A Discrete Laplace–Beltrami Operator for Simplicial Surfaces |
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Authors: | Alexander I Bobenko Boris A Springborn |
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Institution: | 1.Institut für Mathematik,Technische Universit?t Berlin,Berlin,Germany |
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Abstract: | We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry
of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian
(the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface.
This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The
definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in
piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed,
Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the
edge flipping algorithm terminates also in the setting of piecewise flat surfaces.
Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. |
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Keywords: | Laplace operator Delaunay triangulation Dirichlet energy Simplicial surfaces Discrete differential geometry |
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