Advanced Lectures in Mathematics (ALM 3): Computational Conformal Geometry

Author: Xianfeng David Gu
Language: English
ISBN/ISSN: 7040231892
Published on: 2008-01
Hardcover

Computational conformal geometry is an emerging inter-disciplinary field, which applies algebraic topology, differential geometry and Riemann surface theories in geometric modeling, computer graphics, computer vision, medical imaging, visualization, scientific computation and many other engineering fields. This textbook is the first one to thoroughly introduce both theoretical foundations and practical algorithms of Computational conformal geometry, which have the direct applications in engineering and digital geometric processing, such as surface parameterization, surface matching, brain mapping, 3D face recognition and identification, facial expression animation, dynamic face tracking, mesh-spline conversion and so on.

1.1 Overview of Theories
1.2 Algorithms for Computing Conformal Mappings
1.3 Applications
Further Readings
Part I Theories
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
Homology and Cohomology
3.1 Simplicial Homology
3.2 Cohomology
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
5.3 Orthonormal Movable Frame
5.4 Covariant Differentiation
5.5 Gauss-Bonnet Theorem
5.6 Index Theorem of Tangent Vector Field
5.7 Minimal Surface
Problems
Riemann Surface
6.1 Riemann Surface
6.2 Riemann Mapping Theorem
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
6.10 Teichm011er Space and Modular Space
Problems
Harmonic Maps and Surface Ricci Flow
7.1 Harmonic Maps of Surfaces
7.2 Surface Ricci Flow
Problems
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
Topological Algorithms
9.1 Triangular Meshes
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
l 1.5 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
12.10 Hyperbolic Ricci Flow for Surfaces with Boundaries
Further Readings
Problems
A Major Algorithms
B Acknowledgement
Reference
Index