Geometric Differentiation: For the Intelligence of Curves and Surfaces

by I. R. Porteous

Overview

This is a revised and extended version of the popular first edition. Inspired by the work of Thom and Arnold 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 author has included many exercises and examples to illustrate the results proved.

Product Details

ISBN-13:	9780521810401
Publisher:	Cambridge University Press
Publication date:	12/13/2001
Edition description:	Revised Edition
Pages:	350
Product dimensions:	6.22(w) x 9.33(h) x 0.91(d)

Table of Contents

1. Plane curves; 2. Some elementary geometry; 3. Plane kinetics; 4. The derivatives of a map; 5. Curves on the unit sphere; 6. Space curves; 7. k-times linear forms; 8. Probes; 9. Contact; 10. Surfaces in R3; 11. Ridges and ribs; 12. Umbilics; 13. The parabolic line; 14. Involutes of geodesic foliations; 15. The circles of a surface; 16. Examples of surfaces; 17. Flexicords of surfaces; 18. Duality.

Editorial Reviews

"Porteous' approach is distinguished by the extreme richness of the examples he treats, and his novel emphasis on certain concepts—ridges, ribs, umbilics—where third-and-higher-order derivates are crucial. The book's illustrations are helpful and elegant...highly recommended." Choice

From the Publisher

The second edition of this textbook by Porteous (pure mathematics, U. of Liverpool), for advanced undergraduate and beginning postgraduate students of mathematics, centers on the local differential geometry of smooth curves and surfaces in Euclidean space. The text has been revised and expanded, and includes a new chapter incorporating the work of Richard Morris, Mike Puddephat and Tom Barrick on what happens to flexcords at generic and near-generic births of umbilics, and the work of Bill Bruce, Peter Giblin and Farid Tari, using the machinery of singularity theory, and also includes new information on the geometry of curves on 2 and new results on curves and surfaces in 3. The material is relevant to scientists and engineers in a variety of disciplines. Annotation c. Book News, Inc., Portland, OR (booknews.com)