Geometric Methods and Applications: For Computer Science and Engineering

Jean Gallier (auth.)0387950443, 9780387950440

This book is an introduction to the fundamental concepts and tools needed for solving problems of a geometric nature using a computer. It attempts to fill the gap between standard geometry books, which are primarily theoretical, and applied books on computer graphics, computer vision, robotics, or machine learning.

This book covers the following topics: affine geometry, projective geometry, Euclidean geometry, convex sets, SVD and principal component analysis, manifolds and Lie groups, quadratic optimization, basics of differential geometry, and a glimpse of computational geometry (Voronoi diagrams and Delaunay triangulations). Some practical applications of the concepts presented in this book include computer vision, more specifically contour grouping, motion interpolation, and robot kinematics.

In this extensively updated second edition, more material on convex sets, Farkas’s lemma, quadratic optimization and the Schur complement have been added. The chapter on SVD has been greatly expanded and now includes a presentation of PCA.

The book is well illustrated and has chapter summaries and a large number of exercises throughout. It will be of interest to a wide audience including computer scientists, mathematicians, and engineers.

Reviews of first edition:

“Gallier’s book will be a useful source for anyone interested in applications of geometrical methods to solve problems that arise in various branches of engineering. It may help to develop the sophisticated concepts from the more advanced parts of geometry into useful tools for applications.” (Mathematical Reviews, 2001)

“…it will be useful as a reference book for postgraduates wishing to find the connection between their current problem and the underlying geometry.” (The Australian Mathematical Society, 2001)

Table of contents :
Front Matter….Pages i-xxvii
Introduction….Pages 1-5
Basics of Affine Geometry….Pages 7-63
Basic Properties of Convex Sets….Pages 65-83
Embedding an Affine Space in a Vector Space….Pages 85-101
Basics of Projective Geometry….Pages 103-175
Basics of Euclidean Geometry….Pages 177-212
Separating and Supporting Hyperplanes….Pages 213-229
The Cartan–Dieudonné Theorem….Pages 231-280
The Quaternions and the Spaces S 3 , SU(2), SO(3), and ℝ ℙ 3 ….Pages 281-300
Dirichlet–Voronoi Diagrams and Delaunay Triangulations….Pages 301-319
Basics of Hermitian Geometry….Pages 321-342
Spectral Theorems in Euclidean and Hermitian Spaces….Pages 343-365
Singular Value Decomposition (SVD) and Polar Form….Pages 367-385
Applications of SVD and Pseudo-inverses….Pages 387-410
Quadratic Optimization Problems….Pages 411-430
Schur Complements and Applications….Pages 431-437
Quadratic Optimization and Contour Grouping….Pages 439-457
Basics of Manifolds and Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras….Pages 459-528
Basics of the Differential Geometry of Curves….Pages 529-583
Basics of the Differential Geometry of Surfaces….Pages 585-654
Appendix….Pages 655-658
Back Matter….Pages 659-680