Computational Aspects of Discrete Minimal Surfaces

In differential geometry the study of smooth submanifolds with distinguished curvature properties has a long history and belongs to the central themes of this Þeld. Modern work on smooth submanifolds, and on surfaces in particular, relies heavily on geometric and analytic machinery which has evolved over hundreds of years. However, non-smooth surfaces are also natural mathematical objects, even though there is less machinery available for studying them. Consider, for example , the pioneering work on polyhedral surfaces by the Russian school around Alexandrov [1], or Gromov's approach of doing geometry using only a set with a measure and a measurable distance function [10]. Also in other Þelds, for example in computer graphics and numerics, we nowadays encounter a strong need for a discrete differential geometry of arbitrary meshes. These tutorial notes introduce the theory and computation of discrete minimal surfaces which are characterized by variational properties , and are based on a part of the authors Habilitationsschrift [27]. In Section 1 we introduce simplicial surfaces and their function spaces. Laplace-Beltrami harmonic maps and the solution of the discrete Cauchy-Riemann equations are introduced on simplicial surfaces in Section 2. These maps are the basis for an interative algorithm to compute discrete minimal and constant mean curvature surfaces which is discussed in Section 3. There we deÞne the discrete mean curvature operator, derive the associate family of discrete minimal surfaces in terms of conforming and non-conforming triangles meshes, and present some recently discovered complete discrete surfaces, the family of discrete catenoids and helicoids. Polyhedral meshes belong to the most basic structures for the representation of geometric shapes not only in numerics and computer graphics. Especially the Þniteness of the set of vertices and of their combinatorial relation makes them an ideal tool to reduce inÞnite dimensional problems to Þnite problems. In this section we will review the basic combinatorial and topological deÞnitions and state some of their differential geometric properties. In practice, a variety of different triangle and other polyhedral meshes are used. In this introduction we restrict ourselves to simplicial complexes , or conforming meshes, where two polygons must either be disjoint or have a common vertex or a common edge. Or for short, a polygon is not allowed to contain a vertex of another polygon in the interior of one of its edges. This restriction avoids discontinuity problems in the shape, so-called hanging nodes. Further, we restrict our discussion to …