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 Hn
3.3 The Upper Half Space Model of Hn
3.4 The Poincar6 Disc Model Bn of Hn
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 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 AProofofLeibon's Theorem
7.5 A Sketch of a Proof of Theorem 73(c)
7.6 Marden-Rodin's Proof Thurston-Andreev Theorem
8 Non-negative Curvature metrics and Delaunay Polytopes
8.1 Non-negative and Curvature Metrics and Delaunay Condition ..
8.2 Relationship between, Curvature and the Discrete Curvature ko
8.3 The work of Rivin and Leibon on Delaunay Polyhedral Surfaces
9 A Brief Introduction to Teichmiiller 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 Teichmuller Spaces
9.4 New Coordinates for the Teichmuller Space
10 Parameterizatios of Teichmuller 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
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.
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