Mathematische Annalen

, Volume 348, Issue 1, pp 1–24 | Cite as

A curvature theory for discrete surfaces based on mesh parallelity

  • Alexander I. Bobenko
  • Helmut Pottmann
  • Johannes Wallner
Article

Abstract

We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces’ areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discrete isothermic minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature surfaces, Christoffel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic billiards.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Alexander I. Bobenko
    • 1
  • Helmut Pottmann
    • 2
    • 3
  • Johannes Wallner
    • 4
  1. 1.Institut für Mathematik, TU BerlinBerlinGermany
  2. 2.Geometric Modeling and Industrial Geometry, TU WienViennaAustria
  3. 3.King Abdullah University of Science and TechnologyThuwalSaudi Arabia
  4. 4.Institut für Geometrie, TU GrazGrazAustria

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