Advances in Discrete Differential Geometry

Description

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing. Front Matter....Pages i-x Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization....Pages 1-56 Discrete Complex Analysis on Planar Quad-Graphs....Pages 57-132 Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices....Pages 133-149 Numerical Methods for the Discrete Map \(Z^a\) ....Pages 151-176 A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths....Pages 177-195 Complex Line Bundles Over Simplicial Complexes and Their Applications....Pages 197-239 Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes....Pages 241-265 Vertex Normals and Face Curvatures of Triangle Meshes....Pages 267-286 S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature....Pages 287-308 Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization....Pages 309-345 On the Lagrangian Structure of Integrable Hierarchies....Pages 347-378 On the Variational Interpretation of the Discrete KP Equation....Pages 379-405 Six Topics on Inscribable Polytopes....Pages 407-419 DGD Gallery: Storage, Sharing, and Publication of Digital Research Data....Pages 421-439