv v ′ , and also α′′ = xuu u′2 + 2xuv u′ v ′ + xvv v ′2 + xu u′′ + xv v ′′ , so that α′′ × α′ = (xuu × xu )u′3 + (xvv × xv )v ′3 + (xuu × xv + 2xuv × xu )u′2 v ′ +(xvv × xu + 2xuv × xv )u′ v ′2 + (u′′ v ′ − v ′′ u′ )xu × xv . Therefore, (α′′ × α′ ) · (xu × xv ) = (xuu × xu ) · (xu × xv )u′3 (17.19) + (xvv × xv ) · (xu × xv )v ′3 + (xuu × xv + 2xuv × xu ) · (xu × xv )u′2 v ′ + (xvv × xu + 2xuv × xv ) · (xu × xv )u′ v ′2 + (u′′ v ′ − v ′′ u′ )kxu × xv k2 . We next show how to express the coefficients of u′3 , v ′3 , u′2 v ′ , u′ v ′2 on the righthand side of (17.19) in terms of E, F, G and the Christoffel symbols. First, we use the Lagrange identity (7.1), page 193, and Theorem 17.7 to compute (17.20) Similarly, (17.21) (xuu × xu ) · (xu × xv ) = (xuu · xu )(xu · xv ) − (xuu · xv )kxu k2 = 12 Eu F − Fu − 12 Ev E = −(E G − F 2 )Γ211 . (xvv × xv ) · (xu × xv ) = (E G − F 2 )Γ122 . 17.5. GEODESIC TORSION AND FRENET FORMULAS 545 Also, (xuu × xv + 2xuv × xu ) · (xu × xv ) (17.22) and (17.23) = (xuu · xu )kxv k2 − (xuu · xv )(xu · xv ) +2(xuv · xu )(xu · xv ) − 2(xuv · xv )kxu k2 = 12 Eu G − Fu − 21 Ev F + Ev F − Gu E = (E G − F 2 )(Γ111 − 2Γ212 ) (xvv × xu + 2xuv × xv ) · (xu × xv ) = −(E G − F 2 )(Γ222 − 2Γ112 ). When equations (17.20)–(17.23) are substituted into (17.19), we get the required result. 17.5 Geodesic Torsion and Frenet Formulas Just as the geodesic curvature κg [α] of a curve α is a variant of the ordinary curvature κ[α], there is a variant of the torsion of α. Definition 17.20. Let M be an oriented regular surface in R3 . The geodesic torsion τg [α] of a curve α: (c, d) → M is given by (17.24) τg [α](t) = Sα′ (t) · J α′ (t) . kα′ (t)k2 The following result is easily proved (see Exercise 3). Recall (Definition 13.10) that a curve on a surface is called principal if it is everywhere tangent to a principal direction. Lemma 17.21. The definition of geodesic torsion does not depend on the parametrization. Furthermore, a curve on a surface has geodesic torsion zero if and only if it is principal. The study of principal curves is the subject of Chapter 19. There is also an analog of the Frenet formulas for a curve α on an oriented surface M ⊂ R3 . Just as we did in Theorem 7.10, we put T = α′ /kα′ k. However, since we want to study the relation between the geometry of α and M, we use the vector fields J T and U instead of N and B. Note that J T is always tangent to M, and U is always perpendicular to M; in contrast, N and B in general will be neither tangent nor perpendicular to M, and therefore are not useful to describe the geometry of M. 546 CHAPTER 17. INTRINSIC SURFACE GEOMETRY Definition 17.22. Let M be an oriented regular surface in Rn with unit normal U, and let α : (c, d) → M be a curve on M. Denote by J the complex structure of M determined by U, and write T = α′ /kα′ k. The Darboux frame ﬁeld of α with respect to M consists of the triple of vector fields {T, J T, U}. Corresponding to the Frenet formulas are the Darboux formulas; we first describe them for a unit-speed curve β. Theorem 17.23. Let β : (c, d) → M be a unit-speed curve, and let {T, J T, U} be the Darboux frame field of β with respect to M. Then   T′ = κg J T +κn U,   (J T)′ = −κg T +τg U,    ′ U = −κn T −τg J T. Proof. We have T′ = (T′ · J T)J T + (T′ · U)U = (β ′′ · J T)J T + (β′′ · U)U = κg J T + κn U. The other formulas are proved in a similar fashion. The Darboux formulas for an arbitrary-speed curve can be derived in the same way that we derived the Frenet formulas for an arbitrary-speed curve on page 203. Here is the result. Corollary 17.24. Let α : (c, d) → M be a curve with speed v = kα′ k, and let {T, J T, U} be the Darboux frame field of α with respect to M. Then   T′ = v κg J T +v κn U,   ′ (J T) = −v κg T +v τg U,    ′ U = −v κn T −v τg J T. Frame fields constitute a very useful tool for studying curves and surfaces. See the paper [Cartan] of É. Cartan4 for a general discussion of this subject. Cartan’s work has been developed by S. S. Chern and others [BCGH] into the so-called method of moving frames, which is recognized as one of the basic techniques of research in modern differential geometry. 4 Élie Cartan (1869–1951). French mathematician. One of the preeminent mathematicians of the 20th century. His important contributions include the classifications of compact Lie groups and symmetric spaces and the study of various geometries using moving frames and the structure equations. 17.6. EXERCISES 547 17.6 Exercises M 1. Use Brioschi’s formula to compute the Gaussian curvature of the following surfaces, and compare its effectiveness with the programs in Notebook 13. (a) A sphere. (b) A torus. (c) A catenoid. (d) Enneper’s minimal surface. M 2. Compute the Christoffel symbols for each of the surfaces in Exercise 1. 3. Prove Lemma 17.21 and Corollary 17.24. Complete the proof of Theorem 17.23. M 4. Compute the normal and geodesic curvature of a spherical spiral, as defined in Exercise 14 of Chapter 7. 5. Show that the geodesic torsion of any curve on a sphere vanishes. 6. Compute the geodesic curvature and the geodesic torsion of the curve t 7→ (t, at, at2 ) on the hyperbolic paraboloid (u, v) 7→ (u, v, uv), in terms of the constant a. M 7. Compute the geodesic torsion of a winding line on a torus, as discussed in connection with torus knots in Section 7.7. 8. Let p ∈ M where M is a surface in R3 , and let α : (a, b) → M be a curve with α(0) = p. Show that the geodesic torsion of α at p is given by (17.25) τg [α](0) = 1 2 k2 (p) − k1 (p) sin 2θ, where θ denotes the oriented angle from e1 to up . Compare formulas (17.25) and (13.17). 9. Show, as claimed in (v) on page 537, that any rotation about the z-axis preserves the curvature of a monkey saddle, but only rotations of angles ±2π/3 are in fact isometries of the surface. Chapter 18 Asymptotic Curves and Geodesics on Surfaces In this chapter we begin a study of special curves lying on surfaces in R3 . An asymptotic curve on a surface M ⊂ R3 is a curve whose velocity always points in a direction in which the normal curvature of M vanishes. In some sense, M bends less along an asymptotic curve than it does along a general curve. As a simple example, the straight lines on the cylinder (u, v) 7→ (cos u, sin u, v) formed by setting u constant are asymptotic curves. If p is a hyperbolic point of M (meaning that the Gaussian curvature is negative at p), there will be exactly two asymptotic curves passing through p. In Section 18.1, we derive the differential equation that must be satisfied in order that a curve be asymptotic. This is obtained by merely substituting the velocity vector of the curve into the second fundamental form, and setting the result equal to zero. We also prove Theorem 18.7, which relates the Gaussian curvature of M to the torsion (as defined in Section 7.2) of any asymptotic curve lying on M. In Section 18.2 we identify families of asymptotic curves on various classes of surfaces, and construct patches built out of asymptotic curves. Any straight line contained in a surface defines not only an asymptotic curve, but also a geodesic. A geodesic on a surface M is a curve with constant speed and vanishing geodesic curvature (as defined in Section 17.4). Roughly speaking, a geodesic is the shortest curve among all piecewise-differentiable curves on M connecting two points, although this property will not be demonstrated until Chapter 26. 557 558 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Geodesics were first studied by Johann Bernoulli1 in 1697, though the name ‘geodesic’ is due to Liouville in 1844. In Section 18.3, we determine the differential equations for a geodesic. Their formulation allows one to prove that there is a geodesic through a given point p pointing in any direction tangent to M at p. This of course contrasts with the case of asymptotic curves, but is what one expects intuitively from the distance-minimizing property of geodesics. Many examples of parametrizations of surfaces that we have already given in this book are so-called Clairaut patches, as defined in Section 18.5. These are patches for which the geodesics equations simplify and are more readily solvable. Applications are given in Section 18.6. Since we shall be doing mostly local calculations in this chapter, we shall assume that all surfaces are orientable, unless explicitly stated otherwise. This means that we can choose once and for all a globally defined unit normal U for any surface M. 18.1 Asymptotic Curves Let M be a regular surface in R3 . On page 390, we defined asymptotic directions and asymptotic curves on a surface M. The former are defined by the vanishing of the normal curvature at a given p ∈ M. The following lemma is obvious from the definitions. Lemma 18.1. Let M ⊂ R3 be a regular surface. (i) At an elliptic point of M there are no asymptotic directions. (ii) At a hyperbolic point of M there are exactly two asymptotic directions. (iii) At a parabolic point of M there is exactly one asymptotic direction. (iv) At a planar point of M every direction is asymptotic. Hyperbolic points are considered in more detail in Theorem 18.4 below. An asymptotic curve is a curve α in M for which the normal curvature vanishes in the direction α′ , that is k α′ (t) = 0 1 Johann Bernoulli (1667–1748) occupied the chair of mathematics at Groningen from 1695 to 1705, and at Basel, where he succeeded his elder brother Jakob, from 1705 to 1748. Although Johann was tutored by Jakob, he became involved in controversies with him, and also with his own son Daniel (1700–1782). Discoveries of Johann Bernoulli include the exponential calculus, the treatment of trigonometry as a branch of analysis, the determination of orthogonal trajectories and the solution of the brachistochrone problem. Other mathematician sons of Johann were Nicolaus (1695-1726) and Johann (II) (1710–1790). 18.1. ASYMPTOTIC CURVES 559 for all t in the domain of definition of α. Here is an alternative description. Lemma 18.2. A curve α in a regular surface M ⊂ R3 is asymptotic if and only if its acceleration α′′ is always tangent to M. Proof. Without loss of generality, we can assume that α is a unit-speed curve. If U denotes the surface normal to M, then differentiation of α′ · U = 0 yields (18.1) 0 = α′ · U′ + α′′ · U = −k(α′ ) + α′′ · U (see page 387). Hence k(α′ ) vanishes if and only if α′′ is perpendicular to U. Next, we derive the differential equation for the asymptotic curves. Lemma 18.3. Let α be a curve that lies in the image of a patch x. Write α(t) = x(u(t), v(t)). Then α is an asymptotic curve if and only if one of the following equivalent conditions is satisfied for all t: (i) e α(t) u′ (t)2 + 2f α(t) u′ (t)v ′ (t) + g α(t) v ′ (t)2 = 0; (ii) II α′ (t), α′ (t) = 0, where II is the second fundamental form defined on page 401. Proof. Corollary 10.15, page 294, implies that α′ = xu u′ + xv v ′ . Hence by Lemma 13.17 on page 395, we have e α(t) u′ (t)2 + 2f α(t) u′ (t)v ′ (t) + g α(t) v ′ (t)2 ′ (18.2) . k α (t) = E α(t) u′ (t)2 + 2F α(t) u′ (t)v ′ (t) + G α(t) v ′ (t)2 Then (i) follows from (18.2), and (ii) is a restatement of (i). The equation in (i) can be written more succinctly as e u′2 + 2f u′ v ′ + g v ′2 = 0. (18.3) We call (18.3) the differential equation for the asymptotic curves of a surface. Using Lemma 13.31 on page 405, we see that (18.3) is equivalent to eu′2 + 2feu′ v ′ + e e gv ′2 = 0, (18.4) in terms of the triple products ee = [xuu xu xv ], fe = [xuv xu xv ], g = [xvv xu xv ]. e Theorem 18.4. In a neighborhood of a hyperbolic point p of a regular surface M ⊂ R3 there exist two distinct families of asymptotic curves. 560 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Proof. In a neighborhood of a hyperbolic point p, the equation eea2 + 2feab + ge b2 = 0 has real roots. Hence we have the factorization eea2 + 2feab + ge b2 = (Aa + Bb)(Ca + Db), where A, B, C, D are real. Thus the differential equation for the asymptotic curves also factors: (18.5) (Au′ + Bv ′ )(Cu′ + Dv ′ ) = 0, where now A, B, C, D are real functions. One family consists of the solution curves to Au′ + B v ′ = 0, and the other family consists of the solution curves to C u′ + D v ′ = 0. Next, we give a characterization of an asymptotic curve in terms of its curvature as a curve in R3 . Theorem 18.5. Let α be a regular curve on a regular surface M ⊂ R3 , and denote by {T, B, N} the Frenet frame and by κ[α] the curvature of α. Also, let U be the surface normal of M. Then α is an asymptotic curve if and only if at every point α(t), either κ[α](t) = 0 or N(t) · U = 0. Proof. Without loss of generality, we can assume that α has unit speed. The first Frenet formula (see page 197) in the case of nonzero curvature κ[α] is T′ = κ[α]N. (18.6) We can also make sense of (18.6) at points where κ[α] vanishes, by choosing the unit vector N arbitrarily. By (18.1), we have k(α′ ) = α′′ · U = T′ · U = κ[α]N · U. It follows that at each point α(t), either κ[α] = 0 or N is well defined and perpendicular to U. The conclusion follows. The curvature of a straight line vanishes identically, whence Corollary 18.6. A straight line that is contained in a regular surface is necessarily an asymptotic curve. Now we establish an important relation between the torsion of an asymptotic curve and the Gaussian curvature of the surface containing it. 18.1. ASYMPTOTIC CURVES 561 Theorem 18.7. (Beltrami2 -Enneper) Let α be an asymptotic curve on a regular surface M ⊂ R3 , and assume the curvature κ[α] of α does not vanish. Then the torsion τ [α] of α and the Gaussian curvature K of M are related along α by K ◦ α = −τ [α]2 . Proof. Without loss of generality, we can assume that α is a unit-speed curve. Since α is an asymptotic curve with nonvanishing curvature, it follows from Theorem 18.5 that T · U = N · U = 0. Therefore, U = ±B along α. Then the third Frenet formula of (7.12) implies that dU dU (18.7) · . τ [α]2 = B′ · B′ = ds ds On the other hand, since the second fundamental form II of M vanishes along α, equation (13.23), page 402, for the third fundamental form reduces to (18.8) 0 = (III + K I)(α′ , α′ ) = dU dU · + K. ds ds The theorem follows from (18.7) and (18.8). Corollary 18.8. An asymptotic curve on a regular surface M ⊂ R3 with constant negative curvature has constant torsion. This result applies in particular to the examples of surfaces given in Sections 15.4 and 15.6. Next, we consider the problem of building patches from asymptotic curves. Definition 18.9. An asymptotic patch on a regular surface M ⊂ R3 is a patch for which the u- and v-parameter curves are asymptotic curves. Theorem 18.10. Let x be a patch for which f never vanishes. Then x is an asymptotic patch if and only if e and g vanish identically. 2 Eugenio Beltrami (1835–1900). Professor at the Universities of Bologna, Pisa, Padua and Rome. He found the first concrete model of non-Euclidean geometry, published in [Beltr] and in this paper showed how possible contradictions in non-Euclidean geometry would reveal themselves in the Euclidean geometry of surfaces. Beltrami also made known the work of the Jesuit mathematician Saccheri, whose work Euclides ab omni naevo vindicatus foreshadowed non-Euclidean geometry. 562 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Proof. If the curve u 7→ x(u, v) is asymptotic, it follows from Lemma 18.2 that e = xuu · U = 0. Similarly, if v 7→ x(u, v) is asymptotic, then g = 0. Conversely, if e and g vanish identically, the differential equation for the asymptotic curves becomes f u′ v ′ = 0. Clearly, u = u0 and v = v0 are solutions of this differential equation. Otherwise said, the u- and v-parameter curves are asymptotic. 18.2 Examples of Asymptotic Curves and Patches Ruled surfaces According to Corollary 18.6, ruled surfaces are constructed from asymptotic straight lines. For example, if we parametrize the hyperbolic paraboloid z = xy by (10.12), page 296, it is easy to compute e = g = 0 and f = (1 + u2 + v 2 )−1/2 . Hence, by Theorem 18.10, (u, v) 7→ (u, v, uv) is an asymptotic patch. Of course the resulting curves u 7→ (u, v0 , u v0 ) and v 7→ (u0 , v, u0 v) are the straight-line rulings visible in Figure 14.3 on page 435. Figure 18.2: The elliptical helicoid The helicoid defined on page 376 is a special case of the elliptical helicoid (18.9) helicoid[a, b, c](u, v) = av cos u, b v sin u, cu . 18.2. EXAMPLES OF ASYMPTOTIC CURVES 563 Like the previous example, helicoid[a, b, c] is an asymptotic patch, but this time only the v-parameter curves are straight lines. The u-parameter curves are, of course, elliptical helices. Theorem 18.7 tells us that the Gaussian curvature of the elliptical helicoid, when restricted to one of these helices, equals minus the square of the torsion of the helix. This fact is verified in Notebook 18. The exponentially-twisted helicoid is the surface (18.10) exptwist[a, c](u, v) = av cos u, av sin u, aecu . This is a helicoidal-like surface whose twisting varies exponentially and, unlike (18.9), exptwist[a, c] does not have e = g = 0. Instead, an asymptotic patch needs to be found by integrating the equations; the result is exptwistasym[a, c](p, q) = aec(p−q)/2 cos p, aec(p−q)/2 sin p, aecp . We omit the details, which can be found in Notebook 18, though a similar technique is carried out for the funnel surface on the next page. Figure 18.3: The patches exptwistasym[1, 0.3] and exptwist[1, 0.3] The Monkey Saddle We parametrize the monkey saddle z = Re (x + iy)3 = x3 − 3xy 2 by the Monge patch (10.17), page 303. Because the point (0, 0, 0) is planar, every direction at (0, 0, 0) is asymptotic. This does not mean, however, that 564 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS there are asymptotic curves in every direction. But it is easy to find three straight lines passing through (0, 0, 0) that lie entirely in the surface, namely v 7→ (0, v, 0), √ v 7→ (v 3, v, 0), √ v 7→ (−v 3, v, 0). Independently, it can be checked that each of these curves satisfies the differential equation u u′2 − 2v u′ v ′ − u v ′2 = 0 for asymptotic curves. These three lines are also examples of geodesics, and have been emphasized as such in Figure 18.5. The Funnel Surface Consider the regular surface which is the image of the patch x: (0, ∞) × [0, 2π] −→ R3 (18.11) x(r, θ) = (r cos θ, r sin θ, log r). This is a parametrization of the graph of the function z= 1 log(x2 + y 2 ). 2 We compute r −1 , f = 0, g= √ . e= √ 2 r 1+r 1 + r2 Thus the differential equation for the asymptotic curves becomes − (18.12) r′2 + r θ′2 = 0, r or θ′ = ± r′ . r The two equations θ′ = ± r′ /r have respective solutions θ + 2u = log r, θ + 2v = − log r, where u and v are constants of integration. Solving for r and θ in terms of u and v gives r = eu−v and θ = −u − v. The patch y(u, v) = = x r(u, v), θ(u, v) eu−v cos(u + v), −eu−v sin(u + v), u − v is therefore asymptotic. It is illustrated in Figure 18.4. 18.3. GEODESIC EQUATIONS 565 Figure 18.4: The funnel surface with asymptotic patch 18.3 The Geodesic Equations Our starting point is the following definition of a geodesic on a surface in R3 . Definition 18.11. Let M ⊂ R3 be a surface and α : (a, b) → M a curve. We say that α is a geodesic on M if the tangential component α′′ (t)⊤ of the acceleration of α vanishes. It is important to note that we do not assume that α has unit speed in this definition. Nonetheless, Lemma 18.12. A geodesic α necessarily has constant speed. Proof. We compute d kα′ (t)k2 = 2α′′ (t) · α′ (t). dt Since α′ (t) is tangential to the surface, the right-hand side is zero, and ds = kα′ (t)k dt is constant. The definition of geodesic is in some sense complementary to that of asymptotic curve. Recall (see page 390) that α: (a, b) → M ⊂ R3 is an asymptotic curve provided the normal component of α′′ vanishes. However, there is an important difference, since the definition of geodesic requires no knowledge of 566 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS normal directions, and immediately extends to a surface in Rn . Indeed, we shall soon see that geodesics can be defined instrinsically, in terms of just the first fundamental form. The following result is an immediate consequence of Theorem 17.7 on page 539, and an analogue of Corollary 18.6. Corollary 18.13. A straight line in R3 that is contained in a surface M is a geodesic on M, provided its parametrization has constant speed. As remarked on page 564, the monkey saddle has three such straight lines passing through its center. These form part of the plot in Figure 18.5 that shows geodesics emanating from the center; in fact all those visible are approximately straight lines. Figure 18.5: Geodesics on a monkey saddle z = x3 − 3xy 2 , plotted for z > −0.05 and viewed from above Using the terminology of the previous chapter, we next determine the differential equations that any geodesic on a surface in R3 must satisfy. Lemma 18.14. Let M ⊂ R3 be a surface parametrized by a regular patch x: U → R3 , where U ⊂ R2 . The geodesics on M are determined by the system of two second-order differential equations:   u′′ + Γ111 u′2 + 2Γ112 u′ v ′ + Γ122 v ′2 = 0, (18.13)  ′′ v + Γ211 u′2 + 2Γ212 u′ v ′ + Γ222 v ′2 = 0, where the Γijk are the Christoffel symbols of x. 18.3. GEODESIC EQUATIONS 567 Proof. Let α: (a, b) → M be a curve. We can write (18.14) α(t) = x u(t), v(t) , where u, v : (a, b) → R are differentiable functions. When we differentiate (18.14) twice using the chain rule, we get (18.15) α′′ = xuu u′2 + xu u′′ + 2xuv u′ v ′ + xvv v ′2 + xv v ′′ . Then (17.8), page 538, and (18.15) imply that (18.16) where α′′ (t) = u′′ + Γ111 u′2 + 2Γ112 u′ v ′ + Γ122 v ′2 xu + v ′′ + Γ211 u′2 + 2Γ212 u′ v ′ + Γ222 v ′2 xv + e u′2 + 2f u′ v ′ + g v ′2 U, U= xu × xv kxu × xv k is the unit normal vector field to M. In order that α be a geodesic, the coefficients of xu and xv in (18.16) must each vanish. Hence we get (18.13). Now we can obtain another important property of geodesics. Corollary 18.15. Isometries and local isometries preserve geodesics. Proof. Since the Christoffel symbols are expressible in terms of E, F and G, Corollary 12.8 implies that local isometries preserve equations (18.13). Next, we prove that from each point on a surface there is a geodesic in every direction. More precisely: Theorem 18.16. Let p be a point on a surface M and vp a tangent vector to M at p. Then there exists a geodesic γ parametrized on some interval containing 0 such that γ(0) = p and γ ′ (0) = vp . Furthermore, any two geodesics with these initial conditions coincide on any interval containing 0 on which they are defined. Proof. Let x: U → M be a regular patch with x(0, 0) = p. Then (18.13) is the system of differential equations determining the geodesics on the trace of x. Writing γ(t) = x(u(t), v(t)), the initial conditions translate into ( ( u(0) = 0, u′ (0) = v1 , and (18.17) v(0) = 0, v ′ (0) = v2 , where v1 , v2 are the components of vp . From the theory of ordinary differential equations, we know that the system (18.13) together with the four initial conditions (18.17) has a unique solution on some interval containing 0. Hence the theorem follows. 568 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Theorem 18.16 gives no information concerning the size of the interval on which the geodesic satisfying given initial conditions is defined. Let γ 1 and γ 2 be geodesics satisfying the same initial conditions. Since the two geodesics must coincide on some interval, it is clear that there is a geodesic γ 3 whose domain of definition contains those of both γ 1 and γ 2 and coincides wherever possible with the two geodesics. Applying this consistency result, we obtain a single maximal geodesic. The domain of a maximal geodesic may or may not be the whole real line, and we shall discuss this matter further in Chapter 26. It is important to distinguish between a geodesic and the trace of a geodesic, because not every reparametrization of a geodesic has constant speed. Therefore, we introduce the following notion: Definition 18.17. A curve α in a surface M is called a pregeodesic provided there is a reparametrization α ◦ h of α such that α ◦ h is a geodesic. We can now characterize a pregeodesic by the vanishing of the geodesic curvature, defined in Section 17.4. See also Exercise 11. Lemma 18.18. Let M be a surface and let α: (a, b) → M be a regular curve. Then the following conditions are equivalent: (i) α is a pregeodesic; (ii) there exists a function f : (a, b) → R such that α′′ (t)⊤ = f (t)α′ (t) for a < t < b; (iii) the geodesic curvature of α vanishes. Proof. Suppose β = α ◦ h is a geodesic. Without loss of generality, β′ never vanishes. Then β ′ = (α′ ◦ h)h′ , so that h′ never vanishes. Furthermore, it follows that 0 = (β ′′ )⊤ = h′2 (α′′ ◦ h)⊤ + h′′ (α′ ◦ h), so that ′′ ⊤ (α ) = − h′′ −1 α′ ; ◦h h′2 thus we can take f = −(h′′ /h′2 ) ◦ h−1 . Conversely, assume that (α′′ )⊤ = f α′ and let h be a nonzero solution of the differential equation h′′ + h′2 (f ◦ h) = 0. Put β = α ◦ h; then (β′′ )⊤ = h′2 (α′′ ◦ h)⊤ + h′′ (α′ ◦ h) = (f ◦ h)h′2 + h′′ α′ ◦ h = 0. This establishes the equivalence of (i) and (ii). The equivalence of (ii) and (iii) follows from equation (17.18) on page 542. For κg [α] vanishes if and only if α′′ (t) · J α(t)′ = 0 for all t, which is equivalent to (ii). 18.4. FIRST EXAMPLES OF GEODESICS 569 18.4 First Examples of Geodesics We begin with a general result relating the shape operator of a surface in R3 and the Frenet frame of a curve on the surface. Lemma 18.19. Let M be an orientable surface in R3 , and let α : (a, b) → M be a unit-speed geodesic on M with nonzero curvature. Denote by {T, N, B} the Frenet frame field of α and by κ and τ the curvature and torsion of α. Then it is possible to choose a unit normal vector field U to M such that (18.18) S(T) = κ T − τ B, where S denotes the shape operator of M with respect to U. Proof. Since α has nonzero curvature, the vector field N is defined on all of (a, b). The assumption that α is a geodesic implies that N = α′′ /κ2 is everywhere perpendicular to M. Thus it is possible to choose U so that N(t) = U α(t) for a < t < b. By the definition of the shape operator and by the Frenet Formulas (7.12), page 197, we have S(T) = − d U α(t) = −N′ = κT − τ B. dt Before proceeding further with the general theory, we find geodesics in three important cases. Geodesics on a Sphere We can use Lemma 18.19 to determine the geodesics of a sphere. Lemma 18.20. The geodesics on a sphere of radius c > 0 in R3 are parts of great circles. Proof. Let β be a geodesic on a sphere of radius c. By part (i) of Theorem 8.15, page 242, the curvature of β is nonzero, so that the Frenet frame {T, N, B} of β is well defined. Since the principal curvatures of a sphere are constant and equal, the shape operator of the sphere (with a proper choice of the unit normal) satisfies 1 S(T) = T. (18.19) c It follows from (18.18) and (18.19) that the curvature of β has the constant value 1/c, and that the torsion vanishes. Moreover, part (vi) of Theorem 8.15 implies that β is part of a circle. Since the radius of the circle is the maximum possible, β must be part of a great circle. 570 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Figure 18.6: Geodesics on the sphere Figure 18.6 (left) is an outline of the unit sphere S 2 (1), formed by a number of geodesics each of length 6 (rather than 2π), passing through a given point p ∈ S 2 (1). The curves on the right are the same geodesics, but viewed in the rectangular domain of the standard chart (16.26). Both families of curves were plotted numerically using a program from Notebook 18. Surfaces of Revolution In this subsection, we limit ourselves to proving the following simple result. Theorem 18.21. Any meridian of a surface of revolution M ⊂ R3 can be parametrized as a geodesic. Proof. Let β be a unit-speed parametrization of a meridian of M, and let Π be a plane passing through β and the axis of revolution. Since β is a plane curve in Π , the unit normal N = J β ′ also lies in Π . Theorem 15.26, page 485, implies that the plane Π meets M perpendicularly, and so N is also perpendicular to M. But then β ′′ = κ2 N is perpendicular to M (see (1.12) on page 14). By definition, β is a geodesic. We shall give a criterion for a parallel to be a geodesic in Section 18.6. For a different generalization of Theorem 18.21, see Exercise 12. Describing all the geodesics on a general surface of revolution is complicated. This complexity is illustrated by Figure 18.7, which results when the sphere of the previous figure is replaced by a torus – it shows a selection of geodesics emanating from a point p on the inside top left of the torus. For example, the one that appears horizontal relative to the chart (on the right) corresponds to the innermost circle of the torus. Figure 18.8 displays the boundary or ‘geodesic circle’ consisting of all those points a fixed distance r from p, as measured along 18.4. FIRST EXAMPLES OF GEODESICS 571 these geodesics. As r increases further, this closed curve will soon intersect itself in different places, and the ‘geodesic disk’ of center p and radius r will no longer resemble a disk even topologically. Figure 18.7: Geodesics on the torus of revolution Figure 18.8: Geodesic circle on a torus Geodesics on a Generalized Cylinder Geodesics on a generalized cylinder (defined on page 438) are characterized by Theorem 18.22. Let M be the generalized cylinder whose base curve α is a plane curve perpendicular to the rulings of M, and denote by u a unit vector tangent to M and perpendicular to α, so that M is parametrized as x(u, v) = α(u) + v u. Then the geodesics of M are precisely those constant speed curves on M that have constant slope with respect to u. 572 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Proof. Let γ : (a, b) → M be a curve on the generalized cylinder; we can write γ(t) = α u(t) + v(t)u. (18.20) for some functions t 7→ u(t) and t 7→ v(t). Then (18.21) γ ′ = α′ u′ + v ′ u and γ ′′ = α′′ u′2 + α′ u′′ + v ′′ u. Now assume that γ is a geodesic. Since any normal vector to M is always perpendicular to u, it follows from the second equation of (18.21) that v ′′ = 0, so that v ′ is a constant. But u · α′ = 0, so γ makes a constant angle with u, and γ must have constant slope (see Section 8.5). Conversely, let γ : (a, b) → M be a constant speed curve that has constant slope with respect to u. Then (18.20) holds with v ′ constant, so that α ◦ u is the projection of γ onto the plane perpendicular to u, and γ ′′ is perpendicular to u. Moreover, it follows from (18.20) that γ ′′ = (α ◦ u)′′ . Also, Lemma 8.21 on page 247 implies that α ◦ u has constant speed. Hence γ ′′ is perpendicular to both xu = α′ and u = xv , proving that γ is a geodesic. Figure 18.9: A geodesic on the generalized cylinder (u, v) 7→ (u, sin u, v) We may deduce that the geodesics on a circular or elliptical cylinder are circles, lines or helices. A more general example to illustrate the principle is shown in Figure 18.9. 18.5 Clairaut Patches In general, the geodesic equations (18.13) are difficult to solve explicitly. However, there are two important cases where their solution can be reduced to computing integrals. 18.5. CLAIRAUT PATCHES 573 Definition 18.23. Let M be a surface with metric ds2 = E du2 + 2F dudv + Gdv 2 . (i) A u-Clairaut patch on M is a patch x: U → M for which Ev = Gv = F = 0. (ii) A v-Clairaut patch on M is a patch x: U → M for which Eu = Gu = F = 0. We work mainly with v-Clairaut patches, leaving the corresponding results for u-Clairaut patches to the reader. To cite two examples, consider (i) the standard parametrization of the unit sphere S 2 (1), having E = cos2 v, F = 0 and G = 1. (ii) the isothermal chart on S 2 (1) of Lemma 16.16 on page 519 (with v in place of w), having E = G = sech v and F = 0. Many others can be found from earlier chapters. The Christoffel symbols for a v-Clairaut patch are considerably simpler than those of a general patch. The following lemma is an immediate consequence of Theorem 17.7 on page 539. Lemma 18.24. For a v-Clairaut patch with ds2 = E du2 + Gdv 2 we have (18.22)    Γ111 = 0,       Ev Γ112 = ,  2E        Γ1 = 0, 22 Γ211 = −Ev , 2G Γ212 = 0, Γ222 = Gv , 2G so that the geodesic equations (18.13) reduce to  Ev ′ ′   u′′ + u v = 0,  E (18.23)    v ′′ − Ev u′2 + Gv v ′2 = 0. 2G 2G Some geodesics on a Clairaut patch are easy to determine. Lemma 18.25. Let x: U → M be a v-Clairaut patch. Then: (i) any v-parameter curve is a pregeodesic; 574 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS (ii) a u-parameter curve u 7→ x(u, v0 ) is a geodesic if and only if Ev (u, v0 ) vanishes along u 7→ x(u, v0 ). Proof. To prove (i), consider a v-parameter curve β defined by β(t) = x(u0 , t). It suffices by Lemma 18.18 to show that β ′′ (v)⊤ is a scalar multiple of xv (u0 , v). Since xu and xv are orthogonal, this will be the case provided β ′′ (v), xu (u0 , v) is zero. This equals ′ hxvv (u0 , v), xu (u0 , v)i = hxv , xu i − hxv , xuv i = Fv − 21 Gu = 0, the prime indicating differentiation with respect to t = v. To prove (ii), we observe that if the u-parameter curve α(t) = x(t, v0 ) is a geodesic, then ′ Ev (u, v0 ) = 2 hxuv (u, v0 ), xu (u, v0 )i = 2 hxv , xu i − 2 hxv , xuu i = 2Fu − 2 hxv , α′′ i = 0, where this time the prime denotes d/du. Conversely, if Ev (u, v0 ) = 0, then hxuu , xv i = hxu , xv iu − hxu , xvu i = Fu − 12 Ev = − 21 Ev , so that hα′′ , xv i (u, v0 ) = − 12 Ev (u, v0 ) = 0. But also, hα′′ , xu i = 21 Eu = 0, so that α is a geodesic. We turn now to the problem of finding other geodesics on a Clairaut patch. The key result we need is: Theorem 18.26. (Clairaut’s relation) Let x: U → M be a v-Clairaut patch, and let β be a geodesic whose trace is contained in x(U). Let θ be the angle between β′ and xu . Then √ (18.24) E kβ ′ k cos θ is constant along β. Proof. We write β(t) = x(u(t), v(t)). From the assumption that Eu = 0 and the first equation of (18.23), it follows that (E u′ )′ = E ′ u′ + E u′′ = Ev v ′ u′ + E u′′ = 0. Hence there is a constant c such that (18.25) E u′ = c. 18.5. CLAIRAUT PATCHES 575 Furthermore, kβ ′ k cos θ = √ hxu , xu u′ + xv v ′ i hxu , β ′ i = = kxu ku′ = E u′ . kxu k kxu k When this is combined with (18.25), we obtain (18.24). Definition 18.27. The constant c given by (18.25) is called the slant of the geodesic β in the v-Clairaut patch x. Then the angle between the geodesic and xu is given by c √ . cos θ = kβ′ k E We next obtain information about geodesics on a Clairaut patch other than those covered by Lemma 18.25. Lemma 18.28. Let x: U → M be a v-Clairaut patch, and let β : (a, b) → R be a unit-speed curve whose trace is contained in x(U). Write β(t) = x((u(t), v(t)). If β is a geodesic, then there is a constant c such that  c  u′ = ,   E √ (18.26)  E − c2  ′  v = ± √ . EG Conversely, if (18.26) holds, and if v ′ (t) 6= 0 for a < t < b, or if v ′ (t) = 0 for a < t < b, then β is a geodesic. Proof. Suppose that β is a unit-speed geodesic. The first equation of (18.26) follows from (18.25). Furthermore, 1 = β ′ , β′ = hxu u′ + xv v ′ , xu u′ + xv v ′ i = E u′2 + Gv ′2 = so that 1 v = G ′2 c2 + Gv ′2 , E c2 E − c2 1− = . E EG Thus the second equation of (18.26) also holds. Conversely, suppose that (18.26) holds. The first equation implies that c ′ Ev u′ v ′ cEv v ′ =− =− . u′′ = 2 E E E Thus we see that the first equation of (18.23) is satisfied. Furthermore, β has unit speed because c 2 E − c2 ′ ′ ′2 ′2 = 1. +G β , β = E u + Gv = E E EG 576 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Next, when we differentiate the equation Gv ′2 = 1 − E u′2 , we obtain (18.27) Gv v ′3 + 2Gv ′′ v ′ = −Ev v ′ u′2 − 2E u′ u′′ Ev ′ ′ u v = Ev v ′ u′2 . = −Ev v ′ u′2 + 2E u′ E Then (18.27) implies that the second equation of (18.23) is satisfied on any interval where v ′ is different from zero. On the other hand, on an interval where v ′ vanishes identically, the second equation of (18.26) implies that E is a constant on that interval. Again, the second equation of (18.23) is satisfied. Corollary 18.29. Let x: U → M be a v-Clairaut patch. A curve α: (a, b) → M of the form α(v) = x u(v), v is a pregeodesic if and only if there is a constant c such that s G du (18.28) = ±c . dv E(E − c2 ) Then c is the slant of α with respect to x. Proof. There exists a unit-speed geodesic β reparametrizing α such that β(s) = α v(s) = x u(v(s)), v(s) . Then Lemma 18.28 implies that du c = ds E Hence du ds du = dv dv ds and c E √ E − c2 dv =± √ . ds EG s G √ . = ±c 2 E(E − c2 ) E−c ± √ EG Conversely, if (18.28) holds, we define u′ and v ′ by s G u′ = ±c and E u′2 + Gv ′2 = 1. v′ E(E − c2 ) = An easy calculation then shows that (18.26) holds, and so we get a pregeodesic. 18.6 Use of Clairaut Patches Let us consider several examples of Clairaut patches and find their geodesics. 18.6. USE OF CLAIRAUT PATCHES 577 The Euclidean Plane in Polar Coordinates The polar-coordinate parametrization of the xy-plane is (18.29) x(u, v) = (v cos u, v sin u, 0). It is clear that (18.29) is a v-Clairaut patch; indeed E = v2 , F = 0, G = 1. Equation (18.28) becomes c du = ±p , dv v 2 (v 2 − c2 ) or equivalently cdv p = ±du. v 2 (v 2 − c2 ) Carrying out the integration (done by computer in Notebook 18), we obtain c √ = ±(u − u0 ), or c = ±v sin(u − u0 ) arctan v 2 − c2 for some constant u0 . This is merely the equation of a general straight line in polar coordinates. Solving the geodesic equations in polar coordinates is harder without the techniques of the previous section. Finding the geodesics in the plane using Cartesian coordinates is of course easier; see Exercise 13. Surfaces of Revolution The standard parametrization of a surface of revolution in R3 is x(u, v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) . Let us assume that ϕ(v) > 0; then ϕ(v) can be interpreted as the radius of the parallel u 7→ (ϕ(v) cos u, ϕ(v) sin u, ψ(v)). Since E = ϕ(v)2 , F =0 and G = ϕ′ (v)2 + ψ ′ (v)2 , we see that Eu = Gu = F = 0; thus x is again a v-Clairaut patch, and the equation for the geodesics is p ϕ′2 + ψ ′2 du . =± p dv ϕ ϕ2 − c2 Since the v-parameter curves are meridians, we recover Lemma 18.21 from Lemma 18.25(i). Indeed, Theorem 18.26 is often quoted in the special case of a surface of revolution. From Lemma 18.25, we deduce Corollary 18.30. Let M be a surface of revolution in R3 . Then a parallel is a geodesic if an only if ϕ′ (v0 ) = 0. 578 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Figure 18.10: Geodesics winding around the pseudosphere The Pseudosphere Applying the method above to the pseudosphere, parametrized by (15.6) on page 480, allows us to determine explicitly its geodesics. The equations are √ a2 − 2c2 − a2 cos 2v √ u = u0 ± , 2 c sin v with c, u0 constants3 . We plot u as a function of v in Notebook 18, so as to compare the results with the numerical solutions illustrated in Figure 18.10. For each geodesic on the pseudosphere, there are at most two points with equal ‘height’, having the same value of v, but there may be many values of v with the same value of u modulo 2π. 3 For more information on geodesics on the pseudosphere and interesting pictures of them, see F. Schilling’s Die Pseudosphäre und die nichteuklidische Geometrie [Schill]. 18.7. EXERCISES 579 18.7 Exercises 1. Show that the differential equation for the asymptotic curves on a Monge patch x(u, v) = (u, v, h(u, v)) is huu u′2 + 2huv u′ v ′ + hvv v ′2 = 0. 2. Show that the differential equation for the asymptotic curves on a polar patch of the form x(r, θ) = (r cos θ, r sin θ, h(r)) is h′′ (r)r′2 + h′ (r)r θ′2 = 0. M 3. Show that an asymptotic parametrization of a catenoid is given by −p + q p+q −p + q −p + q p+q . cosh , sin cosh , y(p, q) = cos 2 2 2 2 2 Plot this surface for 0 < p, q < 2π. 4. Find the differential equation for the asymptotic curves on the torus x(u, v) = (a + b cos v) cos u, (a + b cos v) sin u, b sin v . M 5. Find the asymptotic curves of the generalized hyperbolic paraboloid z = x2n − y 2n . Display some particular cases. M 6. Find the asymptotic curves of the surface defined by z = xm y n . Display some particular cases. 7. Show that the Gaussian and mean curvatures of the (circular) helicoid defined on page 376 are given by K =− c2 , (c2 + a2 v 2 )2 H = 0. Conclude that each u-parameter curve has constant torsion. Is this fact known from an earlier chapter? 8. Find an asymptotic patch for the surface z = (x2 + y 2 )α , where α is a real number with 0 6= α < 1/2. Figure 18.11 is an ordinary plot of the case α = −1/2 colored by Gaussian curvature. 580 CHAPTER 18. ASYMPTOTIC CURVES AND GEODESICS Figure 18.11: Part of the surface z = (x2 + y 2 )−1/2 with z < 3 2 9. Find an asymptotic patch for the surface z = e−α(u number with 0 6= α < 1. +v 2 ) , where α is a real M 10. An asymptotic parametrization of the ‘shoe’ surface is given by shoe[a, b](p, q) = 1 2 b 2 3b 3 p + 14pq + q 2 . − (p − q) 3 , −p − q, 4a 4 Plot this surface for a = 1, b = −1 and −1 < p, q < 1. 11. Show that a pregeodesic is a geodesic if and only if it has constant speed. 12. Show that a meridian on a generalized helicoid can be parametrized as a geodesic. [Hint: Generalize Theorem 18.21.] 13. Use Lemma 18.28 to determine the geodesics on the plane parametrized by Cartesian coordinates. M 14. Find a parametrization of a general geodesic on a catenoid and draw several of them. 18.7. EXERCISES 581 15. Let β be a geodesic on a catenoid other than the center circle, and suppose the initial velocity of β is perpendicular to the axis of revolution. Use Clairaut’s relation (Theorem 18.26) to show that β approaches the center curve asymptotically, but never reaches it. M 16. Plot geodesics and geodesic circles on ellipsoids with two distinct axes and on ellipsoids with three distinct axes. 17. Let α be an asymptotic curve defined by α(t) = surfrev[α] u(t), v(t) , where surfrev[α] is the standard parametrization of a surface of revolution given by (15.1). Show that u and v satisfy the differential equation ′′ ϕ (t)ψ ′ (t) − ϕ′ (t)ψ ′′ (t) ′ 2 v ′ (t)2 . u (t) = ϕ(t)ψ ′ (t) 18. Fill in the proof of Lemma 18.24. Chapter 19 Principal Curves and Umbilic Points In this chapter we continue our study of special curves on surfaces. A principal curve on a surface is a curve whose velocity always points in a principal direction, that is, a direction in which the normal curvature is a maximum or a minimum. In Section 19.1, we derive the differential equation for the principal curves on a patch in R3 and give examples of its solution. As in Section 18.2, the idea then is to reparametrize a surface with specified coordinate curves. An umbilic point on a surface is a point at which the principal curvatures are equal, so that at such a point it is not possible to distinguish principal directions. Every point of a sphere is an umbilic point. We discuss umbilic points in Section 19.2 and show, for example, that if a regular surface M ⊂ R3 consists entirely of umbilic points, then it is (perhaps not surprisingly) part of a plane or sphere. For surfaces such as an ellipsoid y2 z2 x2 + + =1 a2 b2 c2 with a, b, c distinct, the umbilic points are isolated and can be considered to be degenerate principal curves. In fact, each looks very much like a navel, hence the name. The four umbilic points on an ellipsoid (with a, b, c distinct) are easy to locate visually, provided one draws the ellipsoid so that the principal curves can be seen, as in Figure 19.6. The coefficients of the first and second fundamental forms of a regular surface M in R3 are not independent of one another. The Peterson–Mainardi– Codazzi equations consist of two equations that relate the derivatives of the coefficients of the second fundamental form to the coefficients of the first and second fundamental forms themselves. They simplify considerably when the coordinate curves are principal, and this justifies their inclusion in this chapter. 593 594 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS The derivation of these equations in Section 19.3 gives rise to formulas that foresee the definition of the Riemann curvature tensor in Section 25.5. In Section 19.4 we use the Peterson–Mainardi–Codazzi equations to prove Hilbert’s Lemma (Lemma 19.12), which in turn is used to prove a so-called rigidity result, namely Liebmann’s Theorem. For certain surfaces, such as the ellipsoid, there is a much more effective way to find principal curves than solving the differential equation discussed in Section 19.1. It is based on the notion of a triply orthogonal system of surfaces, something introduced by Lamé to study equations of mathematical physics (see, for example, [MoSp]). The corresponding notion in the plane, of an orthogonal system of curves, is discussed in the Exercises. The connection between principal curvatures and triply orthogonal systems will become clear after we prove Dupin’s Theorem, which states that surfaces from different families intersect in principal curves. Section 19.5 begins with some examples of triply orthogonal systems, and then introduces curvilinear patches in order to prove Dupin’s Theorem 19.21. In Section 19.6, we define elliptic coordinates and use them to find and draw the principal curves on ellipsoids, hyperboloids of one sheet and hyperboloids of two sheets. In Section 19.7, we succeed in using parabolic coordinates to find the principal curves of elliptic and hyperbolic paraboloids, and this raises the question of how widely applicable our method is. In fact, we show that there exists a triply orthogonal system containing any given surface. The construction uses the concept of parallel surface, and in Section 19.8 we show how to construct the parallel surface M(t) to a given surface M ⊆ R3 . We conclude the chapter with a study of the shape operator of parallel surfaces, and prove the important theorem of Bonnet on the subject. 19.1 The Differential Equation for Principal Curves In this section we first determine when a tangent vector to a patch is a principal vector. Then we derive the differential equation for the principal curves. Lemma 19.1. Let x: U → R3 be a regular patch. A tangent vector vp = v1 xu + v2 xv is a principal vector if and only if (19.1)  v22   det  E  e −v1 v2 F f v12    = 0. G   g 19.1. DIFFERENTIAL EQUATION FOR PRINCIPAL CURVES 595 Proof. Using the notation of Theorem 13.16, page 394, we compute S(vp )× vp and apply Lemma 15.5, page 466: S(vp ) × vp = S(v1 xu + v2 xv ) × (v1 xu + v2 xv ) = a21 v12 − (a11 − a22 )v1 v2 − a12 v22 xu × xv f F − eG f F − gE = − v1 v2 − EG − F2 EG − F2 gF − f G v22 xu × xv − EG − F2 xu × xv 2 2 = − (f E − e F )v1 + (g E − e G)v1 v2 + (g F − f G)v2 . EG − F2 eF − f E EG − F2 v12 From Lemma 15.5, page 466, we know that a vector vp is principal if and only if S(vp ) × vp = 0. It now follows that vp is principal if and only if (19.2) (f E − e F )v12 + (g E − e G)v1 v2 + (g F − f G)v22 = 0. This is just another way of writing (19.1). The lemma and its proof enable us to write down the differential equation for the principal curves, in analogy to the differential equation on page 559 for the asymptotic curves. Substituting the velocity vector (u′ (t), v ′ (t)) in place of (v1 , v2 ) in (19.2) gives Corollary 19.2. Let α be a curve that lies on the trace of a patch x. Write α(t) = x(u(t), v(t)). Then α is a principal curve if and only if (f E − e F ) α(t) u′ (t)2 + (g E − e G) α(t) u′ (t)v ′ (t) +(g F − f G) α(t) v ′ (t)2 = 0 for all t. More informally, we may write the equation as (f E − eF )u′2 + (gE − eG)u′ v ′ + (gF − f G)v ′2 = 0. Let us use it to find the principal curves on the surface helicoid[a, b](u, v) = (av cos u, av sin u, b u). Even better, we shall find a principal patch that reparametrizes the helicoid. The coefficients of the first and second fundamental forms of helicoid[a, b] are easily computed to be F = e = g = 0, E = b 2 + a2 v 2 , G = a2 , f= ab . b 2 + a2 v 2 596 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS The differential equation reduces to f E u′2 − f Gv ′2 = 0, whence a du = √ . 2 dv a v 2 + b2 Integrating, we discover the two solutions av and u − 2p = arcsinh b u − 2q = − arcsinh av , b where p and q are constants of integration. Hence to reparametrize helicoid[a, b], we set b u = p + q, v = sinh(−p + q), a so as to obtain y(p, q) = − b cos(p + q) sinh(p − q), −b sin(p + q) sinh(p − q), b(p + q) . The mapping y is called a principal patch, since by construction the curves p = constant and q = constant are principal. The two families of such curves are visible in Figure 19.1. Figure 19.1: Principal patch on a helicoid It is interesting to compare this plot of a helicoid by principal curves with that by asymptotic curves in Figure 18.1 on page 562. 19.2. UMBILIC POINTS 597 19.2 Umbilic Points In this section we consider those points on a surface in R3 at which all the normal curvatures are equal. Definition 19.3. Let M be a regular surface in R3 . A point p ∈ M is called an umbilic point provided the principal curvatures at p are equal: k1 (p) = k2 (p). It is clear that all points of a plane or a sphere are umbilic points. The vertex (0, 0, 0) of the paraboloid z = x2 + y 2 is an umbilic point, since the symmetry there prevents us from choosing a principal direction. Such a symmetry does not apply to other points, and the vertex is in fact an isolated umbilic point. By contrast, an elliptical paraboloid z = ax2 +by 2 with a > b > 0 has two umbilic points. Figure 19.2 is a rather uninformative plot of these two points made in Notebook 19, and it is the purpose of this chapter to better explain the existence of such umbilic points on paraboloids and similar surfaces. Figure 19.2: Two umbilic points on part of an elliptical paraboloid Lemma 19.4. A point p on a regular surface M ⊂ R3 is an umbilic point if and only if the shape operator of M at p is a multiple of the identity. Proof. Let Sp denote the shape operator of M at p. Then Sp is a multiple of the identity if and only if all of the normal curvatures of M at p coincide. This is true if and only if the maximum and minimum of the normal curvatures, namely, k1 (p) and k2 (p), coincide. Write k(p) = k1 (p) = k2 (p); then Sp = k(p)I, where I denotes the identity map on Mp . First, let us determine the regular surfaces that consist entirely of umbilic points. 598 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Lemma 19.5. If all of the points of a connected regular surface M ⊂ R3 are umbilic, then M has constant Gaussian curvature K > 0. Proof. Let k : M → R be the function such that k(q) is the common value of the principal curvatures at q ∈ M. Let p ∈ M, and let x: U → M be a regular patch such that p ∈ x(U). We can assume that U is connected. Then we have (19.3) Uu = −S(xu ) = −k xu , (19.4) Uv = −S(xv ) = −k xv . We differentiate (19.3) with respect to v and (19.4) with respect to u, then subtract the results. Since Uuv = Uvu and xuv = xvu , we obtain kv xu − ku xv = 0. The linear independence of xu and xv now implies that kv = 0 = ku , and k is constant on the neighborhood U of p. Moreover, the Gaussian curvature K is also constant on U, because it is given by K = k 2 > 0. Let V = { q ∈ M | K(q) = K(p) }. Then V is obviously closed, and by the previous argument it is also open. Since M is connected, it follows (see page 334) that V = M. Hence K is a constant nonnegative function on M. To determine the connected regular surfaces consisting entirely of umbilic points, we imitate the proof of Theorem 1.22 on page 16. Theorem 19.6. Let M be a connected regular surface in R3 consisting entirely of umbilic points. Then M is part of a plane or sphere. Proof. We know from Lemma 19.5 that M has constant curvature K > 0, and that K = k 2 , where k is the common value of the principal curvatures. If K = 0, then both k and S vanish identically. Hence Dv U = 0 for any tangent vector v to M (see page 386). Therefore, for each p ∈ M we have U(p) = (n1 , n2 , n3 )p , where the components ni of the unit normal vector are constants. Then M is contained in the plane perpendicular to (n1 , n2 , n3 )p . Next, suppose K > 0. Choose a point p ∈ M and a unit normal vector U(p) to M at p. We shall show that the point c defined by c=p+ 1 U(p) k is equidistant from all points of M. To this end, let q be any point of M, and let α : (a, b) → M be a curve with a < 0 < 1 < b such that α(0) = p and α(1) = q. We extend U(p) to a unit normal vector field U ◦ α along α. Define a new curve γ : (a, b) → R3 by γ(t) = α(t) + 1 U α(t) , k 19.3. PETERSON–MAINARDI–CODAZZI EQUATIONS so that (19.5) But γ ′ (t) = α′ (t) + 599 1 (U ◦ α)′ (t). k (U ◦ α)′ (t) = −S α′ (t) = −kα′ (t), so that (19.5) reduces to γ ′ = 0. Hence γ must be a constant, that is, a single point in R3 , in fact, the point c. Thus 1 c = γ(0) = γ(1) = q + U(q). (19.6) k From (19.6) it is immediate that kc − qk = 1 . |k| Thus M is part of a sphere of radius 1/|k| with center c. Next, we give a useful criterion for finding umbilic points. Lemma 19.7. A point p on a patch x: U → R3 is an umbilic point if and only if there is a number k such that at p we have (19.7) E = k e, F = k f, G = k g. Proof. The point p is umbilic if and only if equation (19.2) holds for all v1 , v2 . It is obvious that (19.7) implies this. For the converse, see Exercise 10. 19.3 The Peterson–Mainardi–Codazzi Equations The nine coefficients in the Gauss equations (17.8), page 538, are not independent. It turns out that there is a relation among e, f, g, their derivatives and the Christoffel symbols. These equations were proved by Gauss ([Gauss2, paragraph 11]) using obscure notation, then reproved successively by Peterson1 in his thesis [Pson], Mainardi2 [Main] and Codazzi3 [Codaz]. See [Reich, page 1 2 Gaspare Karl Mikhailovich Peterson (1828–1881). A student of Minding and Senff in Dorpat (Tartu). Peterson was one of the founders of the Moscow Mathematical Society. Peterson’s derivation of the equations of Theorem 19.8 was not generally known in his lifetime. Mainardi (1800–1879). Professor at the University of Padua. 3 Delfino Codazzi (1824–1873). Professor at the University of Padua. The equations (19.8) were proved by Mainardi [Main] in 1856. However, Codazzi’s formulation was simpler because he was careful that his expressions had geometric meaning, and his applications were wider. Since Codazzi submitted an early version of his work to the French Academy of Sciences for a prize competition in 1859, his contribution was known long before those of Peterson and Mainardi. Codazzi also published papers on geodesic triangles, equiareal mappings and the stability of floating bodies. 600 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS 303–305] and [Cool2] for details on the history of these equations. Theorem 19.8. (The Peterson–Mainardi–Codazzi Equations) Let x: U → R3 be a regular patch.     (19.8)    Then ∂e ∂f − = e Γ112 + f Γ212 − Γ111 − g Γ211 , ∂v ∂u ∂g ∂f − = e Γ122 + f Γ222 − Γ112 − g Γ212 . ∂v ∂u Proof. The idea of the proof is to differentiate the equations of (17.8) and use the fact that xuuv = xuvu and xuvv = xvvu . Thus from the first two equations of (17.8), it follows that 0 = xuuv − xuvu ∂ ∂ Γ111 xu + Γ211 xv + e U − Γ112 xu + Γ212 xv + f U ∂v ∂u 2 1 1 ∂Γ11 ∂Γ12 ∂Γ212 ∂Γ11 xu + xv + (ev − fu )U − − = ∂v ∂u ∂v ∂u −Γ112 xuu + Γ111 − Γ212 xuv + Γ211 xvv + e Uv − f Uu . = We use (17.8) again to expand xuu , xuv , xvv , and the Weingarten equations (13.10) on page 394 to expand Uv , Uu . Two oefficients of xu cancel out, and collecting the remainding terms we obtain: f F − eG ∂Γ112 ∂Γ111 gF − f G xu − f − − Γ212 Γ112 + Γ211 Γ122 + e ∂v ∂u EG − F2 EG − F2 2 2 ∂Γ11 ∂Γ212 + − + Γ111 Γ212 + Γ211 Γ222 − Γ112 Γ211 − Γ212 ∂v ∂u f F − gE eF − f E xv +e − f EG − F2 EG − F2 + ev − fu − e Γ112 + f Γ111 − Γ212 + g Γ211 U 1 ∂Γ112 eg − f2 ∂Γ11 xu − − Γ212 Γ112 + Γ211 Γ122 + F = ∂v ∂u EG − F2 2 2 ∂Γ11 ∂Γ212 eg − f2 + xv − + Γ111 Γ212 + Γ211 Γ222 − Γ112 Γ211 − Γ212 − E ∂v ∂u EG − F2 + ev − fu − e Γ112 + f Γ111 − Γ212 + g Γ211 U. 0 = Taking the normal component in the last line yields the first equation of (19.8). A similar argument exploiting xuvv = xuvu and the second and third equations of (17.8) gives the second equation. 19.3. PETERSON–MAINARDI–CODAZZI EQUATIONS 601 More generally, let us consider the consequences of the equations   xuuv − xuvu = 0,    (19.9) xuvv − xvvu = 0,     Uuv − Uvu = 0. for a regular patch x: U → R3 . When the Gauss equations (17.8) are substituted into (19.9), the result is three equations of the form   A1 xu + B1 xv + C1 U = 0,    (19.10) A2 xu + B2 xv + C2 U = 0,     A x + B x + C U = 0. 3 u 3 v 3 Since xu , xv , U are linearly independent, we must have Aj = Bj = Cj = 0 for j = 1, 2, 3. In the proof of Theorem 19.8, we obtained the Peterson–Mainardi– Codazzi equations (19.8) from C1 = C2 = 0. In fact, A3 = B3 = 0 also yield (19.8). Furthermore, C3 = 0 because when it is computed, all terms cancel. From A1 = B1 = A2 = B2 = 0 we get the four equations  2 ∂Γ212 ∂Γ211   − + Γ111 Γ212 + Γ211 Γ222 − Γ112 Γ211 − Γ212 , E K =   ∂v ∂u       ∂Γ112 ∂Γ111   −F K = − − Γ212 Γ112 + Γ211 Γ122 ,  ∂v ∂u (19.11)  2 ∂Γ122 ∂Γ112    GK = − + Γ122 Γ111 + Γ222 Γ112 − Γ112 − Γ212 Γ122 ,   ∂u ∂v     2 2    −F K = ∂Γ22 − ∂Γ12 + Γ1 Γ2 − Γ1 Γ2 . 22 11 12 12 ∂u ∂v The first two are visible on the previous page. But all four equations turn out to be equivalent to one another. Since each of them (when the Christoffel symbols are expanded) expresses the Gaussian curvature in terms of E, F, G, any one of them provides a new proof of Gauss’ Theorema Egregium (Theorem 17.5 on page 536). The Peterson–Mainardi–Codazzi equations simplify greatly in the case of principal patches, discussed on page 467. Corollary 19.9. Let x: U → R3 be a principal patch. Then (19.12)  Ev e g ∂e   , = +  ∂v 2 E G  ∂g Gu e g   . = + ∂u 2 E G Proof. Equation (19.12) follows from Lemma 17.8 and Theorem 19.8. 602 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS We now state a formula linking principal curvatures and their derivatives: Corollary 19.10. Let x: U → R3 be a regular principal patch with principal curvatures k1 and k2 . Then  Ev    k1v = 2E (k2 − k1 ), (19.13)    k2u = Gu (k1 − k2 ). 2G For a proof of this result, see Exercise 3. 19.4 Hilbert’s Lemma and Liebmann’s Theorem We need to start this section by quoting a lemma that can be found in [dC1, page 185]. It is actually a corollary of a more general result that implies that lines of curvature can be used as coordinate curves. Lemma 19.11. Let p be a nonumbilical point of a regular surface M in R3 . Then there exists a principal patch that parametrizes a neighborhood of p. We shall use this lemma to prove the next more celebrated one, that is a stepping stone to the theorem that follows. Lemma 19.12. (Hilbert4 ) Let M be a regular surface in R3 , and let p ∈ M be a point such that (i) k1 has a local maximum at p; (ii) k2 has a local minimum at p; (iii) k1 (p) > k2 (p). Then K(p) 6 0. Proof. At p we have k1v = k2u = 0, k1vv 6 0 and k2uu > 0 because of (i) and (ii). Condition (iii) and Lemma 19.11 imply that there exists a principal patch x parametrizing a neighborhood of p. Let E, F, G denote the 4 David Hilbert (1862–1943). Professor at Göttingen, the leading German mathematician of his time. Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. Hilbert’s most famous contribution to differential geometry is [Hil], in which he proved that no regular surface of constant negative curvature in R3 can be complete. 19.4. HILBERT’S LEMMA AND LIEBMANN’S THEOREM 603 coefficients of the first fundamental form of x. From the Peterson–Codazzi– Mainardi equations for principal patches, that is (19.13), it follows that (19.14) Ev (p) = Gu (p) = 0. Furthermore, if we differentiate (19.13) we obtain  Ev E Evv − Ev2   (k2 − k1 ) + (k2 − k1 )v ,  k =  1vv 2E 2 2E (19.15)  Gu GGuu − G2u   (k1 − k2 ) + (k1 − k2 )u .  k2uu = 2G2 2G When we evaluate (19.15) at p, we obtain 0 > k1vv (p) = Evv (k2 − k1 ) 2E and 0 6 k2uu (p) = p Guu (k1 − k2 ) . 2G p It now follows from (iii) that Evv (p) > 0 and Guu (p) > 0. Thus from (19.14) and (17.6), page 534, we get K(p) = − Evv + Guu 2E G 6 0. p Now, we can prove the main result of this section, which is sometimes referred to as ‘Rigidity of the Sphere’. 3 Theorem 19.13. (Liebmann5 ) Let M be a compact surface √ in R with constant Gaussian curvature K. Then M is a sphere of radius 1/ K. Proof. We have (19.16) 2 H −K = k1 − k2 2 2 . Since M is compact, the function H 2 − K assumes its maximum value (which must be nonnegative) at some point p ∈ M. Assume that this maximum is positive; we shall obtain a contradiction. Choose an oriented neighborhood U containing p for which H 2 −K is positive on U. The choice of orientation gives rise to a unit normal vector field U defined on all of U. Let k1 and k2 be the principal curvatures defined with respect to U; since K > 0, we can arrange that k1 > k2 > 0 on U. In fact, 5 Heinrich Liebmann (1874–1939). German mathematician, and professor at Munich and Heidelberg. 604 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS because of the identity (19.16), we have k1 > k2 > 0 on U. Since K = k1 k2 is constant, it follows that k1 has a maximum at p and k2 has a minimum at p. Hence Lemma 19.12 implies that K(p) 6 0, contradicting the assumption that K(p) > 0. Therefore, H 2 − K is zero at p. Equation (19.16) then implies that H 2 − K vanishes identically on M, so all points of M are √ umbilic. Now Theorem 19.6 implies that M is part of a sphere of radius 1/ K; since M is compact, M must be an entire sphere. 19.5 Triply Orthogonal Systems of Surfaces This subject is based on the following notion. Definition 19.14. A triply orthogonal system of surfaces on an open set U ⊆ R3 consists of three families A, B, C, of surfaces such that (i) each point of U lies on one and only one member of each family; (ii) each surface in each family meets every member of the other two families orthogonally. Three simple examples of such a system of surfaces are inherent in the use of Cartesian, cylindrical and spherical coordinates in space. Cartesian coordinates are effectively defined by three families of planes parallel to the yz-, zx- and xy-planes, as shown in Figure 19.3. To give the second example, let U be the complement of the vertical axis in R3 ; that is, U = { (p1 , p2 , p3 ) ∈ R3 | (p1 , p2 ) 6= (0, 0)}. Then the system consisting of A = planes through the z-axis, B = planes parallel to the xy-plane, C = circular cylinders around the z-axis constitutes a triply orthogonal system of surfaces on U. See Figure 19.4. For the third case, set V = R3 \ {(0, 0, 0)}, and A = planes containing the z-axis, B = circular cones with vertex (0, 0, 0) and common axis x = 0 = y, C = spheres centered at the origin (0, 0, 0). This gives rise to a triply orthogonal system of surfaces on V. See Figure 19.5. 19.5. TRIPLY ORTHOGONAL SYSTEMS 605 Figure 19.3: Three families of planes Figure 19.4: Two families of planes and one of cylinders Many classical differential geometry books have a section on triply orthogonal systems, see for example [Bian], [Eisen1] and [Wea]. An extensive list of triply orthogonal systems is given in [MoSp]. We shall study more interesting examples in Sections 19.6 and 19.7. 606 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Figure 19.5: One family of planes, one of cones and one of spheres A curve in R3 has one parameter, a surface two. It is also useful to consider a function of three parameters in order to generate a triply orthogonal system. Definition 19.15. A curvilinear patch for R3 is merely a differentiable function x: U → R3 , where U is some open set of R3 . If x: U → R3 is a curvilinear patch, we can write x(u, v, w) = x1 (u, v, w), x2 (u, v, w), x3 (u, v, w) . Differentiation of a curvilinear patch is defined in the same way as differentiation of a patch in R3 . Thus ∂x1 ∂x2 ∂x3 xu = , , , ∂u ∂u ∂u and so forth. Definition 19.16. A curvilinear patch x: U → R3 is regular provided the vector fields xu , xv , xw are linearly independent throughout U or (equivalently) the vector triple product [xu xv xw ] is nowhere zero. We shall say that x is orientation preserving if [xu xv xw ] > 0, and orientation reversing if [xu xv xw ] < 0. Note that if x is orientation reversing, then the curvilinear patch y defined by y(u, v, w) = x(u, w, v) is orientation preserving. 19.5. TRIPLY ORTHOGONAL SYSTEMS 607 Having introduced this terminology, a triply orthogonal system of surfaces can be considered to be a special kind of curvilinear patch. Lemma 19.17. A regular curvilinear patch x: U → R3 satisfying (19.17) xv · xw = xw · xu = xu · xv = 0 determines a triply orthogonal system of surfaces, each of which is regular. Proof. For fixed u, the mapping (v, w) 7→ x(u, v, w) is a 2-dimensional patch. It is regular because xv × xw is everywhere nonzero. Each of its tangent spaces is spanned by xv and xw . The analogous properties hold for the mappings (w, u) 7→ x(u, v, w) and (u, v) 7→ x(u, v, w). Then (19.17) implies that the tangent spaces to the traces of the three patches (19.18) (v, w) 7→ x(u, v, w), (w, u) 7→ x(u, v, w), (u, v) 7→ x(u, v, w) are mutually perpendicular. Thus if we set A = B = C = (v, w) 7→ x(u, v, w) , (w, u) 7→ x(u, v, w) , (u, v) 7→ x(u, v, w) , then A, B, C form a triply orthogonal system of surfaces. That regularity of the curvilinear patch implies regularity of each of the surfaces in A, B, C results from the following identities, which are consequences of (7.1): [xu xv xw ]2 = kxu k2 kxv k2 kxw k2 , kxv × xw k = kxv kkxw k, kxw × xu k = kxw kkxu k, kxu × xv k = kxu kkxv k. Since we shall deal only with local questions concerning triply orthogonal systems of surfaces, we need only consider those systems determined by a curvilinear patch x: U → R3 satisfying (19.17). We shall not prohibit such a curvilinear patch from having isolated singularities and nonregular points. Definition 19.18. Let U be an open set in R3 . A triply orthogonal patch is an orientation-preserving regular curvilinear patch x: U → R3 such that (19.17) holds. Given such a patch, set p = kxu k, q = kxv k and r = kxw k. Then the unit normals determined by x are the vector fields A= xu , p B= xv , q C= xw . r 608 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Using this definition, we can compute xv × xw = q B × r C = q r A = (19.19) qr xu . p This enables us to state Lemma 19.19. A triply orthogonal patch x satisfies the following relations: xv × xw = qr xu , p xw × xu = rp xv , q xu × xv = pq xw , r [xu xv xw ] = pq r, xu · xvw = 0, xv · xwu = 0, xw · xuv = 0. Proof. The triple product formula also follows from (19.19). For the dot products, we first differentiate (19.17), so as to obtain   xvu · xw + xv · xwu = 0,    (19.20) xwv · xu + xw · xuv = 0,     x ·x +x ·x uw v u vw = 0. The result follows by combining these equations with appropriate signs, exploiting symmetry of the mixed partial derivatives. Next, we show that the coefficients of the first and second fundamental forms of each of the surfaces determined by a triply orthogonal patch are especially simple. Lemma 19.20. Let x: U → R3 be a triply orthogonal patch. The coefficients E, F, G of the first fundamental form, the coefficients e, f, g of the second fundamental form, the principal curvatures k1 , k2 , the Gaussian curvature K and the mean curvature H of the patches (19.18) are given by the following table: P atch E F G e f g k1 k2 K H (v, w) 7→ · · · q 2 0 r2 − q qu p 0 − r ru p − qu ru − pq rp qu ru p2 q r − (q r)u 2 pq r (w, u) 7→ · · · r2 0 p2 − r rv q 0 − ppv q − rv qr rv pv pq 2 r − (r p)v 2 pq r (u, v) 7→ · · · p2 0 q 2 − ppw r 0 − pw q qw qw − − r rp qr − pv pq pw qw (pq)w − 2 pq r 2 pq r 19.6. ELLIPTIC COORDINATES 609 Proof. That F = 0 for each of the surfaces is a consequence of the assumption that the surfaces are pairwise orthogonal. Furthermore, the formulas for E and G are just the definitions of p, q, r. As for the second fundamental forms, we first note that (19.20) implies that f = 0 for each of the surfaces in (19.18). Next, we compute e for the surface (v, w) 7→ x(v, w). Since the unit normal is xu /kxu k, we have e = xvv · −kxv k2u −q qu xu = = . kxu k 2kxu k p The computation of the other coefficients is similar. Now we can give a simple proof of the following important theorem: Theorem 19.21. (Dupin6 ) The curves of intersection of the surfaces (19.18) of a triply orthogonal patch are principal curves on each. Proof. Lemma 19.20 implies that F = f = 0, the condition that appears in Lemma 15.5 on page 466. Thus, each of the patches given by (19.18) has the property that the parameter curves are principal curves. 19.6 Elliptic Coordinates In this section we show how to find triply orthogonal patches that give rise to principal patches on ellipsoids, hyperboloids of one sheet and hyperboloids of two sheets. Each of these surfaces can be rotated and translated so that it is described by a nonparametric equation of the form (19.21) x2 y2 z2 + + =1 a b c for appropriate a, b, c. (In contrast to page 312, the constants are not squared, so that (19.21) can represent three different types of nondegenerate quadrics.) Fix a, b, c, and define (19.22) F [λ](x, y, z) = y2 z2 x2 + + − 1. a−λ b−λ c−λ The surfaces defined implicitly by the equations F [λ] = 0 are called the confocals of the surface defined by (19.21), which corresponds to λ = 0. (See [HC-V, 1.4].) 6 Baron Pierre-Charles-François Dupin (1784–1873). French mathematician, student of Monge. In addition to this theorem, Dupin’s contributions to differential geometry include the theory of cyclides presented in Chapter 20, and the Dupin indicatrix that gives an indication of the local behavior of a surface in terms of its power series expansion. In 1830 Dupin was elected deputy for Tarn and continued in politics until 1870. He also wrote extensively on economics and industry. 610 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS For each fixed λ, the gradient of (19.22) is given by (19.23) grad F [λ] = 2x 2y 2z . , , a−λ b−λ c−λ If we now choose to fix x, y, z, λ for which F [λ](x, y, z) = 0, then (19.23) is a normal vector to the confocal surface at (x, y, z) (see page 10.48). The link with triple orthogonal ststems is provided by Lemma 19.22. If λ, µ are distinct real numbers, then F [λ] − F [µ] = λ−µ 1 4 grad F [λ] · grad F [µ]. Proof. It suffices to compute F [λ] − F [µ] and use (19.23). The formulation of the lemma is easy to remember, since if we vary λ and let it approach µ, the equation becomes the identity F ′ [λ] = 14 k grad F [λ]k2 . In order to exploit Lemma 19.22, we fix a point P = (x, y, z) and solve the equation F [λ](x, y, z) = 0, which becomes a cubic equation in λ by the time we have multiplied both sides by (a − λ)(b − λ)(c − λ). Suppose that this equation has three real roots u, v, w all distinct from a, b, c, so that (19.24) (u − λ)(v − λ)(w − λ) = (a − λ)(b − λ)(c − λ) −x2 (b − λ)(c − λ) − y 2 (a − λ)(c − λ) − z 2 (a − λ)(b − λ). Not only does this mean that the confocal surfaces F [u], F [v], F [w] all pass through P , but they are mutually perpendicular at P . For Lemma 19.22 tells us that their normal vectors are orthogonal. Carrying out this procedure at each point in R3 gives rise to elliptic coordinates, which we now describe. To find the values of x, y, z, or rather x2 , y 2 , z 2 , such that (19.24) holds for all λ, we give λ the values a, b, c in succession. The result is: (19.25)  (a − u)(a − v)(a − w)  x2 = ,    (b − a)(c − a)      (b − u)(b − v)(b − w) y2 = ,  (a − b)(c − b)         z 2 = (c − u)(c − v)(c − w) . (a − c)(b − c) The solutions to equations (19.25) form eight patches. 19.6. ELLIPTIC COORDINATES 611 Definition 19.23. Let a > b > c. An elliptic coordinate patch is one of the eight curvilinear patches defined by s (a − u)(a − v)(a − w) , (u, v, w) 7→ ± (b − a)(c − a) ± (19.26) s (b − u)(b − v)(b − w) , ± (c − b)(a − b) s (c − u)(c − v)(c − w) (a − c)(b − c) ! . The proof of the next result formalizes what we have already said, with xu playing the role of grad F [λ]. Lemma 19.24. For a > b > c, each mapping (19.26) is triply orthogonal. It is regular on the set (p1 , p2 , p3 ) ∈ R3 | p1 , p2 , p3 are distinct, p1 6= a, p2 6= b, p3 6= c . Proof. If we indicate the mapping in question by x: (u, v, w) 7→ (x, y, z), then it follows from (19.25) that 2x ∂x (a − v)(a − w) x2 =− = . ∂u (b − a)(c − a) (u − a) Similar formulas tell us that  x y z   , 2xu = , ,    (u − a) (u − b) (u − c)      y z x , , , (19.27) 2xv =  (v − a) (v − b) (v − c)       x y z   , , .  2xw = (w − a) (w − b) (w − c) Then (19.25) and (19.27) imply that 4 xu · xv = x2 y2 z2 + + (u − a)(v − a) (u − b)(v − b) (u − c)(v − c) = a−w b−w c−w + + (b − a)(c − a) (c − b)(a − b) (a − c)(b − c) = −(a − w)(b − c) − (b − w)(c − a) − (c − w)(a − b) (a − b)(b − c)(c − a) = 0. 612 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Similarly, xw · xu = 0 = xv · xw , and (19.28) p2 = kxu k2 = (u − v)(u − w) , 4(a − u)(b − u)(c − u) q 2 = kxv k2 = (v − w)(v − u) , 4(a − v)(b − v)(c − v) r2 = kxw k2 = (w − u)(w − v) . 4(a − w)(b − w)(c − w) Hence the Jacobian matrix of x has rank 3 on the set stated. If we fix w, then (19.26) determines a patch (u, v) 7→ (x, y, z) as explained on page 607. In these circumstances, u and v are called elliptic coordinates, and we next determine their possible ranges and associated surfaces. Lemma 19.25. Let a > b > c > 0. (i) Suppose c > w so that x2 y2 z2 + + =1 a−w b−w c−w is an ellipsoid. In order that x, y, z in (19.25) be real, either (19.29) a>v>b>u>c or a > u > b > v > c. (ii) Suppose b > w > c, so that y2 z2 x2 + + =1 a−w b−w c−w is a hyperboloid of one sheet. In order that x, y, z be real, either (19.30) a>v>b>c>u or a > u > b > c > v. (iii) Suppose a > w > b, so that y2 z2 x2 + + =1 a−w b−w c−w is a hyperboloid of two sheets. In order that x, y, z be real, either (19.31) a>b>v>c>u or a > b > u > c > v. Proof. If w < c < b < a, it follows from (19.25) that   (a − u)(a − v) > 0,    (19.32) (b − u)(b − v) < 0,     (c − u)(c − v) > 0. Without loss of generality, v > u. Then (19.32) implies (i). The proofs of (ii) and (iii) are similar. 19.6. ELLIPTIC COORDINATES 613 It is complicated to generate an entire ellipsoid using elliptical coordinates, because each of the eight patches must be plotted separately as an octant of the ellipsoid. In Figure 19.6, the lower four octants have been translated down by −2 so that we can peek inside the egg. The new coordinates make it easy to spot the umbilics, and we can determine them analytically using methods from the previous section. Figure 19.6: Principal curves on x2 y2 + + z2 = 1 12 5 Theorem 19.26. Suppose a > b > c > 0. (i) If c > w, then the umbilic points of the ellipsoid (19.29) are the four points s s ! (a − b)(a − w) (b − c)(c − w) . , 0, ± ± (a − c) (a − c) (ii) If b > w > c, then the hyperboloid (19.30) has no umbilic points. (iii) If a > w > b, then the umbilic points of the hyperboloid (19.31) are the four points s s ! (a − c)(a − w) (b − c)(b − w) ± , ± , 0 . (a − b) (a − b) 614 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Figure 19.7: Principal curves on x2 y2 z2 + − =1 9 2 2 Figure 19.8: Principal curves on x2 y2 z2 − − =1 4 3 7 Proof. From Lemma 19.20 and (19.28), it follows that the principal curvatures of the patch (u, v) 7→ (x, y, z) (defined by (19.26) with w fixed) are given by 19.7. PARABOLIC COORDINATES AND A GENERAL CONSTRUCTION 615 1 1 ∂ log(p2 ) = , 2r ∂w 2(u − w)r 1 1 ∂ log(q 2 ) = . k2 = − 2r ∂w 2(v − w)r k1 = − Then k1 = k2 implies that u = v. In case (i) we have u = v = b, and in case (iii) we have u = v = c, but in case (ii) no umbilic points are possible. Hence the umbilic points are as stated. There are no umbilics on a hyperboloid of one sheet (Figure 19.7), but two on each sheet of the two-sheeted variety (Figure 19.8). Figure 19.9 illustrates how an ellipsoid, a hyperboloid of one sheet and a hyperboloid of two sheets intersect orthogonally in principal curves. Figure 19.9: Triply orthogonal system formed by ellipsoids and hyperboloids 19.7 Parabolic Coordinates and a General Construction Parabolic coordinates are defined in a similar way to elliptic coordinates, in order to furnish principal patches on elliptic paraboloids and hyperbolic paraboloids. We proceed in the same way that we did in Section 19.6, starting from the equation x2 y2 (19.33) + = 2z − c a b of a general paraboloid. 616 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS This time, the confocals are defined as the zero sets of (19.34) G[λ] = y2 x2 + − 2z + c + λ. a−λ b−λ This equation is chosen so that the analog of Lemma 19.22 remains valid, and it is also the case that the parabolas y2 − 2z + c + λ = 0 b−λ in the yz-plane have a common focus, namely the point (y, z) = (0, (b + c)/2). The confocal surfaces through a given point (x, y, z) ∈ R3 are given by the cubic equation (a − λ)(b − λ)G[λ] = 0, and the roots u, v, w satisfy (19.35) (u − λ)(v − λ)(w − λ) = (a − λ)(b − λ)(2z − c − λ) − x2 (b − λ) − y 2 (a − λ). In order that (19.35) hold for all λ, we must have  (u − a)(v − a)(w − a)   x2 = ,   a−b      (u − b)(v − b)(w − b) y2 = ,  b−a         z = c − a − b + u + v + w; 2 (19.36) the last equality was obtained by comparing coefficients of λ2 . Expressing x, y, z in terms of u and v gives Definition 19.27. Suppose a > b. A parabolic coordinate patch is one of the four curvilinear patches defined by (u, v, w) (19.37) 7→ ± r (a − u)(a − v)(a − w) , b−a ! r (b − u)(b − v)(b − w) c − a − b + u + v + w . , ± a−b 2 19.7. PARABOLIC COORDINATES AND A GENERAL CONSTRUCTION 617 Figure 19.10: Triply orthogonal system formed by elliptic and hyperbolic paraboloids The analog of Lemma 19.24 is illustrated in Figure 19.10, and the individual surfaces are plotted overleaf. With a different choice of parameters, it is possible to construct a triply orthogonal system involving two distinct families of hyperbolic paraboloids. Lemma 19.28. For a > b, the curvilinear patch (19.37) is triply orthogonal. It is regular on the set (p1 , p2 , p3 ) ∈ R3 | p1 , p2 , p3 are distinct, p1 6= a, p2 6= b . Proof. We skip some of the details, since the method is identical to that of (19.27). It follows from (19.36) that xu · xv = xw · xu = xv · xw = 0, and that p2 = kxu k2 = (u − v)(u − w) , 4(a − u)(b − u) q 2 = kxv k2 = (v − w)(v − u) , 4(a − v)(b − v) r2 = kxw k2 = (w − u)(w − v) . 4(a − w)(b − w) Hence the Jacobian matrix of x has rank 3 on the stated set. 618 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Figure 19.11: Principal curves on x2 y2 x2 y2 + = −2z + 7 and + = 2z 2 3 5 4 Figure 19.12: Principal curves on x2 − y 2 = 4z − 3 We show that any principal patch in R3 is part of a triply orthogonal system of surfaces. We first prove Theorem 19.29. A necessary and sufficient condition that a curve on a surface be a principal curve is that the surface normals along the curve form a flat ruled surface. Proof. Let β : (a, b) → R3 be a unit-speed curve on a surface M. Without loss of generality, we can assume that M is oriented with unit normal U. Let T denote the unit-tangent vector of β, and let N be the surface formed by the normals along β. We parametrize N as y(s, v) = β(s) + v U(s), 19.7. PARABOLIC COORDINATES AND A GENERAL CONSTRUCTION 619 where s is arc length along β. Then (19.38) ys = T + v Us , yv = U, ysv = Us , yvv = 0. According to Corollary 13.32 on page 405, and the last equation of (19.38), the Gaussian curvature of N is given by K= −[ysv ys yv ]2 [yss ys yv ][yvv ys yv ] − [ysv ys yv ]2 = 2 2 . kys k2 kyv k2 − (ys · yv )2 kys k2 kyv k2 − (ys · yv )2 Thus K = 0 if and only if the vector triple product [ysv ys yv ] is zero. From (19.38), we have [ysv ys yv ] = Us × (T + v Us ) · U = Us × T · U. Since both T and Us are perpendicular to U, this implies that K = 0 if and only if (19.39) 0 = Us × T = −S(T) × T, where S denotes the shape operator of M. It follows from Lemma 15.5, page 466, that (19.39) is precisely the condition that β be a principal curve of M. Figure 19.13: A triply orthogonal system constructed from surfaces parallel to a revolved nephroid, planes and cones We are now able to construct the triply orthogonal system containing a given principal patch, such as that shown in Figure 19.13. 620 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Theorem 19.30. Let x: U → R3 be a principal patch. Define a curvilinear patch y : U × R → R3 by (19.40) y(u, v, w) = x(u, v) + w U, where U is a unit normal to x. Put A = B = C = (u, v) 7→ y(u, v, w) , (w, u) 7→ y(u, v, w) , (v, w) 7→ y(u, v, w) ; then A, B, C form a triply orthogonal system of surfaces. Each surface in B and C is a flat ruled surface. Proof. Let k1 and k2 be the Then we have   yu =    yv =     y = w principal curvatures corresponding to xu and xv . xu + w Uu = (1 − w k1 )xu , xv + w Uv = (1 − w k2 )xv , U. Hence yu , yv , yw are mutually orthogonal, so that y is a triply orthogonal patch. Now y determines a triply orthogonal system of surfaces by Lemma 19.17 on page 607. The last statement is a consequence of Theorem 19.29. 19.8 Parallel Surfaces In Section 4.5 we constructed a parallel curve to a given plane curve. There is a similar definition of parallel surface that was implicit in the preceding section. Definition 19.31. Let M ⊂ R3 be a regular surface. The surface parallel to M at a distance t > 0 is the set M(t) = q ∈ R3 distance(q, M) = t . If M is connected and orientable, then M(t) will consist of two components; see, for example, [Gray]. But if t is large, M(t) may not be a regular surface, as can be seen in Figure 19.14 that extends Figure 4.13 on page 112. However, for t close to 0, we shall prove that M(t) is a regular surface. Figure 19.15 shows parallel surfaces to a catenoid for small t. The resulting components (for t 6= 0) are not themselves catenoids, since the ‘stem’ can shrink to nothing, nor are they minimal surfaces. In the next section we shall undertake a quantitive study of the curvature of a parallel surface. 19.8. PARALLEL SURFACES 621 First, let us define the notion of parallel patch. The idea is to move the patch a distance t along its normal U. We want to be able to go in either direction along the normal, so we now allow t to be either positive or negative. Definition 19.32. Let x: U → R3 be a regular patch. Then the patch parallel to x at distance t is the patch given by (19.41) x[t](u, v) = x(u, v) + t U(u, v), where U = xu × xv /kxu × xv k. The similarity with (19.40) allows us to assert that each surface in the family A in Theorem 19.30 is parallel to x(U). We next give conditions for the patch parallel to x at distance t to be regular. Figure 19.14: Parallel surface to the ellipsoid 4y 2 4x2 + + z2 = 1 9 9 Lemma 19.33. Let M ⊂ R3 be a regular surface for which there exists a single regular patch x: U → M. Assume that there is a number t0 > 0 such that det(I − tS) > 0 (19.42) for |t| < t0 on M, where I denotes the identity transformation and S the shape operator. Then x[t] is a regular patch on M(t) for |t| < t0 . Proof. We have (19.43) x[t]u = xu + t Uu = (I − tS)xu ; similarly for x[t]v . Therefore, (19.44) x[t]u × x[t]v = (I − tS)xu × (I − tS)xv = det(I − tS)xu × xv . Hence x[t]u × x[t]v is nonzero, and therefore x[t] is regular. Thus M(t) is a regular surface, since it is entirely covered by the regular patch x[t]. 622 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS Next, we consider parallel surfaces to arbitrary regular surfaces. Lemma 19.34. Let M ⊂ R3 be a regular surface and let M(t) denote the surface parallel to M at a distance t. Assume that there exists t0 > 0 such that (19.42) holds. Then the parallel surface M(t) is a regular surface for |t| < t0 . Proof. Suppose that x: U → M is a regular injective patch on M, and set V = U × { t | |t| < t0 }. Use (19.41) to define y : V → R3 by merely writing y(u, v, t) = x(u, v)[t]. The resulting Jacobian matrix J(y) = (I − tS)xu , (I − tS)xv , U , computed using (19.43), is obviously nonzero on V. Hence y is injective on a neighborhood of each point of V; in particular, x[t] is a regular injective patch on a neighborhood of each point of U. Patches of the form x[t] cover M(t), so M(t) is a regular surface. Figure 19.15: Parallel surfaces to a catenoid (in the middle) 19.9 The Shape Operator of a Parallel Surface First, we show that the shape operator of a regular surface M ⊆ R3 determines the shape operator of each of its parallel surfaces. Lemma 19.35. Let M ⊂ R3 be a regular surface and let M(t) denote the surface parallel to M at a distance t. Assume that condition (19.42) holds. Let S be the shape operator of M and S(t) the shape operator of M(t). Then 19.9. SHAPE OPERATOR OF A PARALLEL SURFACE 623 (i) the matrix-valued function t 7→ S(t) satisfies S ′ (t) = S(t)2 ; (ii) S(t) = S(I − tS)−1 ; (iii) the principal curvatures of M(t) are given by ki (t) = (19.45) ki , 1 − tki for i = 1, 2, where k1 and k2 are the principal curvatures of M. (iv) The Gaussian and mean curvatures of M(t) are given by (19.46) K(t) = K 1 − 2t H + t2 K H(t) = and H −tK , 1 − 2t H + t2 K where K and H denote the Gaussian and mean curvatures of M. Proof. Without loss of generality, M is parametrized by a single regular patch x: U → M. Then the patch x[t] on M(t) defined by (19.41) is regular, with unit normal x[t]u × x[t]v U[t] = . x[t]u × x[t]v But, given that det(I − tS) > 0, (19.44) implies that U[t] coincides with U. Therefore, (19.47) S(t)x[t]u = −U[t]u = −Uu = Sxu , proving that S(t)x[t]u does not depend on t. Hence (19.48) S(t)x[t]u ′ = 0, where the prime denotes differentiation with respect to t. Also, from (19.41), we obtain (19.49) x[t]′u = Uu = −Sxu . Then (19.47)–(19.49) imply that S ′ (t)x[t]u = −S(t)x[t]′u = S(t)Sxu = S(t)2 x[t]u . Similarly, S ′ (t)x[t]v = S(t)2 x[t]v . Equation (19.44) implies that x[t]u and x[t]v are linearly independent. Hence S ′ (t) = S(t)2 , proving (i). e = S(I − tS)−1 , and use the binomial theorem To prove (ii), we define S(t) to expand it as a convergent geometrical series for small t: (19.50) e = S(t) ∞ X k=0 tk S k+1 . 624 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS e for all t, and so It is clear from (19.50) that S commutes with S(t) e 2. Se′ (t) = S 2 (I − tS)−2 = S(t) e Since S(0) = S, we conclude that Se coincides with S. For (iii) we compute the eigenvalues of S(I − tS)−1 and use (ii). To prove (iv), we use (19.45). In particular, K(t) = k1 (t)k2 (t) = K k1 k2 = (1 − t k1 )(1 − t k2 ) 1 − 2tH + t2 K and calculation of H(t) is similar . Now we can prove an important result due to Bonnet. As Chern has remarked (see [Chern3]), this theorem tells us that finding surfaces of constant mean curvature is equivalent to finding surfaces of constant positive Gaussian curvature. Theorem 19.36. (Bonnet) Let M ⊂ R3 be a regular surface. (i) If M has constant mean curvature H ≡ 1/(2c), then the parallel surface M(c) has constant Gaussian curvature 1/c2 , and the parallel surface M(2c) has constant mean curvature −1/(2c). (ii) If M has constant Gaussian curvature K ≡ 1/c2 , then the parallel surfaces M(±c) have constant mean curvature ∓1/(2c). Proof. If H ≡ 1/(2c), then it follows from (19.46), that K(c) = K 1 = 2. 1 − 2c/(2c) + K c2 c The rest of (i) and (ii) follow from (19.46) in a similar fashion (see Exercise 15). Finally, we prove a partial converse to Theorem 19.36. Theorem 19.37. Let M ⊂ R3 be a regular surface with constant positive Gaussian curvature 1/a2 , where a > 0. Let M(t) denote the surface parallel to M at a distance t. Suppose that the umbilic points of M are isolated. If M(t) has constant mean curvature, then t = ±a. Proof. Fix t, and suppose that H(t) is constant on M(t). The second equation of (19.46) implies that k1 (1 − tk2 ) + k2 (1 − tk1 ) = 2H(t)(1 − tk1 )(1 − tk2 ), or k1 + k2 − 2t t2 = 2H(t) 1 − t(k1 + k2 ) + 2 . 2 a a 19.10. EXERCISES 625 Hence 2t t2 (k1 + k2 ) 1 + 2tH(t) = 2 H(t) + 2 H(t) 2 + 2 . a a By hypothesis, the right-hand of (19.51) is constant. But if the left-hand of (19.51) is constant, Lemma 19.12 forces it to vanish at the nonumbilic points of M. Hence 1 + 2tH(t) = 0 everywhere, and (19.51) now implies that (19.51) t2 2t 1 1 t t2 2t 0 = 2 H(t) 1 + 2 + 2 = − 1 + 2 + 2 = − + 2 . a a t a a t a Therefore, t = ±a. We shall investigate surfaces with constant nonzero Gaussian curvature in Chapter 21. 19.10 Exercises 1. Prove the second equation of (19.8). 2. Verify that if x: U → R3 is a regular patch, then ∂Γ112 ∂Γ111 − − Γ212 Γ112 + Γ211 Γ122 = −F K, ∂v ∂u as stated in the proof of Theorem 19.8. 3. Prove Corollary 19.10. 4. The Peterson–Mainardi–Codazzi equations also simplify for an asymptotic patch x: U → R3 (see page 561). Show that ∂ log f = Γ111 − Γ212 , ∂u ∂ log f = Γ222 − Γ112 . ∂v 5. Find a patch x: U → R3 for which the coefficients of the first and second fundamental forms are E = a2 cos2 v, F = 0, G = a2 , e = −a cos2 v, f = 0, g = −a. 6. Find a patch x: U → R3 for which the coefficients of the first and second fundamental forms are E = a2 , F = 0, G = 1, e = −a, f = 0, g = 0. 626 CHAPTER 19. PRINCIPAL CURVES AND UMBILIC POINTS 7. Find the most general patch x: U → R3 for which xuv = 0. 8. Show that there is no patch having du2 + dv 2 and du2 − dv 2 as its first and second fundamental forms. 9. Show that there is no patch having du2 + cos2 u dv 2 and cos2 u du2 + dv 2 as its first and second fundamental forms. 10. Let p be a point of an open set U in R2 at which a patch x: U → R3 satisfies f E = e F, g E = eG and g F = f G. Show that there exists a number k such that at p we have e = kE f = kF and g = k G. M 11. Show that the surface of revolution generated by a parallel curve β to a curve α is the same as a parallel surface y to the surface of revolution x generated by α. 12. The notion of a system of orthogonal curves in R2 corresponds to the notion of a system of triply orthogonal surfaces in R3 : Definition 19.38. An orthogonal system of curves on an open set U of R2 consists of two families A, B, of curves such that (i) each point of U lies on one and only one member of each family; (ii) each curve in each family meets every member of the other family orthogonally. Let U ⊆ R2 be an open subset and u = u(x, y), v = v(x, y) two functions defined on U so that (x, y) 7→ (u, v) is a regular patch U → R2 . Suppose that the partial derivatives of u, v satisfy (19.52) ux = vy , uy = −vx . Show that the images of the families H and V of horizontal and vertical lines in R2 constitute a system of orthogonal curves. M 13. In the previous exercise, (19.52) are the so-called Cauchy-Riemann equations that express the fact that u+iv is a holomorphic function (such functions will be discussed in Section 22.1). Use methods from Notebook 19 to study the case in which u + iv = tan(x + iy), illustrated in Figure 19.16. 19.10. EXERCISES 627 Figure 19.16: Image of horizontal and vertical lines by z 7→ tan z Figure 19.17: Orthogonal ellipsoid and hyperboloid of revolution 14. Let A and B be a system of orthogonal curves in R2 that are symmetric e and B e be the families of with respect to reflection in a line ℓ. Let A surfaces of revolution in R3 generated by A and B, using ℓ as axis (see e and B, e together with the family C of planes Figure 19.17). Show that A through ℓ, constitute a triply orthogonal system of surfaces. 15. Complete the proof of Theorem 19.36. Chapter 20 Canal Surfaces and Cyclides of Dupin Let M be a regular surface in R3 and let W ⊆ M be the image of a regular patch on which a unit normal U is defined and differentiable. Denote by k1 and k2 the principal curvatures of M with respect to U, and suppose that they are ordered so that k1 > k2 on W. Deﬁnition 20.1. The reciprocals ρ1 = 1 k1 and ρ2 = 1 k2 of the principal curvatures are called the principal radii of curvature of M. This chapter is concerned with the geometry arising from this definition, and analogs of the evolute of a curve, defined in Section 4.1. For a point q in a regular surface M, we denote by ℓq the line normal to M at q; then the surface normal U(q) is a vector in ℓq . Each normal section (see page 391) is a plane curve C in a plane Π containing ℓq . Since U(q) is perpendicular to C at q, the center of curvature (see page 99) of C lies on ℓq . Indeed, the centers of curvature at q fill out a connected subset F of ℓq called the focal interval of M at q; the focal interval reduces to a point if q is an umbilic point; otherwise, it is a line segment, or the complement of a line segment. The extremities of F are called the focal points of M at q. They coincide if and only if q is an umbilic point. If the Gaussian curvature of M vanishes at q, then at least one focal point is at infinity. Figure 20.1 displays the typical behavior at an elliptic and a hyperbolic point; only in the latter case does the focal interval contain 0. 639 640 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Figure 20.1: Focal interval at the vertex of both an elliptic and a hyperbolic paraboloid It is clear from (13.22), page 400, that each ki is continuous on W. But it can happen that k1 or k2 is not differentiable at an umbilic point; this happens for the monkey saddle (see Figure 13.7 on page 404). But it is known that both k1 and k2 are differentiable provided W contains no umbilic points; for details see [Ryan, page 371], [CeRy, page 134]. In this chapter we assume that both k1 and k2 are differentiable on W, except possibly at isolated points, and for convenience we take W = M. Deﬁnition 20.2. The focal set of a regular surface M ⊂ R3 is focal(M) = { p ∈ R3 | p is a focal point of some q ∈ M }. In the case that M ⊂ R3 is connected, orientable and free of umbilic points, the set focal(M) has two components, one corresponding to each principal curvature. There are three possibilities: Case 1. Each component of focal(M) is a surface. Case 2. One component of focal(M) is a curve and the other a surface. Case 3. Each component of focal(M) is a curve. Sections 20.1, 20.2 and 20.3 deal successively with each of these cases. The generic situation is dealt with first, and we investigate properties of the focal surfaces. The second case characterizes envelopes of a 1-parameter family of 20.1. 2-DIMENSIONAL FOCAL SETS 641 spheres, called canal surfaces, and the third gives rise to the cyclides of Dupin. A cyclide of Dupin can be considered to be a canal surface of either of the components of its focal set. Perhaps the simplest type of mapping between open subsets of Rn , other than an isometry or affine transformation, is an inversion. In Section 20.4 we give the precise definition of inversion and prove that it is a conformal map. Inversion of surfaces is studied in Section 20.5, the main aim being to exhibit cyclides of Dupin as inversions of tori. 20.1 Surfaces whose Focal Sets are 2-Dimensional In this section, we consider Case 1. The first order of business is to make each component of focal(M) into a surface, possibly with singularities. This will be done by showing that each regular principal patch x: U → M gives rise to patches z1 : U → focal(M) and z2 : U → focal(M) parametrizing each of the components of focal(M). Deﬁnition 20.3. Let x: U → M be a patch on a surface M ⊂ R3 . With the notation from page 639, the focal patches corresponding to x are defined by (20.1) z1 (u, v) = x(u, v) + ρ1 U(u, v) and z2 (u, v) = x(u, v) + ρ2 U(u, v). By definition, the image of each focal patch zi is contained in focal(M). We want to compute the Gaussian curvatures of each component of the focal set of M, and also the first and second fundamental forms. For this, we need to parametrize M by patches and then compute the first and second fundamental forms of the associated patches z1 , z2 . The calculations are considerably easier when we use principal patches. Theorem 20.4. Let x: U → R3 be a regular principal patch, let z1 and z2 be the corresponding focal patches, and let Ui be the unit normal of zi . Then   z1u = ρ1u U,       ρ1 xv + ρ1v U, z1v = 1 − (20.2) ρ2     ρ1 x   √u ,  U1 = − sign ρ1u 1 − ρ2 E and (20.3)  ρ2   xu + ρ2u U, z2u = 1 −   ρ1   z2v = ρ2v U,     ρ xv   U2 = − sign ρ2v 1 − 2 √ . ρ1 G 642 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN It is a consequence of this theorem that z1 is regular on the set (u, v) ∈ U | ρ1 (u, v) 6= ρ2 (u, v) and ρ1u (u, v) 6= 0 , and z2 is regular on (u, v) ∈ U | ρ1 (u, v) 6= ρ2 (u, v) and ρ2v (u, v) 6= 0 . Proof. We have (20.4) −Uu = S(xu ) = 1 xu ρ1 and − Uv = S(xv ) = 1 xv . ρ2 Differentiating (20.1) and using (20.4), we get (20.5) Similarly, 1 z1u = xu + ρ1u U + ρ1 Uu = xu + ρ1u U + ρ1 − xu = ρ1u U. ρ1 z1v = (20.6) Using (20.5) and (20.6), z1u × z1v = ρ1u U × ρ1 1− xv + ρ1v U. ρ2 ρ1 1− ρ2 ρ1 xv + ρ1v U = ρ1u 1 − U × xv . ρ2 Hence ρ1 U × xv ρ1u 1 − ρ z1u × z1v xu ρ = − sign ρ1u 1 − 1 2 √ . U1 = = √ ρ1 kz1u × z1v k ρ2 E G ρ1u 1 − ρ2 This proves (20.2). Similar calculations yield (20.3). A point of the principal curve corresponding to ρ1 for which ρ1u vanishes is called a ridge point of the principal curve on M. A similar definition holds for the principal curve corresponding to ρ2 . From (20.2) and (20.3) we see that ridge points on M give rise to singular points on the focal set of M. The coefficients of the first fundamental forms of the focal patches are easily computed from Theorem 20.4. We omit the verification, that is carried out in Notebook 20. Lemma 20.5. Let x: U → R3 be a regular principal patch. Then the coefficients of the first fundamental form of z1 are given by (20.7) E1 = ρ21u , F1 = ρ1u ρ1v , G1 = 2 ρ1 1− G + ρ21v , ρ2 and the coefficients of the first fundamental form of z2 are given by 2 ρ2 (20.8) E + ρ22u , F2 = ρ2u ρ2v , G2 = ρ22v . E2 = 1 − ρ1 20.1. 2-DIMENSIONAL FOCAL SETS 643 Before computing the second fundamental forms of the focal patches, we note the following easy consequence of Corollary 19.10, page 602 (see Exercise 2). Lemma 20.6. Let x: U → R3 be a regular principal patch, and denote by ρ1 and ρ2 the principal radii of      (20.9)     curvature of x. Then Ev 1 1 ρ1v , = − ρ21 2E ρ1 ρ2 ρ2u Gu 1 1 . = − ρ22 2G ρ2 ρ1 Now we can compute the second fundamental forms of the focal patches. Lemma 20.7. Let x: U → R3 be a regular principal patch such that the two branches of the focal set of x are parametrized by (20.1). Then the coefficients of the second fundamental form of z1 are given by  √ E ρ1u   e1 = sign ρ1u 1 − ρ1  ,   ρ ρ1 2   (20.10) f1 = 0,     ρ1 Gu ρ    g1 = sign ρ1u 1 − 1 √ , 1− ρ2 ρ2 2 E and the coefficients of    e2      f2 (20.11)        g2 the second fundamental form of z2 are given by ρ2 Ev ρ2 √ , 1− = sign ρ2v 1 − ρ1 ρ1 2 G = 0, √ ρ2 G ρ2v . = sign ρ2v 1 − ρ1 ρ2 Furthermore, g1 and e2 are given by the alternate formulas  ρ1 G ρ1 ρ2u   √ , g = − sign ρ1u 1 −   1 ρ2 ρ22 E (20.12)  ρ2 E ρ2 ρ1v   √ .  e2 = − sign ρ2v 1 − ρ1 ρ21 G Proof. From (20.2) we compute the second derivatives of z1 : (20.13) z1uu = ρ1uu U + ρ1u Uu = ρ1uu U − ρ1u xu , ρ1 (20.14) z1uv = ρ1uv U + ρ1u Uv = ρ1uv U − ρ1u xv . ρ2 644 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Then (20.13) and (20.2) imply that ρ1 xu ρ1u √ xu · − sign ρ1u 1 − e1 = z1uu · U1 = ρ1uu U − ρ1 ρ2 E √ E ρ1 = sign ρ1u 1 − ρ1u . ρ2 ρ1 Similarly, (20.14) implies that ρ1 xu ρ1u √ xv · − sign ρ1u 1 − = 0, f1 = z1uv · U1 = ρ1uv U − ρ1 ρ2 E because F = 0. To compute g1 , we use (20.2) and (20.14), together with the fact that xu · xv = xu · U = 0; thus (20.15) g1 = z1vv · U1 ρ1 ρ1 xvv + ρ1vv U + ρ1v Uv xv + 1 − = − ρ2 v ρ2 ρ1 xu √ · − sign ρ1u 1 − ρ2 E ρ1 ρ1 xu = − sign ρ1u 1 − 1− xvv · √ . ρ2 ρ2 E Since xvv · xu = −xv · xuv = −Gu /2, equation (20.15) can be written as ρ1 ρ1 Gu √ . g1 = sign ρ1u 1 − 1− (20.16) ρ2 ρ2 2 E Similar calculations yield (20.11). Finally, (20.12) is a consequence of (20.10), (20.11) and (20.9) (see Exercise 2 for details). Finally, we find concise formulas for the Gaussian curvatures of the focal patches. Corollary 20.8. Let x: U → R3 be a regular principal patch such that the two branches of the focal set of x are parametrized by (20.1). Then the Gaussian curvatures of z1 and z2 are given by  ρ2u  , K = −   1 (ρ1 − ρ2 )2 ρ1u (20.17)  ρ1v   K2 = − . (ρ1 − ρ2 )2 ρ2v Proof. The first equation in (20.17) is an immediate consequence of formulas (20.7), (20.10), (20.12), and (13.20) on page 400. The second equation is found similarly. See Exercise 2 for details. 20.1. 2-DIMENSIONAL FOCAL SETS 645 We conclude this section with an easy example, used in Notebook 20 to illustrate the theory. We merely state the results and plot the focal surfaces. Focal Sets of a Hyperbolic Paraboloid A hyperbolic paraboloid is parametrized by x(u, v) = (u, v, u v) (see page 296). Its Gaussian and mean curvatures are given by K= and H= Thus eg − f2 1 =− EG − F2 (1 + u2 + v 2 )2 e G − 2f F + g E uv . =− 2 2 2(E G − F ) (1 + u + v 2 )3/2 H2 − K = so that (1 + u2 )(1 + v 2 ) , (1 + u2 + v 2 )3 p  √ (1 + u2 )(1 + v 2 ) −u v +  2  ,   k1 = H + H − K = (1 + u2 + v 2 )3/2 p  √  −u v − (1 + u2 )(1 + v 2 )   k2 = H − H 2 − K = . (1 + u2 + v 2 )3/2 Figure 20.2: Hyperbolic paraboloid and its focal set 646 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Proposition 20.9. The focal patches z1 (u, v), z2 (u, v) of the hyperbolic paraboloid x are given by p − uv 2 ∓ (u2 + 1)(v 2 + 1)v + u, p p −vu2 ∓ (u2 + 1)(v 2 + 1)u + v, 2uv ± (u2 + 1)(v 2 + 1) , with the respective choice of signs. Their traces are illustrated in Figure 20.2. The principal curvatures of x satisfy |k1 (u, v)|, |k2 (u, v)| > 1 . (1 + u2 + v 2 )3/2 20.2 Canal Surfaces An envelope of a 1-parameter family of surfaces is constructed in a similar way that we constructed the envelope of a 1-parameter family of lines on page 175. The family is described by a differentiable function F (x, y, z, t) = 0, where t is a parameter. When t can be eliminated from the equations F (x, y, z, t) = 0 and ∂F (x, y, z, t) = 0, ∂t we get the envelope, as a surface described implicitly. To give a concrete example, consider two first-order polynomials F1 (x, y, z) = ax + by + cz + d, and define (20.18) F2 (x, y, z) = a′ x + b′ y + c′ z + d′ , F (x, y, z, t) = t2 F2 (x, y, z) + tF1 (x, y, z) + f, where f is another constant. Then (20.18) is a 1-parameter family of planes, whose envelope is obtained by solving the system   t2 F2 + tF1 + f = 0,  2tF2 + F1 = 0, and consists of the parabolic cylinder F12 = 4f F2 , assuming that f is nonzero and (a, b, c), (a′ , b′ , c′ ) are not proportional. For an extensive discussion of envelopes, see [BrGi, chapter 5] and [Bolt]. Deﬁnition 20.10. The envelope of a 1-parameter family t 7→ S 2 (t) of spheres in R3 is called a canal surface. The curve formed by the centers of the spheres is called the center curve of the canal surface. The radius of the canal surface is the function r such that r(t) is the radius of the sphere S 2 (t). 20.2. CANAL SURFACES 647 It is easy to find one principal curvature and its corresponding principal curve for a canal surface. Lemma 20.11. Let t 7→ S 2 (t) be the 1-parameter family of spheres that defines a canal surface M. Then for each t the intersection S 2 (t) ∩ M is a circle and a principal curve on M. Proof. Since S 2 (t) and M are tangent along S 2 (t) ∩ M, the angle between their normals is zero. Furthermore, any curve on S 2 (t) is a principal curve, in particular, S 2 (t) ∩ M. Theorem 15.6 on page 466 then implies that S 2 (t) ∩ M is a principal curve on M. Now we can characterize canal surfaces. Theorem 20.12. Let M ⊂ R3 be a regular surface without umbilic points. The following are equivalent: (i) M is a canal surface; (ii) one of the systems of principal curves of M consists of circles; (iii) one of the components of focal(M) is a curve. Proof. That (i) implies (ii) is a consequence of Lemma 20.11. The normal lines passing through points on any principal curve that is a circle meet at a point. As the circular principal curves move on M, their centers generate a curve, which must be one of the components of focal(M). Thus (ii) implies (iii). Finally, if (iii) holds, we can use the component of focal(M) that is a curve to generate a canal surface using the distance between focal(M) and M as the radius of the canal surface. Evidently, this canal surface coincides with M. Hence (iii) implies (i). We can find a parametrization of a canal surface, provided we make mild assumptions about the center curve. Theorem 20.13. Suppose the center curve of a canal surface is a unit-speed curve γ : (a, b) → R3 with nonzero curvature. Then the associated canal surface can be parametrized by the mapping (20.19) y(t, θ) = γ(t) + r(t) − Tr′ (t) p + 1 − r′ (t)2 −N cos θ + B sin θ , where T, N, B denote the tangent, normal, binormal of γ. 648 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Proof. Ignoring (20.19) for the moment, let y denote a patch that parametrizes the envelope of spheres defining the canal surface. Since the curvature of γ is nonzero, the Frenet frame {T, N, B} is well defined, and we can write (20.20) y(t, θ) − γ(t) = a(t, θ)T(t) + b(t, θ)N(t) + c(t, θ)B(t), where a, b and c are differentiable on the interval on which γ is defined. We must have 2 (20.21) y(t, θ) − γ(t) = r(t)2 , expressing analytically the fact that y(t, θ) lies on a sphere S 2 (t) of radius r(t) centered at γ(t). Furthermore, y(t, θ) − γ(t) is a normal vector to the canal surface, so (20.22) y(t, θ) − γ(t) · yt = 0, (20.23) y(t, θ) − γ(t) · yθ = 0. Equations (20.22) and (20.23) assert that the vectors yt and yθ are tangent to the sphere S 2 (t). From (20.20) and (20.21) we get   a 2 + b 2 + c2 = r 2 , (20.24)  aa + b b + cc = r r′ . t t t On the other hand, when we differentiate (20.20) with respect to t and use the Frenet formulas (7.12) on page 197, we obtain (20.25) yt = (1 + at − b κ)T + (aκ − cτ + bt )N + (ct + b τ )B. Then (20.24), (20.25), (20.20) and (20.22) imply that (20.26) a + r r′ = 0, and from (20.24) and (20.26) we get b2 + c2 = r2 (1 − r′2 ). By Lemma 1.23, page 17, we are free to write p p b = −r 1 − r′2 cos θ and c = r 1 − r′2 sin θ, for a suitable parameter θ. Thus (20.20) becomes p p y(t, θ) − γ(t) = −r r′ T − r 1 − r′2 N cos θ + r 1 − r′2 B sin θ, (20.27) which is equivalent to (20.19). The notion of canal surface is a very general concept. In the next three subsections we consider three important subclasses of canal surfaces. 20.2. CANAL SURFACES 649 Tubes as Canal Surfaces It is easy to see that when the radius function r(t) is constant, the definition of canal surface reduces to the definition (7.36) of a tube given on page 209. In fact, we can characterize tubes among all canal surfaces. Theorem 20.14. Let M be a canal surface. Then the following conditions are equivalent: (i) M is a tube parametrized by (20.19); (ii) the radius of M is constant; (iii) the radius vector of each sphere in the family that defines the canal surface M meets the center curve orthogonally. Proof. When we compare (20.19) and (7.36), we see that (i) is equivalent to (ii). The radius vector of each sphere is the left-hand side of (20.27), from where it follows that (ii) is equivalent to (iii). Let us compute the first and second fundamental forms of a tube parametrized by (20.19) with r′ = 0, so that y(t, θ) = γ(t) + r(−N cos θ + B sin θ). We have yθ = r (N sin θ + B cos θ), and yt = T − r(−κ T + τ B) cos θ + r(−τ N) sin θ = (1 + r κ cos θ)T − r τ (N sin θ + B cos θ) = (1 + r κ cos θ)T − τ yθ , using the Frenet formulas (7.12). Hence  2 2 2   E = yt · yt = (1 + rκ cos θ) + r τ ,   F = yt · yθ = −r2 τ ,     G = y · y = r2 . θ θ Furthermore, yt × yθ = (1 + r κ cos θ)T × r(N sin θ + B cos θ) = r(1 + r κ cos θ)(−N cos θ + B sin θ), 650 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN so that kyt × yθ k2 = r2 (1 + r κ cos θ)2 . We can therefore define a unit normal vector to y by (20.28) U = −N cos θ + B sin θ. Next, we compute yθθ = r (N cos θ − B sin θ), yθt = r (−κ T sin θ − τ N cos θ + τ B sin θ), ytt = (rκ′ cos θ)T + (1 + rκ cos θ)(κN) − rτ ′ (N sin θ + B cos θ) −rτ sin θ(−κT + τ B) + cos θ(−τ N) = rκ′ cos θ − rκτ sin θ T + κ + r(κ2 + τ 2 ) cos θ − rτ ′ sin θ N − rτ ′ cos θ + rτ 2 sin θ B. Therefore, relative to the   e    f     g choice (20.28), = −κ cos θ − r κ2 cos2 θ − rτ 2 , = rτ, = −r. From equation (13.20) on page 400, we obtain r κ cos θ + r κ2 cos2 θ + r τ 2 − (r τ )2 κ cos θ K= = 2 2 r (1 + r κ cos θ) r(1 + r κ cos θ) and Hence e G − 2f F + g E 1 =− H= 2 2(E G − F 2 ) 1 1 H −K = rK − 4 r 2 so that 1 +Kr . r 2 ,  √ 1    k1 = H + H 2 − K = − , r √  −κ cos θ   k2 = H − H 2 − K = . 1 + r κ cos θ 20.2. CANAL SURFACES 651 Surfaces of Revolution as Canal Surfaces Almost any surface of revolution is a canal surface. The following theorem characterizes surfaces of revolution among all canal surfaces. Theorem 20.15. The center curve of a canal surface M is a straight line if and only if M is a surface of revolution for which no normal line to the surface is parallel to the axis of revolution. Furthermore, the parametrization (20.19) reduces to the standard parametrization (15.1) of a surface of revolution given on page 462. Proof. Let M be a surface of revolution and let p ∈ M. The normal line ℓp perpendicular to M at p lies in the plane passing through the meridian of M containing p; this plane contains the axis of revolution α of M. Hence either ℓp intersects α or is parallel to it. By hypothesis, the latter possibility is excluded. Let ℓp intersect α at q. The sphere S 2 (q) of radius r(q) = kq − pk and center q meets ℓp perpendicularly at p, and so it is tangent to M at p. Hence M is the canal surface with center curve α and variable radius q 7→ r(q). Conversely, suppose that M is a canal surface whose center curve is a straight line γ. Instead of a Frenet Frame field along γ, we choose a constant frame field {T, N, B}. (See Exercise 5 of Chapter 7.) We can proceed exactly as in the proof of Theorem 20.13; the only difference is that κ = τ = 0 for the center curve. In particular, (20.19) holds. To show that M is a surface of revolution, we use a Euclidean motion to move M so that the center curve is the z-axis and so that T = (0, 0, 1), N = (1, 0, 0), B = (0, 1, 0). We may also assume that γ(t) = (0, 0, t). Then (20.19) becomes p p − r(t) 1 − r′ (t)2 cos θ, r(t) 1 − r′ (t)2 sin θ, t − r(t)r′ (t) . This coincides with (15.1) when we take θ = −u, t = v, p ϕ(v) = −r(v) 1 − r′ (v)2 and ψ(v) = v − r(v)r′ (v). Hence M is a surface of revolution. There is a close connection between evolutes of plane curves and focal sets of surfaces of revolution: Theorem 20.16. One of the components of the focal set of a surface of revolution M generated by a plane curve C is the surface of revolution generated by the evolute of C . 652 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Proof. Let C be parametrized by a unit-speed curve α, and write α = (ϕ, ψ). The standard parametrization of M is x(u, v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) . For simplicity we assume that ϕ > 0. According to Corollary 15.14, page 470, the principal curvatures of M are given by km = (ϕ′′ ψ ′ − ϕ′ ψ ′′ ) and kp = −ψ ′ . ϕ Furthermore, by Lemma 15.10 on page 468, the unit normal is given by U = (ψ ′ cos u, ψ ′ sin u, −ϕ′ ). Hence the formula for the nondegenerate focal patch of M is z1 (u, v) = x(u, v) + ρ1 U(u, v) ψ ′ (v) cos u, ψ ′ (v) sin u, −ϕ′ (v) = ϕ(v) cos u, ϕ(v) sin u, ψ(v) + ϕ′′ (v)ψ ′ (v) − ϕ′ (v)ψ ′′ (v) = ϕ(v) + ψ ′ (v) cos u, ϕ′′ (v)ψ ′ (v) − ϕ′ (v)ψ ′′ (v) ϕ(v) + ψ ′ (v) sin u, ϕ′′ (v)ψ ′ (v) − ϕ′ (v)ψ ′′ (v) ! ϕ′ (v) ψ(v) − ′′ . ϕ (v)ψ ′ (v) − ϕ′ (v)ψ ′′ (v) This is the surface of revolution generated by the curve ψ ′ (t) ϕ′ (t) γ(t) = ϕ(t) + ′′ . , ψ(t) − ϕ (v)ψ ′ (t) − ϕ′ (t)ψ ′′ (t) ϕ′′ (t)ψ ′ (t) − ϕ′ (t)ψ ′′ (t) This curve is precisely the evolute of α, as we see by noting that J α′ (t) = − ψ ′ (t), ϕ(t) and kα′ (t)k = 1, and making use of formulas on pages 15 and 99. For example, we know from Section 4.2 that the evolute of a tractrix is a catenary curve. Hence the 2-dimensional component of the focal set of a pseudosphere is a catenoid. Figure 20.3 displays the pseudosphere together with this 2-dimensional component, corresponding to the plot of a tractrix and its evolute on page 103. 20.2. CANAL SURFACES 653 Figure 20.3: Pseudosphere and the catenoid component of its focal set Canal Surfaces with a Planar Center Curve The parametrization (20.19) may or may not be principal. We have already seen (Lemma 20.11) that any canal surface has a system of circles as principal curves. Let us prove this fact by direct calculation. Lemma 20.17. Let M be a canal surface parametrized by (20.19). For fixed t, the curve θ 7→ y(t, θ) is a principal curve with principal curvature −1/r(t). Proof. Define y by equation (20.19). We can take for the unit normal of the canal surface the vector field (20.29) Thus (20.30) U= 1 y−γ = (y − γ). ky − γk r y = γ + r U. When we differentiate (20.30) and use the fact that neither γ nor r depends on θ, we get (20.31) yθ = r Uθ = −r S(yθ ), where S is the shape operator of the canal surface. The lemma follows. 654 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Next, we characterize canal surfaces whose center curve is planar. Theorem 20.18. Let M be a canal surface parametrized by (20.19). The following conditions are equivalent. (i) For each θ, the curve t 7→ y(t, θ) is a principal curve. (ii) Either the function r is constant and the torsion of the center curve γ vanishes, or both the curvature and torsion of γ vanish. (iii) The center curve γ is a plane curve, and either the function r is constant or the curvature of γ vanishes. Proof. Assume that (i) holds. We first use (20.27) and (20.29) to write unit normal U to the canal surface as p (20.32) U = −r′ T + 1 − r′2 (−N cos θ + B sin θ), so that (20.33) Uθ = p 1 − r′2 N sin θ + B cos θ . Also, (20.32) and the Frenet formulas (7.12) imply that p ′ Ut = −r′′ T − r′ (κN) + 1 − r′2 −N cos θ + B sin θ √ + 1 − r′2 −(−κT + τ B) cos θ + (−τ N) sin θ p = − r′′ + κ cos θ 1 − r′2 T − r′ κ N √ √ ′ + 1 − r′2 −N cos θ + B sin θ − τ 1 − r′2 N sin θ + B cos θ . Combined with (20.33), we get (20.34) Uθ · Ut = −r′ κ sin θ p 1 − r′2 − τ (1 − r′2 ). On the other hand, when we differentiate (20.30) with respect to t, we find that yt = T + r′ U + r Ut . Then (20.30), (20.31), (20.33), (20.34) imply that yt · yθ = T + r′ U + r Ut · (r Uθ ) = r2 Ut · Uθ (20.35) √ = −r2 r′ κ sin θ 1 − r′2 + τ (1 − r′2 ) . If yt is principal, then it must be perpendicular to the principal vector yθ . If this happens, both terms on the right-hand side of (20.35) vanish, since the first term depends on θ and the second does not. Thus τ = 0 and either r′ = 0 or κ = 0. Thus (i) implies (ii). Conversely, if (ii) holds, then (20.35) implies that yt is perpendicular to yθ , and so itself principal. Finally, (ii) is clearly equivalent to (iii). 20.3. CYCLIDES OF DUPIN VIA FOCAL SETS 655 Figure 20.4: The canal surface generated by a sphere of radius 1.8 + 0.48 sin(2t) moving on a circle of radius 5 Figure 20.4 shows the sort of surface that is formed as the envelope of spheres with varying radius. In the next section, we shall study surfaces of this type, but ones which are in a precise sense closer to a torus of revolution, which itself corresponds to constant radius. 20.3 Cyclides of Dupin via Focal Sets In the early 1800s, a class of surfaces all of whose lines of curvature are circles or lines was studied by Charles Dupin. Not only does this class include cones, cylinders and tori, but also distinct new surfaces. Dupin called such surfaces ‘cyclides’ in his book [Dupin2], and the name seems to have been coined by him. Since there is now a more general notion of cyclide, those studied by Dupin are referred to as cyclides of Dupin. Three methods are available for their study: (i) Dupin’s original method consisted of describing a cyclide as an envelope of spheres tangent to three given spheres. (ii) Liouville [Liou] showed that any cyclide of Dupin is the image under an inversion (see page 662) of a torus, circular cylinder or circular cone. (iii) Maxwell1 [Max] gave a direct construction of a cyclide of Dupin as the envelope of spheres whose centers move along ellipses, hyperbolas or parabolas, generalizing the torus of revolution. 1 James Clerk Maxwell (1831–1879). A leading scientist of the 19th century, celebrated for his Treatise on Electricity and Magnetism. The paper [Max] is informative and elegant. Before giving parametrizations of cyclides, Maxwell characterized them by a geometrical argument using almost no algebra. The paper also includes images viewable by a stereoscope together with Maxwell’s recommendations for constructing the stereoscope. 656 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN In this section we follow the approach of Maxwell, as modified by Darboux. Our treatment is based on [Darb4, pages 405–406], though cyclides are also discussed in [CeRy, page 150–166], [Bomp], [Sny] and [Cool1]. The theory of inversion is developed in Sections 20.4 and 20.5. Firstly, we characterize focal sets consisting of two curves. Theorem 20.19. Let M be a surface for which the focal set focal(M) consists of two curves. Then each curve is a conic section (an ellipse, hyperbola, parabola or straight line), and the planes of each component are perpendicular to one another. Proof. We denote the two curves by focal(M)1 and focal(M)2 , and start by considering a point p ∈ focal(M)1 . The set of normal lines to M that pass through p forms a cone, which we call cone(p, M), with p as its vertex. By definition of focal set, each line in this cone also passes through a point of focal(M)2 ; furthermore, for each point q ∈ focal(M)2 , the line containing p and q is in cone(p, M). It follows that the whole curve focal(M)2 must lie on cone(p, M). This construction creates a cone for any p ∈ focal(M)1 . In particular, it works for a point that is nearest to focal(M)2 . For such a point, the corresponding cone is part of a plane. It follows that focal(M)2 is a plane curve. Clearly, the same argument shows that focal(M)1 is also a plane curve lying on a cone. Hence both focal(M)1 and focal(M)2 are conic sections. Indeed, cone(p, M) must be a circular cone for p ∈ focal(M)1 , since it contains a principal curve of M, which is a circle. We are now in a position to define the cyclides of Dupin and give explicit parametrizations of them. Elliptic-Hyperbolic Cyclides Let ellxy be the ellipse in the xy-plane given by y2 x2 + 2 = 1, 2 a a − c2 and let hypxz be the hyperbola in the xz-plane given by z2 x2 − 2 = 1. 2 c a − c2 We assume that a > c > 0. Notice that one focus of the ellipse is (c, 0, 0), which lies on the hyperbola, whereas one focus of the hyperbola is (a, 0, 0), which lies on the ellipse. 20.3. CYCLIDES OF DUPIN VIA FOCAL SETS 657 Figure 20.5: Ellipse and hyperbola in orthogonal planes We first derive a simple formula for the distance between a point on the ellipse and a point on the hyperbola. Lemma 20.20. The distance between a point (x1 , y1 , 0) on ellxy and a point (x2 , 0, z2 ) on the hypxz is given by the formula (20.36) distance (x1 , y1 , 0), (x2 , 0, z2 ) = Proof. We compute a c x1 − x2 . a c = (x1 − x2 )2 + y12 + z22 x22 x21 2 2 2 = (x1 − x2 ) + (a − c ) 1 − 2 + 2 − 1 a c distance2 (x1 , y1 , 0), (x2 , 0, z2 ) = x21 − 2x1 x2 + x22 − x21 + as required. c c2 2 a 2 2 a 2 2 x + x − x = x − x2 , 1 1 2 2 a2 c2 a c Next, we parametrize ellxy and hypxz by  √  α(u) = a cos u, a2 − c2 sin u, 0 , (20.37)  β(v) = c sec v, 0, √a2 − c2 tan v , and we restrict to the branch of hypxz for which −π/2 < v < π/2, so as to state 658 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Lemma 20.21. Let sph1 (u) be a sphere with radius ρ1 (u) centered at α(u) and let sph2 (v) be a sphere with radius ρ2 (v) centered at β(v). Suppose that sph1 (u) and sph2 (v) are tangent to each other. Then there exists k, which is constant with respect to u and v, such that (20.38) ρ1 (u) = −c cos u + k and ρ2 (v) = a sec v − k. Proof. By hypothesis, the two spheres have a unique point in common, say (x3 , y3 , z3 ). Then   distance (x1 , y1 , 0), (x3 , y3 , z3 ) = ρ1 (u), (20.39)  distance (x , 0, z ), (x , y , z ) = ρ (v). 2 2 3 3 3 2 Since the spheres are tangent at (x3 , y3 , z3 ), ρ1 (u) + ρ2 (v) = distance (x1 , y1 , 0), (x2 , 0, z2 ) , and Lemma 20.20 implies that distance equals a c x1 − x2 = |c cos u − a sec v| = a sec v − c cos u. a c Thus, (20.40) ρ1 (u) + c cos u = −ρ2 (v) + a sec v. Since the left-hand side of (20.40) depends only on u, while the right-hand side of (20.40) depends only on v, we let k be the common value, and obtain (20.38). Let us now prove the converse of Lemma 20.21. Lemma 20.22. Let sph1 (u) be a sphere with radius ρ1 (u) centered at α(u), and let sph2 (v) be a sphere with radius ρ2 (v) centered at β(v). Suppose that ρ1 (u) and ρ2 (v) are given by (20.38) for some k. Let (x3 , y3 , z3 ) be the point on the line connecting the points α(u) and β(v) at a distance ρ1 (u) from α(u). Then the distance between the points (x3 , y3 , z3 ) and β(v) is ρ2 (v). Furthermore, sph1 (u) and sph2 (v) are mutually tangent at (x3 , y3 , z3 ). Proof. By Lemma 20.20, the distance from α(u) and β(v) is distance (x1 , y1 , 0), (x2 , 0, z2 ) = c a x1 − x2 = |c cos u − a sec v| a c = a sec v − c cos u, which is precisely ρ1 (u) + ρ2 (v). Therefore, (20.39) holds. This can only happen if sph1 (u) and sph2 (v) are tangent at (x3 , y3 , z3 ). 20.3. CYCLIDES OF DUPIN VIA FOCAL SETS 659 As sph1 (u) moves along the ellipse α(u) it traces out a surface, the envelope of the spheres. This envelope must coincide with the envelope formed by sph2 (v) as it moves along a branch of the hyperbola β(v). The resulting surface is called a cyclide of Dupin of elliptic-hyperbolic type. It depends on three parameters a, c and k. In order to parametrize the cyclide, we define x(u, v) = α(u) + ρ1 (u)V (u, v) = β(v) − ρ2 (v)V (u, v), where V (u, v) is a unit vector. We have whence 0 = α(u) − β(v) + ρ1 (u) + ρ2 (v) V (u, v), x(u, v) = α(u) − ρ1 (u) α(u) − β(v) ρ1 (u) + ρ2 (v) = ρ2 (v)α(u) + ρ1 (u)β(v) . ρ1 (u) + ρ2 (v) Substituting in (20.37) and (20.38), and multiplying top and bottom by cos v gives c(k − c cos u) + a cos u(a − k cos v) x(u, v) = , a − c cos u cos v √ √ a2 − c2 (a − k cos v) sin u a2 − c2 (k − c cos u) sin v . , a − c cos u cos v a − c cos u cos v Figures 20.6–20.8 display one of each of the three classes designated by Deﬁnition 20.23. An elliptic-hyperbolic cyclide M is called a ring cyclide if it has spindle cyclide if it has horn cyclide if it has no self-intersections, 1 self-intersection, 2 self-intersections. Notice also that generically an elliptic-hyperbolic cyclide has two planes of symmetry. If there are additional planes of symmetry, an elliptic-hyperbolic cyclide becomes a torus. Parabolic Cyclides There is another type of cyclide whose focal set consist of two parabolas. Let parxz be the parabola in the xz-plane given by x2 + a, 8a be the parabola in the yz-plane given by z=− and let paryz z= y2 − a. 8a 660 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Figure 20.6: Ring cyclide and its focal set Figure 20.7: Horn cyclide and its focal set Figure 20.8: Spindle cyclide and its focal set 20.4. DEFINITION OF INVERSION 661 We parametrize parxz and paryz by  u2    α(u) = u, 0, − + a , 8a 2  v   β(v) = 0, −v, −a . 8a (20.41) Note that α passes through the focus of β (namely (0, 0, a)) and β passes through the focus of α (namely (0, 0, −a)). The analog of Lemma 20.21 is Lemma 20.24. Let sph1 (u) be a sphere with radius ρ1 (u) centered at α(u), and let sph2 (v) be a sphere with radius ρ2 (v) centered at β(v). Suppose that sph1 (u) and sph2 (v) are tangent to each other. Then there exists k, which is constant with respect to u and v, such that ρ1 (u) = u2 +a+k 8a and ρ2 (v) = v2 + a − k. 8a Proof. This is very similar that of Lemma 20.21. Let (x3 , y3 , z3 ) be the point of intersection of the two spheres. Then α(u) − β(v) 2 = v2 u2 − u, v, 2a − 8a 8a 2 = 1 1 2 (u + v 2 ) + (u2 + v 2 )2 + 4a2 2 64a2 = u2 + v 2 + 16a2 8a 2 . Furthermore, so that ρ1 (u) + ρ2 (v) = distance (x1 , 0, z1 ), (0, y2 , z2 ) = α(u) − β(v) , ρ1 (u) − u2 v2 − a = −ρ2 (v) + + a. 8a 8a Just as with (20.40), both sides must equal a constant k. The equation of a parabolic cyclide can be found in Notebook 20. 20.4 The Deﬁnition of Inversion We want to invert Rn with respect to a point q ∈ Rn ; this means that points near to q are mapped far away in a reasonable way, and vice versa. 662 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Deﬁnition 20.25. The inversion of Rn with respect to a point q ∈ Rn with inversion radius ρ is the map Ψ : Rn \ {q} → Rn \ {q} given by (20.42) Ψ(p) = q + ρ2 We call q the inversion center or pole. p−q . kp − qk2 In the special case q = 0, we see that Ψ(p) = ρ2 p/kpk2. It is easy to check that two applications of an inversion yield the identity map; that is, Ψ ◦ Ψ is the identity map of Rn \ q. This has the important consequence that (20.42) is a bijective mapping. Here is another important property: Lemma 20.26. Let Ψ be an inversion with pole q ∈ Rn . Then Ψ maps the sphere S n−1 (p, a) = {v ∈ Rn | kv − pk = a2 } of dimension n − 1, radius a and center p to another sphere, or a plane. Proof. Without loss of generality, we may suppose that q = 0 is the origin. In an attempt to show that Ψ maps the sphere onto a sphere whose center lies on the line straight line joining 0 to p, let v ∈ S n−1 (p, a) and consider (20.43) kΨ(v) − cpk2 ρ2 v − cp kvk2 = ρ4 = where c is to be determined. Since (20.44) 2 1 2ρ2 c − v · p + c2 kpk2 , 2 kvk kvk2 a2 = kv − pk2 = kvk2 − 2v · p + kpk2 , we may eliminate v · p to find that (20.43) equals ρ2 ρ2 − ckpk2 + ca2 − cρ2 + c2 kpk2 . kvk2 To make the coefficient of 1/kvk2 vanish, we set c= ρ2 , kpk2 − a2 assuming for the moment that kpk 6= a. In this case, Ψ maps the sphere onto a sphere with center cp. If kpk = a then the sphere passes through the inversion center 0. In this case, (20.44) implies that kvk2 = 2p · v, whence Ψ(v) · p = which is the equation of a plane. ρ2 , 2 20.4. DEFINITION OF INVERSION 663 An inversion is characterized by two properties. Lemma 20.27. Let q ∈ Rn . A map Φ: Rn \ {q} → Rn \ {q} is an inversion with inversion radius ρ if and only if there is a number λ > 0 such that kΦ(p) − qk kp − qk = ρ2 (20.45) and Φ(p) − q = λ(p − q) for all p. Proof. It is easy to prove that an inversion satisfies (20.45). Conversely, suppose that Ψ satisfies (20.45). Then λkp − qk2 = ρ2 , and so Ψ(p) − q = ρ2 p−q . kp − qk2 Because of Lemma 20.27, an inversion of Rn is sometimes called a transformation by reciprocal radii in the classical literature. Next, we show that an inversion has the important property that its tangent map preserves angles between tangent vectors. For this, we make the following definition, which corresponds to the notion of conformality for surface mappings given on page 370. Deﬁnition 20.28. Let Φ: U → Φ(U) ⊂ Rn be a map, where U is an open subset of Rn . We say that Φ is a conformal map of U onto Φ(U) provided there is a positive differentiable function λ: U → R such that (20.46) kΦ∗ (vp )k = λ(p)kvp k for all p ∈ U. Proposition 20.29. An inversion is a conformal map. Proof. Let Ψ be the inversion defined by (20.42), and let α: (a, b) → Rn be a curve passing through q. Then (Ψ ◦ α)(t) = q + ρ2 α(t) − q α(t) − q 2 for a < t < b. Differentiating the last equation gives ′ (Ψ ◦ α) (t) = ρ2 α′ (t) α(t) − q − 2 2ρ2 (α(t) − q) · α′ (t) α(t) − q α(t) − q 4 . It follows that ′ (Ψ ◦ α) (t) (20.47) 2 = = ρ2 α′ (t) 2 α(t) − q ρ4 α′ (t) α(t) − q 2 4. − 2ρ2 (α(t) − q) · α′ (t) α(t) − q α(t) − q 4 2 664 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN (A geometrical explanation of why the norm squared simplifies to (20.47) can be found on page 666.) Since any tangent vector can be realized as the velocity vector of a curve, we can replace α(t) by p and α′ (t) by vp , so that kΨ∗ (vp )k = ρ2 kvp k kp − qk2 for all tangent vectors vp . Hence Ψ distorts lengths of all tangent vectors at p by the same factor ρ2 λ(p) = , kp − qk2 in accordance with (20.46). We shall be mainly interesting in finding inversions of surfaces in R3 , but we conclude this section by inverting some plane curves. Inversion provides a useful way to visualize what happens to a curve at infinity. For example, Figure 20.9 compares inverse curves of the catenary and the hyperbola. Although the shape of a hyperbola is not so different from that of a catenary when examined near the origin, the inverse curves are themselves quite different near the origin. Figure 20.9: Inverse curve of a catenary (left) and a hyperbola (right) Similar principles apply to space curves. Figure 20.10 shows an inversion of the twisted cubic on page 202, with pole indicated by the black bead. The fact that the inverted curve rapidly ‘closes up’ indicates that the twisted cubic can be extended to a closed curve in projective space (see forward to page 793). Similar remarks apply to many other space curves which we have studied, including the helix whose inversion is shown in Figure 20.15 on page 671. That figure also serves to outline the result of inverting a cylinder. 20.5. INVERSION OF SURFACES 665 Figure 20.10: Inversion of the twisted cubic with pole (0, 1, 1) 20.5 Inversion of Surfaces The relevance of the present topic to our recent study of surfaces is perhaps best illustrated by the next result. Proposition 20.30. Inversion maps principal curves onto principal curves. Proof. The mapping Ψ in (20.42) is conformal by Proposition 20.29. It therefore preserves orthogonality, and maps a triply orthogonal system of surfaces onto another triply orthogonal system of surfaces. The proposition is now a consequence of Dupin’s Theorem on page 609 and the general construction of Theorem 19.30. This fact can be used to construct many interesting parametrizations of surfaces. For example, Figure 20.11 shows the inverse of an ellipsoid with its lines of curvature. Deﬁnition 20.31. Let x: U → R3 , and let Ψ be the mapping (20.42). The inverse patch of x with respect to q ∈ Rn with radius of inversion ρ is simply the patch Ψ ◦ x. In other words, the inverse patch is the image of the original one by inversion. 666 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Figure 20.11: Inversion of an ellipsoid We proceed to obtain formulas for the first and second fundamental forms of an inverse patch. The notion of inversion is such a natural one that these formulas can be obtained without excessive computation, in a similar spirit to that of Section 19.9. We first observe that the tangent vectors of an inverse patch are obtained by reflection in a mirror perpendicular to x − q. Let y = Ψ ◦ x be the inverse patch. Differentiating (20.42), we get 2ρ2 xu · (x − q) (x − q) ρ2 xu yu = (20.48) − , kx − qk2 kx − qk4 with a similar formula for xv . Using the notation of (5.1) on page 128, yu = reﬂx−q ρ2 xu , kx − qk2 yv = reﬂx−q ρ2 xv . kx − qk2 We give two immediate corollaries of these formulas. Corollary 20.32. Let x be a patch in Rn , and let y be the inverse patch of x with respect to an inversion with center q ∈ Rn and radius ρ. Then the coefficients of the first fundamental form of x are related to those of y by E(y) = ρ4 E(x), kx − qk4 F (y) = ρ4 F (x), kx − qk4 G(y) = ρ4 G(x). kx − qk4 Proof. These equations are consequences of the fact that a reflection preserves the norm of a vector. 20.5. INVERSION OF SURFACES 667 The second is a formula for the unit normal to an inverse patch. Corollary 20.33. Let x be a patch in R3 , and let y be the inverse patch of x with respect to q ∈ R3 with radius of inversion ρ. Denote by Ux and Uy the respective normals. Then 2 Ux · (x − q) (x − q) (20.49) . Uy = −reﬂx−q Ux = −Ux + kx − qk2 Proof. Let reﬂ = reﬂx−q denote the reflection that figured in Corollary 20.32. Since reflections in R3 are orientation-reversing, reﬂ(xu × xv ) = −reﬂ(xu ) × reﬂ(xv ), and more to the point, yu × yv = −reﬂ ρ2 xu ρ2 xv . × kxu − qk2 kxv − qk2 The result follows from the fact that Ux and Uy are proportional to xu × xv and yu × yv respectively, and that a reflection preserves norm. We are now in a position to find the coefficients of the second fundamental form of an inverse patch. Lemma 20.34. Let x be a patch in R3 , and let y be the inverse patch of x with respect to q ∈ R3 with radius of inversion ρ. Then the coefficients of the second fundamental form of y are related to those of x by  2ρ2 E(x)Ux · (x − q) −ρ2 e(x)   − , e(y) =  2  kx − qk kx − qk4      2ρ2 F (x)Ux · (x − q) −ρ2 f (x) (20.50) − , f (y) = 2  kx − qk kx − qk4       −ρ2 g(x) 2ρ2 G(x)Ux · (x − q)   g(y) = − . kx − qk2 kx − qk4 Proof. We first use (20.48) to compute yuu 4ρ2 xu · (x − q xu 2ρ2 xuu · (x − q) (x − q) ρ2 xuu = − − kx − qk4 kx − qk2 kx − qk4 2 2ρ2 kxu k2 (x − q) 8ρ2 xu · (x − q) (x − q) + . − kx − qk4 kx − qk6 Taking the dot product with (20.49) and cancelling the various terms gives the first equation of (20.50); proofs of the other equations are similar. 668 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Next, we determine the relation between the principal curvatures of a surface M and the image of M under an inversion Ψ. Let x be a principal patch on M; then the principal curvatures on the trace of x are given by kp (x) = e(x) E(x) and km (x) = g(x) . G(x) Proposition 20.30 implies that Ψ ◦ y is a principal patch, and so the same equations hold with y in place of x. Lemma 20.35. Let x be a principal patch in R3 , and let y be the inverse patch of x with respect to q ∈ R3 with radius of inversion ρ. Then (20.51)  −kx − qk2 2Ux · (x − q)   kp (x) − ,  kp (y) = ρ2 ρ2 2    km (y) = −kx − qk km (x) − 2Ux · (x − q) . ρ2 ρ2 Proof. (20.51) is a consequence of Corollary (20.32) and (20.50). As a corollary, an inversion of R3 maps umbilic points into umbilic points, a statement already implicit in Proposition 20.30. Combining (20.1), (20.49) and (20.51), we deduce that the inverse patch y itself has focal patches (20.52) zi = y − ρ2 −kx̃k2 Ux + 2(Ux · x̃)x̃ , kx̃k2 kx̃k2 ki + 2Ux · x̃ where x̃ is shorthand for x − q. This formula can be used to prove Proposition 20.36. The inversion of a torus of revolution is a cyclide of Dupin. Proof. Let x = torus[a, b, b] be a torus of revolution, as defined on page 210. Let y = Ψ ◦ x be an inverse patch, depending on the center q and radius ρ of inversion. It suffices to show that both the associated patches (20.52) are curves, a fact that is readily verified by computer. To cite a simple but typical example, let q = (m, 0, 0) and ρ = 1. The following formulas for z1 and z2 were copied from output from Notebook 20: ! m a2 − b2 + m2 − 1 + a − 2am2 cos u a sin u , 2 , 0 , a2 − 2am cos u − b2 + m2 a − 2am cos u − b2 + m2 ! m −2ab − a2 + b2 − m2 + 1 cos v a sin v − . , 0, 2ab + (a2 + b2 − m2 ) cos v 2ab + (a2 + b2 − m2 ) cos v Each one is certainly a curve, indeed a plane conic. 20.6. EXERCISES 669 The last result enables us to reconcile the types of cyclides in Definition 20.23 with those of tori on page 306. For example, Figure 20.12 shows the inversion of the spindle torus torus[4, 4, 4] of Figure 10.11 (right) on page 306, with center q = (0, 5, 0). It evidently corresponds to Figure 20.8. Figure 20.12: Inversion of the inside-out torus 20.6 Exercises 1. Complete the proofs of Theorem 20.4, Lemma 20.5 and Lemma 20.7. 2. Prove Lemma 20.6 and give the details of equations (20.9) and (20.17). 3. Let M be a surface of revolution generated by a plane curve C , not a part of a circle. Show that the surface of revolution generated by the evolute of C is one component of the focal set of M. 4. Describe the envelope of the 1-parameter of spheres (x − 2 cos t)2 + (y − 2 sin t)2 + z 2 = 1, for 0 6 t < 2π. 5. Find the equations of the two focal hypersurfaces of an elliptic paraboloid, all shown together in Figure 20.13. 670 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN Figure 20.13: Elliptic paraboloid and its focal set M 6. Find the two components of the focal set of a circular helicoid defined on page 376, referring also to Exercise 3 of Chapter 12. Figure 20.14: The two components of the focal set of a helicoid 7. Let M be a surface with the property that ρ1 − ρ2 = a, where a is a constant. Assume that each component of the focal set of M is a regular surface. Show that each component of the focal set of M has constant negative curvature −a−2 . 20.6. EXERCISES 671 M 8. Use the technique that led to Definition 20.23 to find the equation of a parabolic cyclide. Plot several examples. M 9. Compute the metric and second fundamental form of a general elliptichyperbolic cyclide. M 10. Determine analytically the inversion of a helix. The one in Figure 20.15 also allows one to imagine the inversion of a cylinder. Figure 20.15: Inverse curve of a helix 11. Show that the inverse curve with respect to the origin of an equilateral hyperbola is a lemniscate. The parametrization of an equilateral hyperbola that corresponds to the lemniscate is t 7→ a sec t, a tan t . M 12. Plot the inverse curves of an ellipse with respect to several points. 13. Use Lemma 20.35 to obtain the following formula for the mean curvature of an inverse patch: H(y) = − kx − qk2 H(x) 2Ux · (x − q) − . ρ2 ρ2 Find an analogous formula for the Gaussian curvature. 672 CHAPTER 20. CANAL SURFACES AND CYCLIDES OF DUPIN M 14. Find the image of the circular cone x(u, v) = v cos u, v sin u, v under an inversion about an arbitrary point (p1 , p2 , p3 ) and radius ρ = 1. Plot several inverted circular cones, like the one in Figure 20.16. Figure 20.16: Inverted circular cone 15. Show that the focal set of each surface parallel to a surface M coincides with the focal set of M. Chapter 21 The Theory of Surfaces of Constant Negative Curvature In 1879, Luigi Bianchi gave for the first time in his thesis a method for deriving from one surface of constant negative Gaussian curvature an infinite number of other surfaces with the same curvature. His work created great interest, and was followed by papers by Darboux, Lie and Bäcklund. In [Bck], Bäcklund gave a more general construction which led to the Bäcklund transform. In spite of the fact that the paper [Bck] dealt exclusively with surfaces of constant negative curvature, the notion of Bäcklund transform has come to have enormous importance in soliton theory (see for example [Lamb2, Remo]). Our treatment is based on the concept of a Tchebyshef1 net. Patches of two interrelated types are introduced in Section 21.1, and are characterized by properties of the first fundamental form and its coefficients E, F, G. We firstly define a Tchebyshef patch for which E and G are equal and constant, and secondly one for which E + G is constant and F = 0. At this stage, the theory proceeds without reference to the second fundamental form or knowledge of the way that M is contained in R3 , so the principal curvatures k1 , k2 are not yet defined. The results can therefore be applied to the more abstract types of surfaces to be studied in Chapter 26. In Section 21.2, we turn attention to surfaces of constant negative curvature 1 Pafnuty Lvovich Tchebyshef (1821–1894). A leading Russian mathematician of his time. He worked in probability theory, number theory and differential geometry. 683 684 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE in R3 . We show that principal patches on such surfaces have special properties, and the two types of Tchebyshef patches can be realized as asymptotic and principal patches respectively in which k1 , k2 are determined by the metric. Section 21.3 introduces the sine–Gordon equation, solutions of which determine angle functions of Tchebyshef patches with constant curvature. Special solutions are used to treat surfaces of revolution in Section 21.4, culminating in explicit parametrizations for the pseudosphere. Section 21.5 is devoted to the Bianchi transform of surfaces of constant negative curvature. Moving frames on surfaces in R3 are discussed in Section 21.6, as well as an analog of the Frenet formulas of Chapter 7. In Section 21.7 we describe Kuen’s surface as the Bianchi transform of the pseudosphere, itself the transform of a degenerate patch. The Bäcklund transform generalizes that of Bianchi; it is defined and applied to the construction of Dini’s surface in Section 21.8. Throughout this chapter, we shall be considering surfaces of constant negative Gaussian curvature −a−2 . We shall express this with the statement ‘K ≡ −a−2 < 0’, in which it is implicitly understood that a is constant. 21.1 Intrinsic Tchebyshef Patches We begin by defining a special kind of patch that will give rise (in the next section) to an asymptotic patch on a surface with constant negative Gaussian curvature. Definition 21.1. Let a be a positive constant. A Tchebyshef patch or net of radius a is a patch y : U → Rn whose metric ds2 = E dp2 + 2F dpdq + Gdq 2 has the property that E = G = a2 . Like that of isothermal patches on page 518, this definition is intrinsic, in that it imposes conditions merely on the coefficients of the first fundamental form. Sometimes it is required in the definition that a = 1, so that the coordinate curves are parametrized by arc length, but we prefer to allow an arbitrary positive constant. The resulting functions p, q on the surface y(U) are called Tchebyshef coordinates. The Cauchy–Schwarz inequality implies that F 2 = hyp , yq i2 6 kyp k2 kyq k2 = E G = a4 , so we can write F = a2 cos ω for some ω. Thus the metric of a Tchebyshef patch of radius a can be written as (21.1) ds2 = a2 dp2 + 2 cos ω dpdq + dq 2 . 21.1. INTRINSIC TCHEBYSHEF PATCHES 685 We say that the metric given by (21.1) has angle function ω. Note that cos ω = hyp , yq i , kyp k kyq k so that geometrically, ω represents an unoriented angle between the coordinate curves p 7→ y(p, q) and q 7→ y(p, q). Before giving examples of such patches, we compute their Gaussian curvature and Christoffel symbols. Lemma 21.2. The Gaussian curvature of a Tchebyshef patch of radius a > 0 is given by ωpq , a2 sin ω where the subscripts denote partial derivatives. K =− Proof. Apply Brioschi’s formula (17.3), page 533, to the metric (21.1). Since E and G are constant,   Fpq 0 Fp 1   det Fq a2 F  K = (E G − F 2 )2 0 F a2 = = = Fpq (a4 − F 2 ) + Fp Fq F 2 a4 − a4 cos2 ω a6 (−ωp ωq cos ω − ωpq sin ω) sin2 ω + ωp ωq sin2 ω cos ω (a2 sin ω)4 −ωpq . a2 sin ω Easy calculations using (17.7), page 539, yield the following formulas for the Christoffel symbols (see Exercise 1 and Notebook 21). Lemma 21.3. The Christoffel symbols of a Tchebyshef patch are given by Γ111 = (cot ω)ωp , Γ211 = −(csc ω)ωp , Γ112 = 0, Γ212 = 0, Γ122 = −(csc ω)ωq , Γ222 = (cot ω)ωq . A key feature is the vanishing of the ‘mixed’ Christoffel symbols, and this provides a situation in which it is relatively easy to find Tchebyshef coordinates. Lemma 21.4. Let x: U → M be a regular patch on an abstract surface M. A sufficient condition that there exist a Tchebyshef patch that reparametrizes x is that (21.2) Γ112 = Γ212 = 0. 686 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE Proof. Let x be a patch with ds2 = E du2 + 2F dudv + Gdv 2 for which Γ112 = Γ212 = 0. From (17.7), we get 0 = GEv − F Gu = EGu − F Ev . (21.3) When we regard (21.3) as two algebraic equations in the unknowns Ev and Gu , we see that the unique solution of (21.3) is Ev = Gu = 0. Thus E is a function of u alone and G is a function of v alone, so we can define functions p, q by Z Z 1 √ 1 √ (21.4) E du and q(v) = G dv. p(u) = a a The constants of integration are immaterial. Since p′ and q ′ never vanish, the inverse function theorem assures us of the existence of inverse functions u = u(p) and v = v(q). We define a patch y by y(p, q) = x u(p), v(q) . From (21.4) we have a du = √ , dp E a dv = √ dq G dv du = = 0. dq dp and Now Lemma 12.4 on page 365 implies that Ey = Gy = a2 . A closely-related type of patch is defined by Lemma 21.5. Let y be a Tchebyshef patch, and set u+v u−v , x(u, v) = y . 2 2 Then the first fundamental form of x is given by (21.5) ds2x = (a cos θ)2 du2 + (a sin θ)2 dv 2 , where θ = ω/2. Conversely, if the coefficients of the first fundamental form of x satisfy F = 0 and E + G constant, then y(p, q) = x(p + q, p − q) defines a Tchebyshef patch. Proof. Use subscripts to specify the patch in question. Since xu = 21 (yp + yq ) and xv = 12 (yp − yq ), we have Ex = 41 (yp + yq ) · (yp + yq ) = 14 (Ey + 2Fy + Gy ). Thus if Ey = Gy = a2 and Fy = a2 cos ω, then Ex = 12 a2 (1 + cos ω) = a2 cos2 ω = a2 cos2 θ, 2 as stated; Gx similarly. Conversely, suppose that F = 0 and that E + G is constant. Since E and G must both be positive, there exists θ ∈ R for which E du2 + Gdv 2 equals (21.5), where a2 = E + G. The rest then follows by setting ω = 2θ. 21.2. CONSTANT NEGATIVE CURVATURE 687 In the light of Lemma 21.5, we call a patch x whose metric is given by (21.5) a Tchebyshef patch of the second kind with angle function θ. The latter is determined up to sign and addition of integer multiplies of π. With this terminology, we have Lemma 21.6. Let x: U → Rn be a Tchebyshef patch of the second kind with angle function θ. Then the Gaussian curvature of x is given by K= −θuu + θvv . a2 sin θ cos θ Proof. Note that the expression for K is unaffected by the ambiguity in θ. We have √  1 ∂(a sin θ) 1 ∂ G   √ = θu ,   E ∂u = a cos θ ∂u √   1 ∂ E 1 ∂(a cos θ)   √ = = −θv . ∂v a sin θ ∂v G Hence by Corollary 17.4, page 534, ( √ ! √ !) ∂ −θuu + θvv 1 ∂ G 1 ∂ E ∂ −1 √ √ K=√ + = 2 . ∂u ∂u ∂v ∂v a sin θ cos θ EG E G Once again, the Christoffel symbols can readily be found from (17.7) (see Exercise 2 and Notebook 21). Lemma 21.7. The Christoffel symbols of the metric (21.5) are given by Γ111 = −(tan θ)θu = Γ122 , Γ112 = −(tan θ)θv , Γ211 = (cot θ)θv = Γ222 , Γ212 = (cot θ)θu . In the next section, we proceed to construct a Tchebyshef patch of the second kind, starting from a principal patch. 21.2 Patches on Surfaces of Constant Negative Curvature We can establish some important facts about the principal curvatures of surfaces of constant negative curvature in R3 , using the Peterson-Mainardi-Codazzi equations (Theorem 19.8 on page 600). Lemma 21.8. Let y : U → R3 be a principal patch whose metric is given by ds2 = E dp2 + Gdq 2 , and let k1 , k2 denote the principal curvatures. Suppose that y has constant Gaussian curvature −a−2 . Then   E(1 + a2 k12 ) is a function of p alone,  G(1 + a2 k 2 ) 2 is a function of q alone. 688 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE Proof. The first equation of (19.13), page 602, together with the assumption that k1 k2 = −a−2 imply that (21.6) k1q Eq = = 2E k2 − k1 k1q −a2 k1 k1q . = 1 1 + a2 k12 − 2 − k1 a k1 We integrate (21.6) with respect to q, obtaining − 12 log 1 + a2 k12 = 1 2 log E + A(p), where A(p) denotes the constant of integration, a function of p. Exponentiation yields (21.7) (1 + a2 k12 )−1 = e2A(p) E, and the lemma’s first conclusion. The second equation of (19.13) yields the second conclusion in the same way. The next important existence result shows that one can choose the two functions of Lemma 21.8 to be constant and equal to a2 . Theorem 21.9. Suppose that y : U → R3 is a principal patch with metric ds2 = E dp2 + Gdq 2 , and K ≡ −a−2 < 0. Then there is a principal patch x reparametrizing y of the form (21.8) x(u, v) = y p(u), q(v) such that the principal curvatures of x satisfy (21.9)  a2 − E  2  ,  k1 = a2 E 2    k2 = a − G . 2 a2 G Proof. We define x by (21.8) and determine functions p and q so that (21.9) holds. Since xu = p′ (u)yp and xv = q ′ (v)yq , the coefficients E = Ex , F = Fx , G = Gx of the first fundamental form of x are related to those of y by  ′ 2    E(u, v) = p (u) Ey p(u), q(v) ,  (21.10) F (u, v) = 0,     G(u, v) = q ′ (v)2 G p(u), q(v). y Similarly, the coefficients of the second fundamental form of x are xuu = p′′ (u)yp + p′ (u)2 ypp , xuv = p′ (u)q ′ (v)ypq , xvv = q ′′ (v)yq + q ′ (v)2 yqq , 21.2. CONSTANT NEGATIVE CURVATURE and 689  e(u, v) = p′ (u)2 ey p(u), q(v) ,      f (u, v) = fy p(u), q(v) = 0,      g(u, v) = q ′ (v)2 gy p(u), q(v) (21.11) (see equations (13.41), page 419). Denote by e k1 , e k2 the principal curvatures of y; these are patch-independent, so (21.12) k1 (u, v) = e k1 p(u), q(v) and k2 (u, v) = e k2 p(u), q(v) , equations that also follow directly from (21.10) and (21.11). From (21.7), (21.10) and (21.12), we get  p′ (u)2 1  2    k1 (u, v) = a2 E(u, v)eA(p(u)) − 1 ,  q ′ (v)2 1  2   k2 (u, v) = 2 −1 . a G(u, v)eB(q(v)) To finish the proof, it suffices to choose p and q so that p′ (u)2 e−A(p(u)) = q ′ (v)2 e−B(q(v)) = a2 . Then (21.9) follows. The time has come to give a concrete realization of the angle functions of the previous section. Recall the discussion in the proof on page 401. Let M be a surface in R3 , and fix a point p ∈ M. Let k1 , k2 be the principal curvatures at p, and let e1 , e2 be corresponding unit principal vectors. If u1 is a unit asymptotic vector, there exists an angle θ such that (21.13) By Lemma 13.27, (21.14) which shows that (21.15) u1 = e1 cos θ + e2 sin θ. 0 = k(u1 ) = k1 cos2 θ + k2 sin2 θ, u2 = e1 cos θ − e2 sin θ is also an asymptotic vector. It follows that ω = 2θ is the (or rather, an) angle between the two asymptotic directions. Theorem 21.10. Let M ⊂ R3 be a surface with K ≡ −a−2 < 0. Denote by 2θ the angle between asymptotic vectors. In a neighborhood of each point of M there exists a principal patch x: U → M such that 690 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE (i) the metric of x is given by (21.5) on page 686. (ii) The principal curvatures satisfy (21.16) a2 k12 = tan2 θ, and a2 k22 = cot2 θ. 1 . a sin 2θ (iv) The second fundamental form of x is given by (iii) The mean curvature is H = ± II = ±a sin θ cos θ(du2 − dv 2 ). Proof. Choose a principal patch and normalize it according to Theorem 21.9. Choose θ so that tan2 θ = a2 k12 ; then E tan2 θ = a2 Ek12 = a2 − E, and so E sec2 θ = a2 , or E = a2 cos2 θ. The second part of (ii) is a consequence of the equation k1 k2 = −a−2 (see (21.14)). It also follows that G = a2 sin2 θ. Hence (i) holds. For (iii), observe that tan θ cot θ 2 =± + . 2H = k1 + k2 = ± a a a sin(2θ) Let e, f, g be the coefficients of the second fundamental form of x. Lemma 15.5 on page 466 implies that f = 0. To prove (iii), we use Lemma 13.33 on page 405 to compute ae = ak1 E = ±a2 tan θ cos2 θ = ±a sin θ cos θ. Similarly, g = ∓a sin θ cos θ. Part (i) of the above theorem asserts that x is a Tchebyshef patch of the second kind, and we record Definition 21.11. A Tchebyshef principal patch of radius a > 0 and angle function θ is patch satisfying (i) and (ii). Ordinarily, the principal curvatures of a patch x depend on the choice of immersion of the patch into R3 . But for a Tchebyshef principal patch of constant Gaussian curvature −a−2 , the metric determines the principal curvatures. Given a Tchebyshef principal patch x, the associated map y in Lemma 21.5 will be an asymptotic patch. For (21.13) and (21.15) are unit vectors tangent to the coordinate curves defining the metric (21.1). We call y a Tchebyshef asymptotic patch. Thus, one feature of Tchebyshef patches is the easy way in which an asymptotic patch can be converted to a principal patch, and vice versa; this feature is not shared by general patches. When each parametrization is plotted, the resulting images give us distinct visual information. 21.3. SINE–GORDON EQUATION 691 21.3 The Sine–Gordon Equation Finding surfaces of constant negative curvature in R3 is closely related to solving a famous partial differential equation, called the sine–Gordon equation2 . Let θ = θ(u, v) be a function of two variables. There are two versions. The first, (21.17) θuu − θvv = sin θ cos θ, is related by Lemma 21.6 to finding principal patch parametrizations of constant negative curvature. Let u = p + q and v = p − q; then ∂ ∂ ∂ = + ∂p ∂u ∂v and ∂ ∂ ∂ = − , ∂q ∂u ∂v so that ∂2 ∂2 ∂2 − 2. = 2 ∂p∂q ∂u ∂v Hence, if ω = 2θ, we have θpq = θuu − θvv , and (21.18) ωpq = sin ω. This is the version of the sine–Gordon equation which is related, by Lemma 21.2, to finding asymptotic patch parametrizations of constant negative curvature. Clearly, a change of coordinates converts solutions of one version into solutions of the other version, as in Lemma 21.5. Here is a basic existence theorem for the sine–Gordon equation. The proof ([Bian, volume 1, pages 660–664]) resembles the method of Picard iteration used to establish the existence of solutions to certain first-order differential equations. Theorem 21.12. Let φ, ψ : R → R be functions possessing first derivatives such that φ(0) = ψ(0). Then the sine–Gordon equation (21.18) has a unique solution ω satisfying ω(p, 0) = φ(p) and ω(0, q) = ψ(q). It is natural to look for solutions of the sine–Gordon equation (21.17) that depend on only one variable. Lemma 21.13. Let θ be a solution of (21.17), depending on u alone. Then there exists a constant b such that (21.19) θu2 = b2 − cos2 θ. Proof. Since θ is a function of u alone, (21.17) can be multiplied by θu and written as (21.20) 2θuu θu = −(−2 cos θ sin θ)θu . Integrating (21.20) yields (21.19). 2a variant of the Klein–Gordon equation θuu − θvv = θ 692 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE To assist in subsequent applications of this lemma, we record some useful trigonometric identities. Lemma 21.14. Suppose u = log tan(θ/2). Then sin θ = sech u and cos θ = − tanh u. Proof. We have sech u = eu 2 2 θ θ = = 2 sin cos = sin θ, −u 2 2 +e tan(θ/2) + cot(θ/2) and tanh u = eu − e−u tan(θ/2) − cot(θ/2) θ θ = = sin2 − cos2 = − cos θ. 2 2 eu + e−u tan(θ/2) + cot(θ/2) 21.4 Tchebyshef Patches on Surfaces of Revolution We first determine the general form of a Tchebyshef principal parametrization of a surface of revolution with constant negative Gaussian curvature. Theorem 21.15. Let M be a surface of revolution with K ≡ −a−2 < 0, and let θ be an angle function satisfying θv = 0 and (21.19). A Tchebyshef principal patch on M is given by a a x(u, v) = (21.21) sin(θ(u)) cos(b v), sin(θ(u)) sin(b v), ψ(u) , b b where a (21.22) ψ ′ (u) = ± cos2 θ(u) . b Proof. The surface of revolution M can be parametrized locally as (21.23) x(u, v) = α sin(θ(u)) cos(βv), α sin(θ(u)) sin(βv), ψ(u) , for some constants α, β and another function ψ. For this is equivalent to the standard parametrization on page 462, with u and v reversed. We compute   xu (u, v) = α cos θ cos(βv)θu , α cos θ sin(βv)θu , ψ ′ (u) ,  x (u, v) = −αβ sin θ sin(βv), αβ sin θ cos(βv), 0. v Next we impose the condition that x be a Tchebyshef principal patch. Firstly, a2 sin2 θ = G = α2 β 2 sin2 θ implies that α2 β 2 = a2 ; similarly, a2 cos2 θ = E = α2 (cos2 θ)θu2 + ψ ′ (u)2 . The condition that M have constant negative curvature is (21.19), and we get (21.24) ψ ′ (u)2 = cos2 θ a2 − α2 b2 + α2 cos2 θ . When we choose α = a/b and β = b, we get (21.22). 21.4. SURFACES OF REVOLUTION 693 Next, we examine the special case of the preceding theorem in which b2 = 1. Theorem 21.16. Let x: U → R3 be a Tchebyshef principal patch of radius a > 0 whose angle function θ satisfies the partial differential equations θu2 = sin2 θ (21.25) and for which (21.26) Then and θv = 0, x(u, v) = ϕ(u) cos v, ϕ(u) sin v, ψ(u) . (i) x has constant negative curvature −a−2 . (ii) ϕ and ψ are given by ϕ(u) = α a sech(u + C) and ψ(u) = β a u + B − tanh(u + C) , where α and β are ±1, and B, C are constants. (iii) Up to a Euclidean motion, x is a reparametrization of a pseudosphere of radius a. (iv) Any angle function is given by θ = 2δ arctan(eγ(u+C) ), where γ and δ are ±1. Proof. From (21.25), we get θuu = sin θ cos θ; Lemma 21.6 implies that x has K ≡ −a−2 < 0, proving (i). Furthermore, the first equation of (21.25) can be rewritten as (21.27) θu = γ sin θ, where γ = ±1, and the solution to 21.25 is tan 1 θ = eγ(u+C) , = 2 |csc θ + cot θ | where C is a constant. Whence (iv). From (21.27), (iv) and Lemma 21.14 we get (21.28) and (21.29) γ θu = sin θ = δ sin(δ θ) = δ sech γ(u + C) = δ sech(u + C) cos θ = cos(δ θ) = − tanh γ(u + C) = −γ tanh(u + C). Now suppose that x is given by (21.26), so that G = ϕ(u)2 and E = ϕ′ (u)2 + ψ ′ (u)2 . 694 CHAPTER 21. SURFACES OF CONSTANT NEGATIVE CURVATURE For x to be a Tchebyshef principal patch, we must have ϕ(u)2 = G = a2 sin2 θ, so that (21.30) ϕ(u) = α a sin θ, where α = ±1. Then (21.28) and (21.30) imply that (21.31) ϕ(u) = α δ a sech(u + C). Without loss of generality, we can assume δ = 1 in (21.31), and so we obtain the first equation of (ii). Similarly, the fact that ϕ′ (u)2 + ψ ′ (u)2 = E = a2 cos2 θ, combined with (21.30) and (21.27), implies that (21.32) ψ ′ (u)2 = E − ϕ′ (u)2 = a2 cos2 θ − a2 (cos2 θ)θu2 = a2 (cos2 θ) 1 − θu2 = a2 cos4 θ. 4 From (21.32) and (21.29) we get ψ ′ (u)2 = a2 tanh(u + C) , so that 2 ψ ′ (u) = β a tanh(u + C) , (21.33) where β = ±1. Integrating (21.33), we find that (21.34) ψ(u) = β a u + B − tanh(u + C) , where B is the constant of integration, completing the proof of (ii). Written out, (21.26) now becomes α sin v α cos v x(u, v) = a , , β u + B − tanh(u + C) , cosh(u + C) cosh(u + C) where both α and β are ±1. A translation along the z-axis combined with the change of variable u → 7 u − C transforms this into α cos v α sin v x(u, v) = a (21.35) , , β (u − tanh u) . cosh u cosh u Let u = log tan(q/2) and v = p + π(α

Log In

GeoDif de CyS con Mathematica

GeoDif de CyS con Mathematica

Related Papers

RELATED PAPERS