Geometric Differentiation

For the Intelligence of Curves and Surfaces

By: I. R. Porteous

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Hardcover | 13 December 2001 | Edition Number 2

At a Glance

Hardcover
350 Pages

Edition Type
Revised

Dimensions(cm)
22.86 x 15.24 x 2.06

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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.

Industry Reviews

'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

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Introduction	p. xi
Plane curves	p. 1
Introduction	p. 1
Regular plane curves and their evolutes	p. 9
Curvature	p. 14
Parallels	p. 22
Equivalent parametric curves	p. 26
Unit-speed curves	p. 27
Unit-angular-velocity curves	p. 28
Rhamphoid cusps	p. 29
The determination of circular points	p. 33
The four-vertex theorem	p. 35
Exercises	p. 37
Some elementary geometry	p. 42
Introduction	p. 42
Some linear facts	p. 42
Some bilinear facts	p. 44
Some projective facts	p. 46
Projective curves	p. 46
Spaces of polynomials	p. 48
Inversion and stereographic projection	p. 48
Exercises	p. 49
Plane kinematics	p. 51
Introduction	p. 51
Instantaneous rotations and translations	p. 51
The motion of a plane at t = 0	p. 52
The inflection circle and Ball point	p. 53
The cubic of stationary curvature	p. 54
Burmester points	p. 58
Rolling wheels	p. 59
Polodes	p. 61
Caustics	p. 62
Exercises	p. 63
The derivatives of a map	p. 67
Introduction	p. 67
The first derivative and C[superscript 1] submanifolds	p. 67
Higher derivatives and C[superscript k] submanifolds	p. 80
The Faa de Bruno formula	p. 83
Exercises	p. 85
Curves on the unit sphere	p. 88
Introduction	p. 88
Geodesic curvature	p. 89
Spherical kinematics	p. 91
Exercises	p. 94
Space curves	p. 95
Introduction	p. 95
Space curves	p. 95
The focal surface and space evolute	p. 100
The Serret--Frenet equations	p. 105
Parallels	p. 107
Close up views	p. 113
Historical note	p. 116
Exercises	p. 116
k-times linear forms	p. 119
Introduction	p. 119
k-times linear forms	p. 119
Quadratic forms on R[superscript 2]	p. 122
Cubic forms on R[superscript 2]	p. 124
Use of complex numbers	p. 129
Exercises	p. 134
Probes	p. 138
Introduction	p. 138
Probes of smooth map-germs	p. 138
Probing a map-germ V: R[superscript 2]--R	p. 141
Optional reading	p. 145
Exercises	p. 151
Contact	p. 152
Introduction	p. 152
Contact equivalence	p. 152
K-equivalence	p. 154
Applications	p. 155
Exercises	p. 156
Surfaces in R[superscript 3]	p. 158
Introduction	p. 158
Euler's formula	p. 167
The sophisticated approach	p. 169
Lines of curvature	p. 172
Focal curves of curvature	p. 173
Historical note	p. 177
Exercises	p. 178
Ridges and ribs	p. 182
Introduction	p. 182
The normal bundle of a surface	p. 182
Isolated umbilics	p. 183
The normal focal surface	p. 184
Ridges and ribs	p. 187
A classification of focal points	p. 189
More on ridges and ribs	p. 191
Exercises	p. 195
Umbilics	p. 198
Introduction	p. 198
Curves through umbilics	p. 199
Classifications of umbilics	p. 201
The main classification	p. 202
Darboux's classification	p. 203
Index	p. 208
Straining a surface	p. 208
The birth of umbilics	p. 210
Exercises	p. 212
The parabolic line	p. 214
Introduction	p. 214
Gaussian curvature	p. 214
The parabolic line	p. 217
Koenderink's theorems	p. 221
Subparabolic lines	p. 223
Uses for inversion	p. 229
Exercises	p. 230
Involutes of geodesic foliations	p. 233
Introduction	p. 233
Cuspidal edges	p. 234
The involutes of a geodesic foliation	p. 240
Coxeter groups	p. 248
Exercises	p. 252
The circles of a surface	p. 253
Introduction	p. 253
The theorems of Euler and Meusnier	p. 253
Osculating circles	p. 255
Contours and umbilical hill-tops	p. 260
Higher order osculating circles	p. 263
Exercises	p. 263
Examples of surfaces	p. 265
Introduction	p. 265
Tubes	p. 265
Ellipsoids	p. 266
Symmetrical singularities	p. 270
Bumpy spheres	p. 271
The minimal monkey-saddle	p. 280
Exercises	p. 285
Flexcords of surfaces	p. 286
Introduction	p. 286
Umbilics of quadrics	p. 287
Characterisations of flexcords	p. 288
Birth of umbilics	p. 290
Bumpy spheres	p. 298
Exercises	p. 301
Duality	p. 302
Introduction	p. 302
Curves in S[superscript 2]	p. 303
Surfaces in S[superscript 3]	p. 306
Curves in S[superscript 3]	p. 313
Exercises	p. 316
Further reading	p. 317
References	p. 320
Index	p. 327
Table of Contents provided by Syndetics. All Rights Reserved.

Geometric Differentiation

For the Intelligence of Curves and Surfaces

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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