Discrete Differential Geometry (Oberwolfach Seminars Book 38)

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field. Front Matter....Pages i-x Front Matter....Pages 1-1 Surfaces from Circles....Pages 3-35 Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples....Pages 37-56 Designing Cylinders with Constant Negative Curvature....Pages 57-66 On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces....Pages 67-93 Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow....Pages 95-115 The Discrete Green’s Function....Pages 117-133 Front Matter....Pages 135-135 Curves of Finite Total Curvature....Pages 137-161 Convergence and Isotopy Type for Graphs of Finite Total Curvature....Pages 163-174 Curvatures of Smooth and Discrete Surfaces....Pages 175-188 Front Matter....Pages 189-189 Polyhedral Surfaces of High Genus....Pages 191-213 Necessary Conditions for Geometric Realizability of Simplicial Complexes....Pages 215-233 Enumeration and Random Realization of Triangulated Surfaces....Pages 235-253 On Heuristic Methods for Finding Realizations of Surfaces....Pages 255-260 Front Matter....Pages 261-261 What Can We Measure?....Pages 263-273 Convergence of the Cotangent Formula: An Overview....Pages 275-286 Discrete Differential Forms for Computational Modeling....Pages 287-324 A Discrete Model of Thin Shells....Pages 325-337 Back Matter....Pages 339-341 This is the first book on a newly emerging field of discrete differential geometry providing an excellent way to access this exciting area. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. The carefully edited collection of essays gives a lively, multi-facetted introduction to this emerging field. Presents A Collection Of Essays That Cover The Main Topics Of The 2004 Oberwolfach Seminar Which Focused On Discrete Differential Geometry.