Geometric methods and applications. For computer science and engineering. 2nd ed. (English) Zbl 1247.53001

Gallier, Jean

Texts in Applied Mathematics 38. Dordrecht: Springer (ISBN 978-1-4419-9960-3/hbk; 978-1-4419-9961-0/ebook). xxvii, 680 p. (2011).

In the second edition of Geometric Methods and Applications, Jean Gallier expanded some issues and added quite a few topics (for the first edition see Zbl 1031.53001):

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The chapter on convex sets is extended by Carathéodory’s theorem for convex cones and Radon’s theorem for pointed cones. The introduction of centerpoints is also new.

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Separating and supporting hyperplanes are introduced, and the separation of open or closed convex sets by hyperplanes is discussed. Farkas’s lemma and its application to linear programming is now presented. Moreover, the existence of supporting hyperplanes for boundary points of closed convex sets is proved.

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Principal component analysis (PCA) and its relation to singular value decomposition (SVD) is extensively discussed.

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Quadratic optimization problems (also with linear, affine or quadratic constraints) are treated much more extensively than in the first edition. The task of contour grouping stemming from the field of computer vision and its connection with quadratic optimization is dealt with.

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Schur complements and their connection with symmetric positive semidefinite matrices are explained.

Summarizing, one can say that the book contains a valuable collection of modern geometric methods and algorithms readily prepared for solving problems occurring in computer science and engineering.
In the last decade the first edition of Gallier’s monograph has turned out to be a benchmark in its field. The second edition is even more comprehensive and puts more emphasis on the links between different fields. It can be recommended to anybody who is interested in modern geometry and its applications.

Reviewer: Anton Gfrerrer (Graz)

MSC:

53-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
22-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
51-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
65D18	Numerical aspects of computer graphics, image analysis, and computational geometry
51N10	Affine analytic geometry
51N15	Projective analytic geometry
51N20	Euclidean analytic geometry
53A04	Curves in Euclidean and related spaces
53A05	Surfaces in Euclidean and related spaces

Keywords:

affine geometry; projective geometry; Euclidean geometry; convex set; separation theorem; Voronoi diagram; Delaunay triangulation; quadratic optimization; singular value decomposition; Schur complement; Lie group; Lie algebra; differential geometry; quaternions

Citations:

Zbl 1031.53001