计算共形几何

Introduction

1.1 Overview of Theories

1.1.1 RiemannMapping

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 Grou

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 Teichmiiller Space

6.9.1 Quasi-Conformal Map

6.9.2 Extremal Quasi-Conformal Map

6.10 Teichm011er Space and Modular Space

6.10.1 Fricke Space Model

6.10.2 Geodesic Spectrum

Problems

……

Part II Algorithms

A Major Algorithms

B Acknowledgement

Reference

Index

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