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A curvature theory for discrete surfaces based on mesh parallelity

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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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Authors and Affiliations

  1. Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Alexander I. Bobenko

  2. Geometric Modeling and Industrial Geometry, TU Wien, Vienna, Austria

    Helmut Pottmann

  3. King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia

    Helmut Pottmann

  4. Institut für Geometrie, TU Graz, Kopernikusgasse 24, 8010, Graz, Austria

    Johannes Wallner

Authors
  1. Alexander I. Bobenko
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  2. Helmut Pottmann
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  3. Johannes Wallner
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Correspondence to Johannes Wallner.

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Bobenko, A.I., Pottmann, H. & Wallner, J. A curvature theory for discrete surfaces based on mesh parallelity. Math. Ann. 348, 1–24 (2010). https://doi.org/10.1007/s00208-009-0467-9

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  • DOI: https://doi.org/10.1007/s00208-009-0467-9

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