離散曲面的變分原理

前言

This book consists of mathematical and algorithmic studies of geometry of polyhedralsurfaces based on the variations principles. The part of mathematics is based on a lectureseries given by Feng Luo at the Center of Mathematical sciences at Zhejiang Univer-sity, China, in June and July 2006. The algorithmic theory and applications to computergraphic are based on the work of Xianfeng Gu and are written by him. The task of writingthe part of mathematics of the note was done by Junfei Dai who prepared them with greatcare and made a number of improvements in the exposition.The aim of this book is to introduce to the students and researchers an emergingfield of polyhedral surface geometry and computer graphics based on variation princi-ples. These variational principles are derived from the derivatives of the cosine law fortriangles. From mathematical point of view, one of the most fascinating identity in low-dimensional polyhedral geometry is the Schlaefli formula. It relates in a simple and el-egant to way the volume, edge lengths and dihedral angles of tetrahedra in the spheresand hyperbolic spaces in dimension 3. The formula can be considered as a foundation of3-dimensional variational principles for triangulated objects. For a long time, mathemati-cians have been considering the Gauss-Bonnet formula as the 2-dimensional counterpartof Schlaefli. The recent breakthrough in this area was due to the work of Colin de Verdierein 1995 who found the first 2-dimensional identity relating edge lengths and inner anglessimilar to the Schlaefli identity. The mathematical work produced in this book can beconsidered as establishing all 2-dimensional counterparts of Schaefli formula. It turns outthere are continuous families of Schlaefli type identities in dimension 2. These identi-ties produce many interesting variational principles for polyhedral surfaces. In the partof mathematics of the book, we are focusing on a study of the rigidity phenomena onpolyhedral surfaces. Some moduli space problems are also discussed in the book.In the part of algorithm of the book, we introduce discrete curvature flow from boththeoretical and practical points of view. Discrete curvature flow is a powerful tool fordesigning metrics by prescribed curvatures. The algorithm maps general surfaces with ar-bitrary topologies to three canonical spaces. Therefore, all geometric problems of surfacesin 3D space are converted to 2D ones. This greatly improves the efficiency and accuracyfor engineering applications. The discrete Ricci flow algorithm, and Ricci energy opti-mization algorithm are rigorous, robust, flexible and efficient. They have been applied forsurface matching, registration, shape classification, shape analysis and many fundamentalaoolications in oractice.

內容概要

This book intends to lead its readers to some of the current topics of research in the geometry of polyhedral surfaces with applications to computer graphics. The main feature of the book is a systematic introduction to geometry of polyhedral surfaces based on the variational principle. The authors focus on using analytic methods in the study of some of the fundamental results and problems on polyhedral geometry, e. g., the Cauchy rigidity theorem, Thurston's circle packing theorem, rigidity of circle packing theorems and Colin de Verdiere's variational principle. With the vast development of the mathematics subject of polyhedral geometry, the present book is the first complete treatment of the subject.

書籍目錄

1 Introduction1.1 Variational Principle and Isoperimetric Problems1.2 Polyhedral Metrics and Polyhedral Surfaces1.3 A Brief History on Geometry of Polyhedral Surface1.4 Recent Works on Polyhedral Surfaces1.5 Some of Our Results1.6 The Method of Proofs and Related Works2 Spherical Geometry and Cauchy Rigidity Theorem2.1 Spherical Geometry and Spherical Triangles2.2 The Cosine law and the Spherical Dual2.3 The Cauchy Rigidity Theorem3 A Brief Introduction to Hyperbolic Geometry3.1 The Hyperboloid Model of the Hyperbolic Geometry3.2 The Klein Model of Hn3.3 The Upper Half Space Model of Hn3.4 The Poincar6 Disc Model Bn of Hn3.5 The Hyperbolic Cosine Law and the Gauss-Bonnet Formula4 The Cosine Law and Polyhedral Surfaces4.1 Introduction4.2 Polyhedral Surfaces and Action Functional of Variational Framework5 Spherical Polyhedral Surfaces and Legendre Transformation5.1 The Space of All Spherical Triangles5.2 A Rigidity Theorem for Spherical Polyhedral Surfaces5.3 The Legendre Transform5.4 The Cosine Law for Euclidean Triangles6 Rigidity of Euclidean Polyhedral Surfaces6.1 A Local and a Global Rigidity Theorem6.2 Rivin's Theorem on Global Rigidity of Curvature7 Polyhedral Surfaces of Circle Packing Type7.1 Introduction7.2 The Cosine Law and the Radius Parametrization7.3 Colin de Verdiere's Proof of Thurston-Andreev Rigidity Theorem7.4 AProofofLeibon's Theorem7.5 A Sketch of a Proof of Theorem 73(c)7.6 Marden-Rodin's Proof Thurston-Andreev Theorem8 Non-negative Curvature metrics and Delaunay Polytopes8.1 Non-negative and Curvature Metrics and Delaunay Condition ..8.2 Relationship between, Curvature and the Discrete Curvature ko8.3 The work of Rivin and Leibon on Delaunay Polyhedral Surfaces9 A Brief Introduction to Teichmiiller Space9.1 Introduction9.2 Hyperbolic Hexagons, Hyperbolic 3-holed Spheres and the Cosine law9.3 Ideal Triangulation of Surfaces and the Length Coordinate of the Teichmuller Spaces9.4 New Coordinates for the Teichmuller Space10 Parameterizatios of Teichmuller spaces10.1 A Proof of Theorem 10.110.2 Degenerations of Hyperbolic Hexagons10.3 A Proof of Theorem 10.211 Surface Ricci Flow11.1 Conformal Deformation11.2 Surface Ricci Flow12 Geometric Structure12.1 (X, G) Geometric Structure12.2 Affine Structures on Surfaces12.3 Spherical Structure12.4 Euclidean Structure12.5 Hyperbolic Structure12.6 Real Projective Structure13 Shape Acquisition and Representation13.1 Shape Acquisition13.2 Triangular Meshes13.3 Half-Edge Data Structure14 Discrete Ricci Flow14.1 Circle Packing Metric14.2 Discrete Gaussian Curvature14.3 Discrete Surface Ricci Flow14.4 Newton's Method14.5 Isometric Planar Embedding14.6 Surfaces with Boundaries14.7 Optimal Parameterization Using Ricci flow15 Hyperbolic Ricci Flow15.1 Hyperbolic Embedding15.1.1 Embedding One Face15.1.2 Hyperbolic Embedding of the Universal Covering Space15.2 Surfaces with BoundariesReferenceIndex

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《離散曲面的變分原理(英文版)》由高等教育出版社出版。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.

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