This book intends to lead its readers to some of the current topics of research in the geometry of polyhedral surfaces with applications to computer graphics. The main feature of the book is a systematic introduction to geometry of polyhedral surfaces based on the variational principle. The authors focus on using analytic methods in the study of some of the fundamental results and problems on polyhedral geometry, e. g., the Cauchy rigidity theorem, Thurston's circle packing theorem, rigidity of circle packing theorems and Colin de Verdiere's variational principle. With the vast development of the mathematics subject of polyhedral geometry, the present book is the first complete treatment of the subject.
- Front Matter
- 1 Introduction
- 1.1 Variational Principle and Isoperimetric Problems
- 1.2 Polyhedral Metrics and Polyhedral Surfaces
- 1.3 A Brief History on Geometry of Polyhedral Surface
- 1.4 Recent Works on Polyhedral Surfaces
- 1.5 Some of Our Results
- 1.6 The Method of Proofs and Related Works
- 2 Spherical Geometry and Cauchy Rigidity Theorem
- 2.1 Spherical Geometry and Spherical Triangles
- 2.2 The Cosine law and the Spherical Dual
- 2.3 The Cauchy Rigidity Theorem
- 3 A Brief Introduction to Hyperbolic Geometry
- 3.1 The Hyperboloid Model of the Hyperbolic Geometry
- 3.2 The Klein Model of
- 3.3 The Upper Half Space Model of
- 3.4 The Poincaré Disc Model of
- 3.5 The Hyperbolic Cosine Law and the Gauss-Bonnet Formula
- 4 The Cosine Law and Polyhedral Surfaces
- 4.1 Introduction
- 4.2 Polyhedral Surfaces and Action Functional of Variational Framework
- 5 Spherical Polyhedral Surfaces and Legendre Transformation
- 5.1 The Space of All Spherical Triangles
- 5.2 A Rigidity Theorem for Spherical Polyhedral Surfaces
- 5.3 The Legendre Transform
- 5.4 The Cosine Law for Euclidean Triangles
- 6 Rigidity of Euclidean Polyhedral Surfaces
- 6.1 A Local and a Global Rigidity Theorem
- 6.2 Rivin's Theorem on Global Rigidity of φ0 Curvature
- 7 Polyhedral Surfaces of Circle Packing Type
- 7.1 Introduction
- 7.2 The Cosine Law and the Radius Parametrization
- 7.3 Colin de Verdiere's Proof of Thurston-Andreev Rigidity Theorem
- 7.4 A Proof of Leibon's Theorem
- 7.5 A Sketch of a Proof of Theorem 7.3(c)
- 7.6 Marden-Rodin's Proof Thurston-Andreev Theorem
- 8 Non-negative Curvature metrics and Delaunay Polytopes
- 8.1 Non-negative φh and ψh Curvature Metrics and Delaunay Condition
- 8.2 Relationship between φ0, ψ0 Curvature and the Discrete Curvature k0
- 8.3 The work of Rivin and Leibon on Delaunay Polyhedral Surfaces
- 9 A Brief Introduction to Teichmüller Space
- 9.1 Introduction
- 9.2 Hyperbolic Hexagons, Hyperbolic 3-holed Spheres and the Cosine law
- 9.3 Ideal Triangulation of Surfaces and the Length Coordinate of the Teichmüller Spaces
- 9.4 New Coordinates for the Teichmüller Space
- 10 Parameterizatios of Teichmüller spaces
- 10.1 A Proof of Theorem 10.1
- 10.2 Degenerations of Hyperbolic Hexagons
- 10.3 A Proof of Theorem 10.2
- 11 Surface Ricci Flow
- 11.1 Conformal Deformation
- 11.2 Surface Ricci Flow
- 12 Geometric Structure
- 12.1 (X, G) Geometric Structure
- 12.2 Affine Structures on Surfaces
- 12.3 Spherical Structure
- 12.4 Euclidean Structure
- 12.5 Hyperbolic Structure
- 12.6 Real Projective Structure
- 13 Shape Acquisition and Representation
- 13.1 Shape Acquisition
- 13.2 Triangular Meshes
- 13.3 Half-Edge Data Structure
- 14 Discrete Ricci Flow
- 14.1 Circle Packing Metric
- 14.2 Discrete Gaussian Curvature
- 14.3 Discrete Surface Ricci Flow
- 14.4 Newton's Method
- 14.5 Isometric Planar Embedding
- 14.6 Surfaces with Boundaries
- 14.7 Optimal Parameterization Using Ricci flow
- 15 Hyperbolic Ricci Flow
- 15.1 Hyperbolic Embedding
- 15.1.1 Embedding One Face
- 15.1.2 Hyperbolic Embedding of the Universal Covering Space
- 15.2 Surfaces with Boundaries
- Reference
- Index
