Synopsis
1 Introduction.- 1.1 Geometries: Their Origin, Their Uses.- 1.2 Prerequisit es and Notation.- 2 Basics of Affine Geometry.- 2.1 Affine Spaces.- 2.2 Examples of Affine Spaces.- 2.3 Chasles's Identity.- 2.4 Affine Combinations, Barycenters.- 2.5 Affine Subspaces.- 2.6 Affine Independence and Affine Frames.- 2.7 Affine Maps.- 2.8 Affine Groups.- 2.9 Affine Geometry: A Glimpse.- 2.10 Affine Hyperplanes.- 2.11 Intersection of Affine Spaces.- 2.12 Problems.- 3 Properties of Convex Sets: A Glimpse.- 3.1 Convex Sets.- 3.2 Carathéodory's Theorem.- 3.3 Radon's and Helly's Theorems.- 3.4 Problems.- 4 Embedding an Affine Space in a Vector Space.- 4.1 The "Hat Construction," or Homogenizing.- 4.2 Affine Frames of E and Bases of Ê.- 4.3 Another Construction of Ê.- 4.4 Extending Affine Maps to Linear Maps.- 4.5 Problems.- 5 Basics of Projective Geometry.- 5.1 Why Projective Spaces?.- 5.2 Projective Spaces.- 5.3 Projective Subspaces.- 5.4 Projective Frames.- 5.5 Projective Maps.- 5.6 Projective Completion of an Affine Space, Affine Patches.- 5.7 Making Good Use of Hyperplanes at Infinity.- 5.8 The Cross-Ratio.- 5.9 Duality in Projective Geometry.- 5.10 Cross-Ratios of Hyperplanes.- 5.11 Complexification of a Real Projective Space.- 5.12 Similarity Structures on a Projective Space.- 5.13 Some Applications of Projective Geometry.- 5.14 Problems.- 6 Basics of Euclidean Geometry.- 6.1 Inner Products, Euclidean Spaces.- 6.2 Orthogonality, Duality, Adjoint of a Linear Map.- 6.3 Linear Isometries (Orthogonal Transformations).- 6.4 The Orthogonal Group, Orthogonal Matrices.- 6.5 QR-Decomposition for Invertible Matrices.- 6.6 Some Applications of Euclidean Geometry.- 6.7 Problems.- 7 The Cartan-Dieudonné Theorem.- 7.1 Orthogonal Reflections.- 7.2 The Cartan-Dieudonné Theorem for Linear Isometries.- 7.3 QR-Decomposition Using Householder Matrices.- 7.4 Affine Isometries (Rigid Motions).- 7.5 Fixed Points of Affine Maps.- 7.6 Affine Isometries and Fixed Points.- 7.7 The Cartan-Dieudonné Theorem for Affine Isometries.- 7.8 Orientations of a Euclidean Space, Angles.- 7.9 Volume Forms, Cross Products.- 7.10 Problems.- 8 The Quaternions and the Spaces S3, SU(2), SO(3), and ?P3.- 8.1 The Algebra ? of Quaternions.- 8.2 Quaternions and Rotations in SO(3).- 8.3 Quaternions and Rotations in SO(4).- 8.4 Applications of Euclidean Geometry to Motion Interpolation.- 8.5 Problems.- 9 Dirichlet-Voronoi Diagrams and Delaunay Triangulations.- 9.1 Dirichlet-Voronoi Diagrams.- 9.2 Simplicial Complexes and Triangulations.- 9.3 Delaunay Triangulations.- 9.4 Delaunay Triangulations and Convex Hulls.- 9.5 Applications of Voronoi Diagrams and Delaunay Triangulations.- 9.6 Problems.- 10 Basics of Hermitian Geometry.- 10.1 Sesquilinear and Hermitian Forms, Pre-Hilbert Spaces and Hermitian Spaces.- 10.2 Orthogonality, Duality, Adjoint of a Linear Map.- 10.3 Linear Isometries (Also Called Unitary Transformations).- 10.4 The Unitary Group, Unitary Matrices.- 10.5 Problems.- 11 Spectral Theorems in Euclidean and Hermitian Spaces.- 11.1 Introduction: What's with Lie Groups and Lie Algebras?.- 11.2 Normal Linear Maps.- 11.3 Self-Adjoint, Skew Self-Adjoint, and Orthogonal Linear Maps.- 11.4 Normal, Symmetric, Skew Symmetric, Orthogonal, Hermitian, Skew Hermitian, and Unitary Matrices.- 11.5 Problems.- 12 Singular Value Decomposition (SVD) and Polar Form.- 12.1 Polar Form.- 12.2 Singular Value Decomposition (SVD).- 12.3 Problems.- 13 Applications of Euclidean Geometry to Various Optimization Problems.- 13.1 Applications of the SVD and QR-Decomposition to Least Squares Problems.- 13.2 Minimization of Quadratic Functions Using Lagrange Multipliers.- 13.3 Problems.- 14 Basics of Classical Lie Groups: The Exponential Map, Lie Groups, and Lie Algebras.- 14.1 The Exponential Map.- 14.2 The Lie Groups GL(n, ?), SL(n, ?), O(n), SO(n), the Lie Algebras gl(n, ?), sl(n, ?), o(n), so(n), and the Exponential Map.- 14.3 Symmetric Matrices, Symmetric Positive Definite Matrices, and the Expo
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About the Author
Jean Gallier is a Professor at the University of Pennsylvania in the Computer and Information Science Department at the School of Engineering and Applied Science.
Review
From the reviews:
"This is a worthwhile book to have in the library of any univeristy which is involved with these areas of computer science." (The Mathematical Gazette, 2002)
"The treatment of each topic is in depth and to the point. It is a rigorous theorem-proof approach on the one hand, but there are plenty of comments and remarks that make for easier reading." (SIAM Review, 2002)
From the reviews of the second edition:
“The book contains a valuable collection of modern geometric methods and algorithms readily prepared for solving problems occurring in computer science and engineering. ... 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.” (Anton Gfrerrer, Zentralblatt MATH, Vol. 1247, 2012)
“Anyone who likes to read about geometry, differential geometry, linear algebra, or Lie theory should find something of interest in this book. ... this is, of course, a text in geometry, and many aspects of it are covered. ... this book is filled with interesting mathematics, superbly presented. Aside from its potential use as a text, the book should be looked at by anyone who uses or is interested in the topics covered. It is highly recommended.” (Mark Hunacek, The Mathematical Association of America, September, 2011)
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