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