Discrete Differential Geometry (Oberwolfach Seminars Book 38)

Издательство Birkhäuser, 2008, -337 pp. Discrete differential geometry (DDG) is a new and active mathematical terrain where differential geometry (providing the classical theory of smooth manifolds) interacts with discrete geometry (concerned with polytopes, simplicial complexes, etc.), using tools and ideas from all parts of mathematics. DDG aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields such as computer graphics. Discrete differential geometry initially arose from the observation that when a notion from smooth geometry (such as that of a minimal surface) is discretized properly, the discrete objects are not merely approximations of the smooth ones, but have special properties of their own, which make them form a coherent entity by themselves. One might suggest many different reasonable discretizations with the same smooth limit. Among these, which one is the best? From the theoretical point of view, the best discretization is the one which preserves the fundamental properties of the smooth theory. Often such a discretization clarifies the structures of the smooth theory and possesses important connections to other fields of mathematics, for instance to projective geometry, integrable systems, algebraic geometry, or complex analysis. The discrete theory is in a sense the more fundamental one: the smooth theory can always be recovered as a limit, while it is a nontrivial problem to find which discretization has the desired properties. The problems considered in discrete differential geometry are numerous and include in particular: discrete notions of curvature, special classes of discrete surfaces (such as those with constant curvature), cubical complexes (including quad-meshes), discrete analogs of special parametrization of surfaces (such as conformal and curvature-line parametrizations), the existence and rigidity of polyhedral surfaces (for example, of a given combinatorial type), discrete analogs of various functionals (such as bending energy), and approximation theory. Since computers work with discrete representations of data, it is no surprise that many of the applications of DDG are found within computer science, particularly in the areas of computational geometry, graphics and geometry processing. Despite much effort by various individuals with exceptional scientific breadth, large gaps remain between the various mathematical subcommunities working in discrete differential geometry. The scientific opportunities and potential applications here are very substantial. The goal of the Oberwolfach Seminar Discrete Differential Geometry held in May–June 2004 was to bring together mathematicians from various subcommunities working in different aspects of DDG to give lecture courses addressed to a general mathematical audience. The seminar was primarily addressed to students and postdocs, but some more senior specialists working in the field also participated. Part I: Discretization of Surfaces: Special Classes and Parametrizations Surfaces from Circles Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples Designing Cylinders with Constant Negative Curvature On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow The Discrete Green’s Function Part II: Curvatures of Discrete Curves and Surfaces Curves of Finite Total Curvature Convergence and Isotopy Type for Graphs of Finite Total Curvature Curvatures of Smooth and Discrete Surfaces Part III: Geometric Realizations of Combinatorial Surfaces Polyhedral Surfaces of High Genus Necessary Conditions for Geometric Realizability of Simplicial Complexes Enumeration and Random Realization of Triangulated Surfaces On Heuristic Methods for Finding Realizations of Surfaces Part IV: Geometry Processing and Modeling with Discrete Differential Geometry What Can We Measure? Convergence of the Cotangent Formula: An Overview Discrete Differential Forms for Computational Modeling A Discrete Model of Thin Shells 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 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.