هندسه دیسکریت دیفرانسیل: ساختار قابل انتگرال

An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best? This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question ''How do we discretize differential geometry?'' arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful. Discrete discrete differential geometry - Integrable Systems 1 Contents 4 Preface 9 Introduction 11 Chapter 1. Classical Differential Geometry 23 1.1. Conjugate nets 24 1.2. Koenigs and Moutard nets 29 1.3. Asymptotic nets 33 1.4. Orthogonal nets 34 1.5. Principally parametrized sphere congruences 19 41 1.6. Surfaces with constant negative Gaussian curvature 42 1.7. Isothermic surfaces 44 1.8. Surfaces with constant mean curvature 48 1.9. Bibliographical notes 50 Chapter 2. Discretization Principles. Multidimensional Nets1 53 2.1. Discrete conjugate nets (Q-nets) 54 2.2. Discrete line congruences 65 2.3. Discrete Koenigs and Moutard nets 69 2.3.1. Notion of dual quadrilaterals 69 2.4. Discrete asymptotic nets 87 2.5. Exercises 300 2.6. Bibliographical notes 103 Chapter 3. Discretization Principles. Nets in Quadrics 108 3.1. Circular nets 109 3.1.3. Analytic description of circular nets 114 3.2. Q-nets in quadrics 120 3.4. Conical nets 124 3.5. Principal contact element nets 127 3.6. Q-congruences of spheres 131 3.7. Ribaucour congruences of spheres 134 3.8. Discrete curvature line parametrization in Lie, Mobius and Laguerre geometries 136 3.9. Discrete asymptotic nets in Pliicker line geometry 139 3.10. Exercises 141 3.11. Bibliographical notes 144 Chapter 4. Special Classes of Discrete Surfaces 148 4.1. Discrete Moutard nets in quadrics 148 4.2. Discrete K-nets 151 4.3. Discrete isothermic nets 166 4.4. S-isothermicnets 182 4.5. Discrete surfaces with constant curvature 191 4.6. Exercises 200 4.7. Bibliographical notes 307 Chapter 5. Approximation 208 5.1. Discrete hyperbolic systems 208 5.2. Approximation in discrete hyperbolic systems 211 5.3. Convergence of Q-nets 217 5.4. Convergence of discrete Moutard nets 218 5.5. Convergence of discrete asymptotic nets 220 5.6. Convergence of circular nets 221 5.7. Convergence of discrete K-surfaces 226 5.8. Exercises 227 5.9. Bibliographical notes 228 Chapter 6. Consistency as Integrability 230 6.1. Continuous integrable systems 231 6.2. Discrete integrable systems 234 6.3. Discrete 2D integrable systems on graphs 236 6.4. Discrete Laplace type equations 238 6.5. Quad-graphs 239 6.6. Three-dimensional consistency 241 6.7. From 3D consistency to zero curvature representations and Backlund transformations 243 6.8. Geometry of boundary value problems for integrable 2D 248 6.9. 3D consistent equations with noncommutative fields 256 6.10. Classification of discrete integrable 2D systems withfields on vertices. I 260 6.11. Proof of the classification theorem 263 6.12. Classification of discrete integrable 2D systems withfields on vertices. II 273 6.13. Integrable discrete Laplace type equations 277 6.14. Fields on edges: Yang-Baxter maps 282 6.15. Classification of Yang-Baxter maps 287 6.16. Discrete integrable 3D systems 293 6.17. Exercises 300 6.18. Bibliographical notes 307 Chapter 7. Discrete Complex Analysis. Linear Theory 312 7.1. Basic notions of discrete linear complex analysis 312 7.2. Moutard transformation for discrete Cauchy-Riemannequations 315 7.3. Integrable discrete Cauchy-Riemann equations 318 7.4. Discrete exponential functions 321 7.5. Discrete logarithmic function 323 7.6. Exercises 328 7.7. Bibliographical notes 329 Chapter 8. Discrete Complex Analysis. Integrable Circle Patterns 332 8.1. Circle patterns 332 8.2. Integrable cross-ratio and Hirota systems 334 8.3. Integrable circle patterns 337 8.5. Linearization 345 8.6. Exercises 347 8.7. Bibliographical notes 348 Chapter 9. Foundations 352 9.1. Projective geometry 352 9.2. Lie geometry 356 9.3. Mobius geometry 362 9.4. Laguerre geometry 371 9.5. Pliicker line geometry 374 9.6. Incidence theorems 378 Appendix. Solutions of Selected Exercises 390 A.1. Solutions of exercises to Chapter 2 390 A.2. Solutions of exercises to Chapter 3 397 A.3. Solutions of exercises to Chapter 4 398 A.4. Solutions of exercises to Chapter 6 402 Bibliography 406 Notation 420 "An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of Integrable systems. One of the main goals of this book Is to reveal this integrable structure of discrete differential geometry." "The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results. The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question "How do we discretize differential geometry?" arising in their specific field."--Jacket.