92 research outputs found
A discrete Laplace-Beltrami operator for simplicial surfaces
We define a discrete Laplace-Beltrami operator for simplicial surfaces. 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.Comment: 18 pages, 6 vector graphics figures. v2: Section 2 on Delaunay
triangulations of piecewise flat surfaces revised and expanded. References
added. Some minor changes, typos corrected. v3: fixed inaccuracies in
discussion of flip algorithm, corrected attributions, added references, some
minor revision to improve expositio
Weighted Triangulations for Geometry Processing
In this article we investigate the use of weighted triangulations as discrete, augmented approximations of surfaces for digital geometry processing. By incorporating a scalar weight per mesh vertex, we introduce a new notion
of discrete metric that defines an orthogonal dual structure for arbitrary triangle meshes and thus extends weighted Delaunay triangulations to surface meshes. We also present alternative characterizations of this primal-dual
structure (through combinations of angles, areas, and lengths) and, in the process, uncover closed-form expressions of mesh energies that were previously known in implicit form only. Finally, we demonstrate how weighted triangulations provide a faster and more robust approach to a series of geometry processing applications, including the generation of well-centered meshes, self-supporting surfaces, and sphere packing
Combinatorial -curvatures and -flows on polyhedral surfaces, I
We introduce combinatorial -curvature for piecewise linear metrics on
polyhedral surfaces, which is a generalization of the classical combinatorial
curvature on polyhedral surfaces. Then we prove the global rigidity of
-curvature with respect to the discrete conformal factors. To study the
corresponding Yamabe problem of -curvature, we introduce the
combinatorial -Yamabe flow and combinatorial -Calabi flow for
piecewise linear metrics on surfaces. To handle the possible singularities
along the flows, we do surgery on the flows by flipping. Then we prove that if
, there exists a piecewise linear metric with constant
combinatorial -curvature on a polyhedral surface , which is a
parameterized generalization of Gu-Luo-Sun-Wu's discrete uniformization
theorem. We further prove that the combinatorial -Yamabe flow and
-Calabi flow with surgery exists for all time and converges to a
piecewise linear metric with constant combinatorial -curvature for any
initial piecewise linear metric if .Comment: arXiv admin note: text overlap with arXiv:1806.0216
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Extremum problems for eigenvalues of discrete Laplace operators
The P1 discretization of the Laplace operator on a triangulated
polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the P1 discretization of the
Laplace operator. Among all triangles, an equilateral triangle has the maximal first positive eigenvalue. Among all cyclic quadrilateral, a square has the
maximal first positive eigenvalue. Among all cyclic n-gons, a regular one has
the minimal value of the sum of all positive eigenvalues and the minimal value
of the product of all positive eigenvalues.Keywords: extremum, discrete Laplace operator, spectra, cyclic polygo
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