Computational Aspects of Discrete Minimal Surfaces

in: Global Theory of Minimal Surfaces, Proc. of the Clay Mathematics Institute 2001 Summer School, David Hoffman (Ed.), CMI/AMS (2005).

Konrad Polthier


In differential geometry the study of smooth submanifolds with distinguished curvature properties has a long history and belongs to the central themes of this field. 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 [Aleksandrov/Zalgaller67Intrinsic], or Gromov's approach of doing geometry using only a set with a measure and a measurable distance function [Gromov99Metric]. Also in other fields, for example in computer graphics and scientific computing, 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 [Polthier02Habilitationsschrift]. In Section 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 . These maps are the basis for an iterative algorithm to compute discrete minimal and constant mean curvature surfaces which is discussed in Section . There we define 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.


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