Discrete surfaces from circles and spheres
Project leader: A. Bobenko
One of the central Problems of discrete differential geometry is to construct proper discrete analogues of special classes of surfaces (such as minimal, constant mean curvature, isothermic etc.). The main goal of this project is to investigate discretizations of surfaces in terms of circles and spheres. In comparison with direct methods, which usually lead to triangle meshes, these more sophisticated conformal discretizations have essential advantages: They respect conformal properties of surfaces, possess a maximum principle, etc.
The combinatorics of such discretizations is far from arbitrary. Instead it strongly reflexts intrinsic (differential-)geometric structure; in exceptional cases, this even leads to uniqueness theorems for geometry of surfaces with (suitably) given combinatorics. The construction of discrete S-isothermic (in particular of discrete minimal) surfaces, with its "passage throug graph theory" is a striking instance of this Koebe phenomenon.
Another central point of the project is the search for and investigation of variational principles for polyhedral surfaces. Variational principles simplify the theory and lead to new theoretical results and to natural geometric constructions. A striking example of this is a discrete version of the Willmore energy for simplical surfaces given in terms of circles. Minimization of this energy gives a new insight to the classical problem of finding a spherical representation for a given combinatorial polyhedron. For simplicial surfaces it leads to a new notion of an optimal triangulation which is a generalization of the Delaunay triangulation for spheres.
A discrete Schwarz minimal surface
This project has expired and is continued by the follow-up project
Discrete differential geometry of surfaces: Special classes and deformations.