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Computational Conformal Geometry


作者:
丘成桐、顾险峰
定价:
49.00元
ISBN:
978-7-04-023189-2
版面字数:
340千字
开本:
16开
全书页数:
276页
装帧形式:
精装
重点项目:
暂无
出版时间:
2007-12-05
读者对象:
学术著作
一级分类:
自然科学
二级分类:
数学与统计
三级分类:
几何学

The launch of this Advanced Lectures in Mathematics series is aimed at keepingmathematicians informed of the latest developments in mathematics, as well asto aid in the learning of new mathematical topics by students all over the world.Each volume consists of either an expository monograph or a collection of signifi-cant introductions to important topics. This series emphasizes the history andsources of motivation for the topics under discussion, and also gives an overviewof the current status of research in each particular field. These volumes are thefirst source to which people will turn in order to learn new subjects and to dis-cover the latest results of many cutting-edge fields in mathematics.

  • 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

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