前辅文
 1 Introduction
  1.1 Overview of Theories
   1.1.1 Riemann Mapping
   1.1.2 Riemann Uniformization
   1.1.3 Shape Space
   1.1.4 General Geometric Structure
  1.2 Algorithms for Computing Conformal Mappings
  1.3 Applications
   1.3.1 Computer Graphics
   1.3.2 Computer Vision
   1.3.3 Geometric Modeling
   1.3.4 Medical Imaging
  Further Readings
 Part I Theories
  2 Homotopy Group
   2.1 Algebraic Topological Methodology
   2.2 Surface Topological Classification
   2.3 Homotopy of Continuous-Mappings
   2.4 Homotopy Group
   2.5 Homotopy Invariant
   2.6 Covering Spaces
   2.7 Group Representation
   2.8 Seifert-van Kampen Theorem
   Problems
  3 Homology and Cohomology
   3.1 Simplicial Homology
    3.1.1 Simplicial Complex
    3.1.2 Geometric Approximation Accuracy
    3.1.3 Chain Complex
    3.1.4 Chain Map and Induced Homomorphism
    3.1.5 Simplicial Map
    3.1.6 Chain Homotopy
    3.1.7 Homotopy Equivalence
    3.1.8 Relation Between Homology Group and Homotopy Group
    3.1.9 Lefschetz Fixed Point
    3.1.10 Mayer-Vietoris Homology Sequence
    3.1.11 Tunnel Loop and Handle Loop
   3.2 Cohomology
    3.2.1 Cohomology Group
    3.2.2 Cochain Map
    3.2.3 Cochain Homotopy
    Problems
  4 Exterior Differential Calculus
   4.1 Smooth Manifold
   4.2 Differential Forms
   4.3 Integration
   4.4 Exterior Derivative and Stokes Theorem
   4.5 De Rham Cohomology Group
   4.6 Harmonic Forms
   4.7 Hodge Theorem
   Problems
  5 Differential Geometry of Surfaces
   5.1 Curve Theory
   5.2 Local Theory of Surfaces
    5.2.1 Regular Surface
    5.2.2 First Fundamental Form
    5.2.3 Second Fundamental Form
    5.2.4 Weingarten Transformation
   5.3 Orthonormal Movable Frame
    5.3.1 Structure Equation
   5.4 Covariant Differentiation
    5.4.1 Geodesic Curvature
   5.5 Gauss-Bonnet Theorem
   5.6 Index Theorem of Tangent Vector Field
   5.7 Minimal Surface
    5.7.1 Weierstrass Representation
    5.7.2 Costa Minimal Surface
   Problems
  6 Riemann Surface
   6.1 Riemann Surface
   6.2 Riemann Mapping Theorem
    6.2.1 Conformal Module
    6.2.2 Quasi-Conformal Mapping
    6.2.3 Holomorphic Mappings
   6.3 Holomorphic One-Forms
   6.4 Period Matrix
   6.5 Riemann-Roch Theorem
   6.6 Abel Theorem
   6.7 Uniformization
   6.8 Hyperbolic Riemann Surface
   6.9 Teichmüller Space
    6.9.1 Quasi-Conformal Map
    6.9.2 Extremal Quasi-Conformal Map
   6.10 Teichmüller Space and Modular Space
    6.10.1 Fricke Space Model
    6.10.2 Geodesic Spectrum
   Problems
  7 Harmonic Maps and Surface Ricci Flow
   7.1 Harmonic Maps of Surfaces
    7.1.1 Harmonic Energy and Harmonic Maps
    7.1.2 Harmonic Map Equation
    7.1.3 Radó's Theorem
    7.1.4 Hopf Differential
    7.1.5 Complex Form
    7.1.6 Bochner Formula
    7.1.7 Existence and Regularity
    7.1.8 Uniqueness
   7.2 Surface Ricci Flow
    7.2.1 Conformal Deformation
    7.2.2 Surface Ricci Flow
   Problems
  8 Geometric Structure
   8.1 (X, G) Geometric Structure
   8.2 Development and Holonomy
   8.3 Affine Structures on Surfaces
   8.4 Spherical Structure
   8.5 Euclidean Structure
   8.6 Hyperbolic Structure
   8.7 Real Projective Structure
   Problems
 Part II Algorithms
  9 Topological Algorithms
   9.1 Triangular Meshes
    9.1.1 Half-Edge Data Structure
    9.1.2 Code Samples
   9.2 Cut Graph
   9.3 Fundamental Domain
   9.4 Basis of Homotopy Group
   9.5 Gluing Two Meshes
   9.6 Universal Covering Space
   9.7 Curve Lifting
   9.8 Homotopy Detection
   9.9 The Shortest Loop
   9.10 Canonical Homotopy Group Generator
   Further Readings
   Problems
  10 Algorithms for Harmonic Maps
   10.1 Piecewise Linear Functional Space, Inner Product and Laplacian
   10.2 Newton's Method for Open Surface
   10.3 Non-Linear Heat Diffusion for Closed Surfaces
   10.4 Riemann Mapping
   10.5 Least Square Method for Solving Beltrami Equation
   10.6 General Surface Mapping
   Further Readings
   Problems
  11 Harmonic Forms and Holomorphic Forms
   11.1 Characteristic Forms
   11.2 Wedge Product
   11.3 Characteristic 1-Form
   11.4 Computing Cohomology Basis
   11.5 Harmonic 1-Form
   11.6 Hodge Star Operator
   11.7 Holomorphic 1-Form
   11.8 Inner Product Among 1-Forms
   11.9 Holomorphic Forms on Surfaces with Boundaries
   11.10 Zero Points and Critical Trajectories
   11.11 Flat Metric Induced by Holomorphic 1-Forms
   11.12 Conformal Invariants
   11.13 Conformal Mappings for Multi-Holed Annuli
   Further Readings
   Problems
  12 Discrete Ricci Flow
   12.1 Circle Packing Metric
   12.2 Discrete Gaussian Curvature
   12.3 Discrete Surface Ricci Flow
   12.4 Newton's Method
   12.5 Isometric Planar Embedding
   12.6 Surfaces with Boundaries
   12.7 Optimal Parameterization Using Ricci Flow
   12.8 Hyperbolic Ricci Flow
   12.9 Hyperbolic Embedding
    12.9.1 Poincaré Disk Model
    12.9.2 Embedding the Fundamental Domain
    12.9.3 Hyperbolic Embedding of the Universal Covering Space
   12.10 Hyperbolic Ricci Flow for Surfaces with Boundaries
   Further Readings
   Problems
 A Major Algorithms
 B Acknowledgement
 Reference
 Index