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Mathematics - Geometry & Topology | Geometric Methods and Applications - For Computer Science and Engineering

Geometric Methods and Applications

For Computer Science and Engineering

Series: Texts in Applied Mathematics, Vol. 38

Gallier, Jean

2001, XXI, 566 p.

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As an introduction to fundamental geometric concepts and tools needed for solving problems of a geometric nature using a computer, this book attempts to fill the gap between standard geometry books, which are primarily theoretical, and applied books on computer graphics, computer vision, or robotics, which sometimes do not cover the underlying geometric concepts in detail. Gallier offers an introduction to affine geometry, projective geometry, Euclidean geometry, basics of differential geometry and Lie groups, and a glimpse of computational geometry (convex sets, Voronoi diagrams and Delaunay triangulations) and explores many of the practical applications of geometry. Some of these applications include computer vision (camera calibration) efficient communication, error correcting codes, cryptography, motion interpolation, and robot kinematics. This comprehensive text covers most of the geometric background needed for conducting research in computer graphics, geometric modeling, computer vision, and robotics and as such will be of interest to a wide audience including computer scientists, mathematicians, and engineers.

Content Level » Research

Keywords » Triangulation - communication - computational geometry - computer graphics - computer vision - cryptography - differential geometry - differential geometry of surfaces - geometry - modeling - robotics - sets

Related subjects » Geometry & Topology

Table of contents 

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 Exponential Map.- 14.4 The Lie Groups GL(n, ?), SL(n, ?), U(n), SU(n), the Lie Algebras gl(n, ?), sl(n, ?), u(n), su(n), and the Exponential Map.- 14.5 Hermitian Matrices, Hermitian Positive Definite Matrices, and the Exponential Map.- 14.6 The Lie Group SE(n) and the Lie Algebra se(n).- 14.7 Finale: Lie Groups and Lie Algebras.- 14.8 Applications of Lie Groups and Lie Algebras.- 14.9 Problems.- 15 Basics of the Differential Geometry of Curves.- 15.1 Introduction: Parametrized Curves.- 15.2 Tangent Lines and Osculating Planes.- 15.3 Arc Length.- 15.4 Curvature and Osculating Circles (Plane Curves).- 15.5 Normal Pl anes and Curvature (3D Curves).- 15.6 The Frenet Frame (3D Curves).- 15.7 Torsion (3D Curves).- 15.8 The Frenet Equations (3D Curves).- 15.9 Osculating Spheres (3D Curves).- 15.10 The Frenet Frame for nD Curves (n ? 24).- 15.11 Applications.- 15.12 Problems.- 16 Basics of the Differential Geometry of Surfaces.- 16.1 Introduction.- 16.2 Paramet rized Surfaces.- 16.3 The First Fundamental Form (Riemannian Metric).- 16.4 Normal Curvature and the Second Fundamental Form.- 16.5 Geodesic Curvature and the Christoffel Symbols.- 16.6 Principal Curvatures, Gaussian Curvature, Mean Curvature.- 16.7 The Gauss Map and Its Derivative dN.- 16.8 The Dupin Indicatrix.- 16.9 The Theorema Egregium of Gauss, the Equations of Codazzi-Mainardi, and Bonnet’s Theorem.- 16.10 Lines of Curvature, Geodesic Torsion , Asymptotic Lines.- 16.11 Geodesic Lines, Local Gauss-Bonnet Theorem.- 16.12 Applications.- 16.13 Problems.- 17 Appendix.- 17.1 Hyperplanes and Linear Forms.- 17.2 Metric Spaces and Normed Vector Spaces.- References.- Symbol Index.

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