Minimal surfaces from circle patterns: Geometry from combinatorics

摘要

The theory of polyhedral surfaces and, more generally, the field of discrete differential geometry are presently emerging on the border of differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with a finite number of elements (polyhedra), the theory of polyhedral surfaces aims at a development of discrete equivalents of the geometric notions and methods of surface theory. The latter appears then as a limit of the refinement of the discretization. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and visualization. One of the central problems of discrete differential geometry is to find proper discrete analogues of special classes of surfaces, such as minimal, constant mean curvature, isothermic surfaces, etc. Usually, one can suggest various discretizations with the same continuous limit which have quite different geometric properties. The goal of discrete differential geometry is to find a discretization which inherits as many essential properties of the smooth geometry as possible.