A curvature theory for discrete surfaces based on mesh parallelity |
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Authors: | Alexander I Bobenko Helmut Pottmann Johannes Wallner |
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Institution: | 1. Institut für Mathematik, TU Berlin, Stra?e des 17. Juni 136, 10623, Berlin, Germany 2. Geometric Modeling and Industrial Geometry, TU Wien, Vienna, Austria 3. King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia 4. Institut für Geometrie, TU Graz, Kopernikusgasse 24, 8010, Graz, Austria
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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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