词条 | 计算共形几何 |
释义 | 图书信息出版社: 人民教育出版社; 第1版 (2008年1月1日) 外文书名: Computational Conformal Geometry 精装: 510页 正文语种: 简体中文 开本: 16 ISBN: 9787040231892 条形码: 9787040231892 尺寸: 24 x 17 x 2 cm 重量: 558 g 内容简介《计算共形几何(英文版)》是首次对中国基础教育均衡发展作理论和实证研究的专著。以广阔的理论视野,首次从宏观、中观、微观三个维度、十五个内涵指标分析入手,对教育均衡发展进行深入的理论探索和科学阐释,提出了具有独到见解的教育均衡发展理论体系。构建了教育均衡发展的基本体系框架,首次提出了基础教育均衡发展四个阶段的发展理论。首次把指数引入教育均衡分析,构建了分析研究教育均衡发展的指数体系和教育均衡发展指数。基础教育均衡发展是现代教育发展的新境界,是教育未来发展的方向。 目录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 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 3.1.1 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 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 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 Rad6'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 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 9.1.1 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 l1.5 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 Poincare 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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