Discrete Minimal Surfaces of Koebe Type

Abstract

There is an increasing interest to find suitable discrete analogs for known geometric notions and shapes like minimal surfaces. In this article, we consider parametrized surfaces which lead to quadrilateral meshes. In particular, we choose a parametrization where the second fundamental form is diagonal. In addition to the discrete surface we consider a line congruence at the vertices which can be interpreted as a discrete Gauss map. This easily leads to parallel offset meshes. Comparing the areas of two such parallel planar quadrilaterals can then be used to define discrete mean and Gaussian curvature analogously as in the smooth case. This approach leads to a simple notion of discrete minimal surfaces which contains several known definitions as special cases. We especially focus on discrete minimal surfaces whose discrete Gauss map is given by a Koebe polyhedron, i.e. a polyhedral surface with edges tangent to the unit sphere. This case is closely connected to the theory of S-isothermic discrete minimal surfaces. We remind the construction scheme and present analogs for several known smooth minimal surface.