Geometric Differentiation: for the Intelligence of Curves and Surfaces: Second Edition I

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Cambridge University Press 978-0-521-81040-1 - Geometric Differentiation: For the intelligence of Curves and Surfaces: Second Edition I. R. Porteous Frontmatter More information

Geometric differentiation

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-81040-1 - Geometric Differentiation: For the intelligence of Curves and Surfaces: Second Edition I. R. Porteous Frontmatter More information

Geometric differentiation for the intelligence curves and surfaces

Second edition

I. R. Porteous

Senior Lecturer, Department of Pure Mathematics University of Liverpool

University Printing House, Cambridge CB2 8BS, United Kingdom

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www.cambridge.org Information on this title: www.cambridge.org/9780521810401 © Cambridge University Press 1994, 2001 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1994 Second edition 2001 A catalogue record for this publication is available from the British Library ISBN 978-0-521-81040-1 Hardback ISBN 978-0-521-00264-6 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Introduction xi

1 Plane curves 1 1.0 Introduction 1 1.1 Regular plane curves and their evolutes 9 1.2 Curvature 14 1.3 Parallels 22 1.4 Equivalent parametric curves 26 1.5 Unit-speed curves 27 1.6 Unit-angular-velocity curves 28 1.7 Rhamphoid cusps 29 1.8 The determination circular points 33 1.9 The four-vertex theorem 35 Exercises 37

2 Some elementary geometry 42 2.0 Introduction 42 2.1 Some linear facts 42 2.2 Some bilinear facts 44 2.3 Some projective facts 46 2.4 Projective curves 46 2.5 Spaces polynomials 48 2.6 Inversion and stereographic projection 48 Exercises 49

3 Plane kinematics 51 3.0 Introduction 51 3.1 Instantaneous rotations and translations 51

vi Contents

3.2 The motion a plane at t 0 52 3.3 The in¯ection circle and Ball point 53 3.4 The cubic stationary curvature 54 3.5 Burmester points 58 3.6 Rolling wheels 59 3.7 Polodes 61 3.8 Caustics 62 Exercises 63

4 The derivatives of a map 67 4.0 Introduction 67 4.1 The ®rst derivative and C1 submanifolds 67 4.2 Higher derivatives and Ck submanifolds 80 4.3 The FaaÁ de Bruno formula 83 Exercises 85

5 Curves on the unit sphere 88 5.0 Introduction 88 5.1 Geodesic curvature 89 5.2 Spherical kinematics 91 Exercises 94

6 Space curves 95 6.0 Introduction 95 6.1 Space curves 95 6.2 The focal surface and space evolute 100 6.3 The Serret±Frenet equations 105 6.4 Parallels 107 6.5 Close up views 113 6.6 Historicalnote 116 Exercises 116

7 k-times linear forms 119 7.0 Introduction 119 7.1 k-times linear forms 119 7.2 Quadratic forms on R2 122 7.3 Cubic forms on R2 124 7.4 Use complex numbers 129 Exercises 134

Contents vii

8 Probes 138 8.0 Introduction 138 8.1 Probes smooth map-germs 138 2 8.2 Probing a map-germ V : R >! R 141 8.3 Optional reading 145 Exercises 151

9 Contact 152 9.0 Introduction 152 9.1 Contact equivalence 152 9.2 K -equivalence 154 9.3 Applications 155 Exercises 156

10 Surfaces in R3 158 10.0 Introduction 158 10.1 Euler's formula 167 10.2 The sophisticated approach 169 10.3 Lines curvature 172 10.4 Focal curves curvature 173 10.5 Historical note 177 Exercises 178

11 Ridges and ribs 182 11.0 Introduction 182 11.1 The normal bundle a surface 182 11.2 Isolated umbilics 183 11.3 The normal focal surface 184 11.4 Ridges and ribs 187 11.5 A classi®cation focal points 189 11.6 More on ridges and ribs 191 Exercises 195

12 Umbilics 198 12.0 Introduction 198 12.1 Curves through umbilics 199 12.2 Classi®cations umbilics 201 12.3 The main classi®cation 202 12.4 Darboux's classi®cation 203 12.5 Index 208

viii Contents

12.6 Straining a surface 208 12.7 The birth umbilics 210 Exercises 212

13 The parabolic line 214 13.0 Introduction 214 13.1 Gaussian curvature 214 13.2 The parabolic line 217 13.3 Koenderink's theorems 221 13.4 Subparabolic lines 223 13.5 Uses for inversion 229 Exercises 230

14 Involutes of geodesic foliations 233 14.0 Introduction 233 14.1 Cuspidal edges 234 14.2 The involutes a geodesic foliation 240 14.3 Coxeter groups 248 Exercises 252

15 The circles of a surface 253 15.0 Introduction 253 15.1 The theorems Euler and Meusnier 253 15.2 Osculating circles 255 15.3 Contours and umbilical hill-tops 260 15.4 Higher order osculating circles 263 Exercises 263

16 Examples of surfaces 265 16.0 Introduction 265 16.1 Tubes 265 16.2 Ellipsoids 266 16.3 Symmetrical singularities 270 16.4 Bumpy spheres 271 16.5 The minimal monkey-saddle 280 Exercises 285

17 Flexcords of surfaces 286 17.0 Introduction 286 17.1 Umbilics quadrics 287

Contents ix

17.2 Characterisations ¯excords 288 17.3 Birth umbilics 290 17.4 Bumpy spheres 298 Exercises 301

18 Duality 302 18.0 Introduction 302 18.1 Curves in S2 303 18.2 Surfaces in S3 306 18.3 Curves in S3 313 Exercises 316

Further reading 317 References 320 Index 327

Introduction

This book is concerned with the local differential geometry smooth curves and surfaces in Euclidean space. This is a topic which is generally compressed into a short introductory chapter in any standard work on differential geometry, but nevertheless has a surprising richness, with new areas still being explored. The initial impetus to look afresh at this subject was ReneÂ Thom's choice KK1975), but circulating in draft for several years previous to this data) the term `umbilic', borrowed from differential geometry, to describe certain the elementary catastrophes, and the realisation that Darboux's classi®cation K1896) the umbilics a surface, itself not suf®ciently known, was only part the story KPorteous, 1971). In particular the ridges a surface are important features that are almost entirely disregarded in the standard texts, though certain them are familiar to structural geologists as the hinge lines folds in strata KRamsay, 1967). One the few places where they get serious mention in the literature is in the work K1904) A. Gullstrand, who was awarded the Nobel Prize for Physiology and Medicine in 1911 for his work on the accommodation the eye lens, in which work he had to create the necessary fourth order differential geometry to explain the relevant optics. His Prize Lecture K1911) makes amusing reading. Recently they have been rediscovered by workers on face recognition and the interpretation magnetic resonance scans the surface the brain. The work singularity theorists, starting with H. Whitney, R. Thom and J. Mather and extended by many others, notably the Russian School under the leadership V. I. Arnol'd, has richly developed the geometry Taylor series smooth functions, making accessible and familiar the classi®cation the higher order critical points families smooth

xii Introduction

functions. What we do here is to provide a treatment surface theory which is in the spirit singularity theory, yet stops short the hard and powerful theorems that subject. For this reason some the things that we do are heuristic only, though with reference to where a fuller treatment is to be found. A unifying theme throughout the book is the use the family distance-squared functions that relate a smooth submanifold Eu- clidean space to the ambient space. The description the critical points these functions in terms invariant expression s involving the higher derivatives these functions provides a high road into the interesting geometry. The ®rst chapter plunges straight into the interrelationship a smooth parametric plane curve with its evolute or focal curve, a topic ®rst studied in detail by Huygens some years before the advent the differential calculus made this a ®rst exercise in that subject. It is one that is still important in the practical theory the offsets or parallels a plane curve. As Huygens realised, these are recoverable from the evolute by unwinding a string from it or, equivalently, by rolling a ruler along it, a pregnant idea that bears fruit later in the theory space curves and surfaces and their evolutes. Several characterisations are given the curvature and the vertices a curve. The latter topic is the ®rst place where third derivatives make their entry. We learn how to locate the vertices without having to differentiate the explicit but somewhat unattractive formula for the curvature. As a digression from the main theme we apply these ideas brie¯y to some basic concepts plane kinematics that have application in the theory mechanisms, namely the in¯ection circle, the Ball point, the cubic of stationary curvature and the Burmester points the instantaneous motion a plane moving rigidly over the plane. A related subject, though perhaps not obviously so, is the description the caustics the light from a point source re¯ected from a curved mirror, a subject richly reexamined recently by Bruce, Giblin and Gibson K1981). The analogous theory spherical kinematics is brie¯y sketched in a later chapter, prefaced by a short chapter in which some basic facts elementary geometry are recalled. Spherical kinematics has application to the description the motion plates over the surface the Earth in modern plate tectonics. Next comes the theory space curves, introducing the space evolute a space curve and the tangent developable the space evolute, the focal surface the original curve. This is presented before the derivation the classical Serret±Frenet equations for a space curve,

Introduction xiii

usually given pride place. The latter require the theoretically convenient but practically inconvenient use a unit-speed parametrisation for the curve. Also de®ned are the parallels to a space curve, the involutes its evolute, and the focal curves a space curve, shown to be geodesics on its focal surface. Surface theory, depending as it does on the differential calculus functions several variables, requires familiarity not only with quadratic but also with cubic forms in two variables. A chapter interpolated here provides the necessary facts. Two further short chapters follow, one on probe analysis as a diagnostic technique for handling singularities maps and the other on contact. The way is then open to describe the focal surface a regular smooth parametric surface in three-dimensional Euclidean space. Topics discussed in detail include the ®rst and second fundamental forms, the principal centres of curvature and principal curvatures the surface and the induced grid on the surface formed by the two mutually orthogonal families lines of curvature. These only involve the ®rst and second derivatives the parametrisation. It is in the description the ridges the surface and the associated ribs or cuspidal edges the focal surface and the rich geometry around the umbilics the surface, those points where the principal curvatures agree and the surface is most nearly spherical, that the third and higher derivatives become important and the insights singularity theory prove greatest value. Other topics discussed include various aspects the Gauss map a surface, including the parabolic line,thecusps of Gauss and Gauss' `excellent' theorem, in Greek±Latin the theorema egregium. The study the focal surface a surface is rounded f by a detailed examination the inverse construction the involutes or evolvents a geodesic foliation on a regular surface in three- dimensional Euclidean space with emphasis on what happens when the curves the foliation possess linear points, a subject full surprises that has had close attention from the Russian school singularity theorists under the leadership V.I. Arnol'd, and where the full story has only very recently become clear. Our treatment, based on the study the family distance-squared functions, has emphasised the role spheres in surface theory. The circles associated to a surface are also great interest. Here again the initial steps recovering the focal surface are classical, but much the higher order material seems to be new, some it to be found in the

xiv Introduction

thesis James Montaldi K1986a). Remarks on the bumpy circular contours round an umbilical hill-top led to the delightful work Stelios Markatis K1980) on bumpy spheres, outlined here in a ®nal chapter that considers some other important examples also. The title the book has been chosen to indicate that its purpose is to make geometrical the basic properties the derivatives differential maps. It is written to be accessible to ®nal-year honours students or to ®rst-year postgraduates, keen to add some geometry to the standard linear algebra and several variables calculus that in one form or another they will already have met. A non-standard feature which browsers must take note is a notational one. For practical reasons that will become evident when working with third and fourth derivatives the traditional d to denote differentiation is only used occasionally. Instead differentiation

is denoted by subscripts. Thus for a smooth map f : X >! Y between real vector spaces X and Y Kwhere the tail on the arrow indicates that the domain is an open subset the source vector space) the ®rst

derivative is the map f1 : X >! LKX, Y), that associates to each a 2 dom f the linear map f1Ka):X ! Y that Kup to a constant) best approximates f at a, the second derivative is f2 : X >! LKX,KX, Y)), associating to each a 2 X the twice linear Krather than bilinear) map f 2Ka) K f1)1Ka), and so on. Much the material has been class tested from time to time over the last ®fteen years. It has been the basis a number expository lectures, for example to the Archimedeans in Cambridge on several occasions, to colleagues in singularity theory at the AMS symposium at Arcata in 1981 KPorteous 1983a,b), to specialists in computer vision at a Rank Prize Fund symposium at Liverpool in 1986 and to the Oxford group in 1988, and to specialists in computer-aided design at symposia on the Theory Surfaces organised by the Institute Mathematics and its Applications at Cardiff in 1986 and Oxford in 1988 KPorteous, 1987a, 1987b, 1989). Much the work has been done in collaboration with students at Liverpool, notably Stelios Markatis K1980), James Montaldi K1983), Alex Flegmann K1985) and Richard Morris K1990). My thanks are also due to numerous colleagues and friends for their interest and comment, especially Bill Bruce, Peter Giblin, Chris Gibson, Peter Newstead and Terry Wall and students Helen Chappell and Neil Kirk. Especial thanks are due to Peter Ackerley, who has crafted many the ®gures, and to Peter Giblin and Richard Morris for those produced on the computer. I am also most grateful to John Robinson Yeovil for

Introduction xv

permitting me to make use his sculpture Eternity to grace this book. For his remarkable story and pictures all his abstract sculptures see Robinson K1992). ß áëàãîäàðåí òàêæå Â. È. Àðíîëüäó, ÷üß ïîìîøü áûëà ìèå î÷åíü ïîëåçíà.

Liverpool, 1993 Ian R. Porteous

Almost as soon as the ®rst edition this book appeared in print there were two developments which I am delighted to be able to incorporate into this new edition, which preserves as far as possible the pagination the original, but provides two chapters additional material. In the ®rst place Richard Morris, Mike Puddephat and Tom Barrick have helped to resolve what happens to the subparabolic lines a surface, or ¯excords as I now prefer to call them, at generic, and near- generic births umbilics, there being related work also by Bill Bruce, Peter Giblin and Farid Tari, using the machinery singularity theory. This corrects and ampli®es statements made on page 228 the ®rst edition and provides the material a new Chapter 17. Secondly, a remarkable 68-page paper by V.I. Arnol'd appeared in 1995 on the geometry curves on S2, which complements and extends the material Chapter 5. Details are to be found in a new Chapter 18, which also contains new results on curves and surfaces in S3. Initial drafts both these chapters have appeared as Porteous and Puddephat K2000) and Porteous K1999), respectively. I am grateful both to Springer and to the Banach Institute for permitting me to reuse this material, substantially unaltered. Interest in the ridges a surface, in particular, has continued among is required �ﳺ those working in medical imaging and in CAD. What is the ability to landmark a surface that may be slowly changing in time. Lines curvature and geodesics are no direct assistance, as they reform rather than deform as the surface changes. Besides the umbilics and the parabolic line, familiar to all differential geometers, it is the lesser-known ridges and hardly known ¯excords, and special points on them, that ®ll the need. The opportunity has course been taken to correct a number misprints in the ®rst printing, many which have been brought to my attention by diligent readers, to all whom I am most grateful.

Liverpool, 2000 Ian R. Porteous