Minimal surfaces from circle patterns: Geometry from combinatorics

Abstract

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, con stant mean curvature, isothermic surfaces, etc. Usually, one can suggest vari ous discretizations with the same continuous limit which have quite different geometric properties. The goal of discrete differential geometry is to find a dis cretization which inherits as many essential properties of the smooth geometry as possible. Our discretizations are based on quadrilateral meshes, i.e. we discretize parametrized surfaces. For the discretization of a special class of surfaces, it is natural to choose an adapted parametrization. In this paper, we investigate conformai discretizations of surfaces, i.e. discretizations in terms of circles and spheres, and introduce a new discrete model for minimal surfaces. See Figures 1 and 2. In comparison with direct methods (see, in particular, [23]), leading