计算共形几何

计算共形几何

顾险峰, 丘成桐, 著

出版社:高等教育出版社

年代:2007

定价:52.0

书籍简介:

本书详尽介绍了计算共形几何的理论与方法及其在计算机图形学、计算机视觉和医学图像方面的应用。计算共形几何是一门结合了代数拓扑、微分几何、黎曼面理论与计算机科学的新兴交叉学科。丘成桐教授及其学生顾险峰是该学科的主要奠基人,他们发明的顾-丘算法在历史上首次使得一般曲面的共形结构可以计算。计算共形几何方法已成为基本的几何工具,已被广泛应用于曲面参数化、曲面分类、曲面匹配、虚拟脑图、虚拟肠镜、流形样条、文理生成等诸多方面。本书配有详尽的程序示例,适合在图形学、几何学领域的高年级本科生、研究生及研究人员参考。

书籍目录:

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

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

内容摘要:

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.

书籍规格:

书籍详细信息
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9787040231892
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出版地北京出版单位高等教育出版社
版次1版印次1
定价(元)52.0语种英文
尺寸23装帧精装
页数印数 3000

书籍信息归属:

计算共形几何是高等教育出版社于2008.01出版的中图分类号为 O18 的主题关于 计算几何:共形微分几何-英文 的书籍。