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Geometric Differentiation : For the Intelligence of Curves and Surfaces - I. R. Porteous

Geometric Differentiation

For the Intelligence of Curves and Surfaces

Hardcover

Published: 28th April 2015
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This is a revised and extended version of the popular first edition. Inspired by the work of Thom and Arnol'd on singularity theory, such topics as umbilics, ridges and subparabolic lines, all robust features of a smooth surface, which are rarely treated in elementary courses on differential geometry, are considered here in detail. These features are of immediate relevance in modern areas of application such as interpretation of range data from curved surfaces and the processing of magnetic resonance and cat-scan images. The text is based on extensive teaching at Liverpool University to audiences of advanced undergraduate and beginning postgraduate students in mathematics. However, the wide applicability of this material means that it will also appeal to scientists and engineers from a variety of other disciplines. The author has included many exercises and examples to illustrate the results proved.

'The very geometric point of view and many exercises induce me to recommend this book for everyone interested in differential geometry of curves and surfaces.' Internationale Mathematische Nachrichten '... a very good and interesting introduction to differential geometry of curves and surfaces, which can be recommended to anybody interested in the subject.' EMS Newsletter

Introductionp. xi
Plane curvesp. 1
Introductionp. 1
Regular plane curves and their evolutesp. 9
Curvaturep. 14
Parallelsp. 22
Equivalent parametric curvesp. 26
Unit-speed curvesp. 27
Unit-angular-velocity curvesp. 28
Rhamphoid cuspsp. 29
The determination of circular pointsp. 33
The four-vertex theoremp. 35
Exercisesp. 37
Some elementary geometryp. 42
Introductionp. 42
Some linear factsp. 42
Some bilinear factsp. 44
Some projective factsp. 46
Projective curvesp. 46
Spaces of polynomialsp. 48
Inversion and stereographic projectionp. 48
Exercisesp. 49
Plane kinematicsp. 51
Introductionp. 51
Instantaneous rotations and translationsp. 51
The motion of a plane at t = 0p. 52
The inflection circle and Ball pointp. 53
The cubic of stationary curvaturep. 54
Burmester pointsp. 58
Rolling wheelsp. 59
Polodesp. 61
Causticsp. 62
Exercisesp. 63
The derivatives of a mapp. 67
Introductionp. 67
The first derivative and C[superscript 1] submanifoldsp. 67
Higher derivatives and C[superscript k] submanifoldsp. 80
The Faa de Bruno formulap. 83
Exercisesp. 85
Curves on the unit spherep. 88
Introductionp. 88
Geodesic curvaturep. 89
Spherical kinematicsp. 91
Exercisesp. 94
Space curvesp. 95
Introductionp. 95
Space curvesp. 95
The focal surface and space evolutep. 100
The Serret--Frenet equationsp. 105
Parallelsp. 107
Close up viewsp. 113
Historical notep. 116
Exercisesp. 116
k-times linear formsp. 119
Introductionp. 119
k-times linear formsp. 119
Quadratic forms on R[superscript 2]p. 122
Cubic forms on R[superscript 2]p. 124
Use of complex numbersp. 129
Exercisesp. 134
Probesp. 138
Introductionp. 138
Probes of smooth map-germsp. 138
Probing a map-germ V: R[superscript 2]--Rp. 141
Optional readingp. 145
Exercisesp. 151
Contactp. 152
Introductionp. 152
Contact equivalencep. 152
K-equivalencep. 154
Applicationsp. 155
Exercisesp. 156
Surfaces in R[superscript 3]p. 158
Introductionp. 158
Euler's formulap. 167
The sophisticated approachp. 169
Lines of curvaturep. 172
Focal curves of curvaturep. 173
Historical notep. 177
Exercisesp. 178
Ridges and ribsp. 182
Introductionp. 182
The normal bundle of a surfacep. 182
Isolated umbilicsp. 183
The normal focal surfacep. 184
Ridges and ribsp. 187
A classification of focal pointsp. 189
More on ridges and ribsp. 191
Exercisesp. 195
Umbilicsp. 198
Introductionp. 198
Curves through umbilicsp. 199
Classifications of umbilicsp. 201
The main classificationp. 202
Darboux's classificationp. 203
Indexp. 208
Straining a surfacep. 208
The birth of umbilicsp. 210
Exercisesp. 212
The parabolic linep. 214
Introductionp. 214
Gaussian curvaturep. 214
The parabolic linep. 217
Koenderink's theoremsp. 221
Subparabolic linesp. 223
Uses for inversionp. 229
Exercisesp. 230
Involutes of geodesic foliationsp. 233
Introductionp. 233
Cuspidal edgesp. 234
The involutes of a geodesic foliationp. 240
Coxeter groupsp. 248
Exercisesp. 252
The circles of a surfacep. 253
Introductionp. 253
The theorems of Euler and Meusnierp. 253
Osculating circlesp. 255
Contours and umbilical hill-topsp. 260
Higher order osculating circlesp. 263
Exercisesp. 263
Examples of surfacesp. 265
Introductionp. 265
Tubesp. 265
Ellipsoidsp. 266
Symmetrical singularitiesp. 270
Bumpy spheresp. 271
The minimal monkey-saddlep. 280
Exercisesp. 285
Flexcords of surfacesp. 286
Introductionp. 286
Umbilics of quadricsp. 287
Characterisations of flexcordsp. 288
Birth of umbilicsp. 290
Bumpy spheresp. 298
Exercisesp. 301
Dualityp. 302
Introductionp. 302
Curves in S[superscript 2]p. 303
Surfaces in S[superscript 3]p. 306
Curves in S[superscript 3]p. 313
Exercisesp. 316
Further readingp. 317
Referencesp. 320
Indexp. 327
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521810401
ISBN-10: 052181040X
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 350
Published: 28th April 2015
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.1
Weight (kg): 0.64
Edition Number: 2
Edition Type: Revised