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Geometric Differentiation: for the Intelligence of Curves and Surfaces: Second Edition I 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 © 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 of curves and surfaces Second edition I. R. Porteous Senior Lecturer, Department of Pure Mathematics University of Liverpool © 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 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. 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. © 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 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 of 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 of 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 v © 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 vi Contents 3.2 The motion of a plane at t 0 52 3.3 The in¯ection circle and Ball point 53 3.4 The cubic of 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 of complex numbers 129 Exercises 134 © 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 Contents vii 8 Probes 138 8.0 Introduction 138 8.1 Probes of 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 of curvature 172 10.4 Focal curves of curvature 173 10.5 Historical note 177 Exercises 178 11 Ridges and ribs 182 11.0 Introduction 182 11.1 The normal bundle of 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 of 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 of umbilics 201 12.3 The main classi®cation 202 12.4 Darboux's classi®cation 203 12.5 Index 208 © 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 viii Contents 12.6 Straining a surface 208 12.7 The birth of 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 of 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 of 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 of quadrics 287 © 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 Contents ix 17.2 Characterisations of ¯excords 288 17.3 Birth of 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 © 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 © 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 Introduction This book is concerned with the local differential geometry of 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 Rene Thom's choice KK1975), but circulating in draft for several years previous to this data) of the term `umbilic', borrowed from differential geometry, to describe certain of the elementary catastrophes, and the realisation that Darboux's classi®cation K1896) of the umbilics of a surface, itself not suf®ciently known, was only part of the story KPorteous, 1971). In particular the ridges of a surface are important features that are almost entirely disregarded in the standard texts, though certain of them are familiar to structural geologists as the hinge lines of folds in strata KRamsay, 1967). One of the few places where they get serious mention in the literature is in the work K1904) of A. Gullstrand, who was awarded the Nobel Prize for Physiology and Medicine in 1911 for his work on the accommodation of 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 of magnetic resonance scans of the surface of the brain. The work of singularity theorists, starting with H. Whitney, R. Thom and J. Mather and extended by many others, notably the Russian School under the leadership of V. I. Arnol'd, has richly developed the geometry of Taylor series of smooth functions, making accessible and familiar the classi®cation of the higher order critical points of families of smooth xi © 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.
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