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Table of Contents
Intro; Preface; Contents; 1 Introduction; 1.1 Riemann Mapping Theorem; 1.2 Holomorphic Differential; 1.3 Uniformization Theorem; 1.4 Shape Space; 1.5 Geometric Structures; Part I Computational Algorithms; 2 Topological Algorithms; 2.1 Halfedge Data Structure; 2.2 Cut Graph and Fundamental Domain; 2.3 Homotopy and Homology Group Basis; 2.4 Canonical Homotopy Group Generators; 3 Harmonic Map; 3.1 Discrete Harmonic Energy; 3.2 Topological Disk Surfaces; 3.3 Topological Sphere Surfaces; 3.4 Riemann Mapping; 4 Harmonic and Holomorphic Forms; 4.1 Characteristic Forms; 4.2 Cohomology Basis
4.3 Harmonic 1-Form4.4 Hodge Star Operator; 4.5 Holomorphic 1-Form; 5 Discrete Ricci Flow; 5.1 Discrete Background; 5.2 Discrete Surface Ricci Flow; 5.3 Isometric Planar Embedding; 5.4 Hyperbolic Ricci Flow; 5.5 Isometric Hyperbolic Embedding; Part II Engineering Applications; 6 Computer Graphics; 6.1 Introduction; 6.2 Global Surface Parametrization; 6.2.1 Optimal Global Conformal Parametrization Using Discrete One-Form; 6.2.2 Global Conformal Parameterization Using Discrete Euclidean Ricci Flow; 6.3 Uniform Remeshing; 6.3.1 Voronoi Diagram in Different Spaces
6.3.2 Centroid Voronoi Tessellation in Different Spaces6.3.3 CVT Energy in Different Spaces; 6.3.4 Computing Centroidal Voronoi Tessellations; 6.3.5 Uniform Remeshing and Partition; 6.4 Metric-Driven RoSy Fields Design; 6.4.1 Theory of Compatibility; 6.4.2 Computing N-RoSy Fields; 6.4.3 Remeshing; 6.4.4 Celtic Knot on Surfaces; 6.4.5 Pen-and-Ink Sketching of Surfaces; 6.5 Computing Shortest Homotopic Cycles on Polyhedral Surfaces; 6.5.1 Geodesic Uniqueness; 6.5.2 Fuchsian Group Method; 6.5.3 Birkhoff Curve Shortening Method; 6.5.4 Extremal Quasiconformal Mapping; 6.5.5 Homotopy Detection
6.6 Summary and Further Reading7 Computer Vision; 7.1 Introduction; 7.2 Teichmuller Shape Space; 7.2.1 Luo Coordinates; 7.2.2 Fenchel-Nielsen Coordinates; 7.2.3 Robustness of Teichmuller Space Coordinates; 7.2.4 Surface Indexing and Classification; 7.3 3D Facial Shape Index; 7.3.1 Generalized Discrete Ricci Flow; 7.3.2 Doubly Connected Domain; 7.3.3 Multiply Connected Domain; 7.4 Shape Signature; 7.4.1 Beltrami Equation; 7.4.2 Conformal Module; 7.4.3 Holomorphic Differentials; 7.4.4 Conformal Welding; 7.4.5 Computing Shape Signatures of Planar Domains; 7.4.6 Shape Clustering
7.5 Summary and Further Reading8 Geometric Modeling; 8.1 Introduction; 8.2 Triangular B-Splines; 8.2.1 Definition; 8.2.2 Properties; 8.2.3 Surface Reconstruction; 8.3 Theoretical Foundation of Manifold Splines; 8.3.1 Definitions and Concepts; 8.3.2 Equivalence to Affine Atlas; 8.3.3 Existence; 8.4 Constructing Affine Atlas; 8.4.1 Computing Affine Structure Using Holomorphic 1-form; 8.4.2 Computing Affine Structure Using Discrete Ricci Flow; 8.4.3 Computing Affine Structure Using Polycube Maps; 8.5 Extending Planar Splines to Manifold Domains; 8.5.1 Manifold Triangular B-spline
4.3 Harmonic 1-Form4.4 Hodge Star Operator; 4.5 Holomorphic 1-Form; 5 Discrete Ricci Flow; 5.1 Discrete Background; 5.2 Discrete Surface Ricci Flow; 5.3 Isometric Planar Embedding; 5.4 Hyperbolic Ricci Flow; 5.5 Isometric Hyperbolic Embedding; Part II Engineering Applications; 6 Computer Graphics; 6.1 Introduction; 6.2 Global Surface Parametrization; 6.2.1 Optimal Global Conformal Parametrization Using Discrete One-Form; 6.2.2 Global Conformal Parameterization Using Discrete Euclidean Ricci Flow; 6.3 Uniform Remeshing; 6.3.1 Voronoi Diagram in Different Spaces
6.3.2 Centroid Voronoi Tessellation in Different Spaces6.3.3 CVT Energy in Different Spaces; 6.3.4 Computing Centroidal Voronoi Tessellations; 6.3.5 Uniform Remeshing and Partition; 6.4 Metric-Driven RoSy Fields Design; 6.4.1 Theory of Compatibility; 6.4.2 Computing N-RoSy Fields; 6.4.3 Remeshing; 6.4.4 Celtic Knot on Surfaces; 6.4.5 Pen-and-Ink Sketching of Surfaces; 6.5 Computing Shortest Homotopic Cycles on Polyhedral Surfaces; 6.5.1 Geodesic Uniqueness; 6.5.2 Fuchsian Group Method; 6.5.3 Birkhoff Curve Shortening Method; 6.5.4 Extremal Quasiconformal Mapping; 6.5.5 Homotopy Detection
6.6 Summary and Further Reading7 Computer Vision; 7.1 Introduction; 7.2 Teichmuller Shape Space; 7.2.1 Luo Coordinates; 7.2.2 Fenchel-Nielsen Coordinates; 7.2.3 Robustness of Teichmuller Space Coordinates; 7.2.4 Surface Indexing and Classification; 7.3 3D Facial Shape Index; 7.3.1 Generalized Discrete Ricci Flow; 7.3.2 Doubly Connected Domain; 7.3.3 Multiply Connected Domain; 7.4 Shape Signature; 7.4.1 Beltrami Equation; 7.4.2 Conformal Module; 7.4.3 Holomorphic Differentials; 7.4.4 Conformal Welding; 7.4.5 Computing Shape Signatures of Planar Domains; 7.4.6 Shape Clustering
7.5 Summary and Further Reading8 Geometric Modeling; 8.1 Introduction; 8.2 Triangular B-Splines; 8.2.1 Definition; 8.2.2 Properties; 8.2.3 Surface Reconstruction; 8.3 Theoretical Foundation of Manifold Splines; 8.3.1 Definitions and Concepts; 8.3.2 Equivalence to Affine Atlas; 8.3.3 Existence; 8.4 Constructing Affine Atlas; 8.4.1 Computing Affine Structure Using Holomorphic 1-form; 8.4.2 Computing Affine Structure Using Discrete Ricci Flow; 8.4.3 Computing Affine Structure Using Polycube Maps; 8.5 Extending Planar Splines to Manifold Domains; 8.5.1 Manifold Triangular B-spline