流形上的几何三角剖分和离散拉普拉斯算子:更新

IF 0.4 4区 计算机科学 Q4 MATHEMATICS
David Glickenstein
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引用次数: 0

摘要

本文利用加权三角剖分技术研究了分段欧几里得流形上拉普拉斯算子的离散形式。给定沿边界粘在一起的简单欧几里得集合,可以通过考虑顶点上的权值来构造庞加莱乌对偶上的几何结构。我们证明了这等价于在顶点处指定球面半径,在边缘处指定广义交角,或者通过指定某种划分边缘的方法。这种几何结构产生了作用于顶点上的函数的离散拉普拉斯算子。我们详细地研究了这些几何结构,考虑了当对偶体积是非简并的,它对应于2维的加权Delaunay三角剖分,以及如何找到这样的非简并加权三角剖分。最后,我们简要地讨论离散黎曼流形的可能性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Geometric triangulations and discrete Laplacians on manifolds: An update

This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the Poincaré dual may be constructed by considering weights at the vertices. We show that this is equivalent to specifying sphere radii at vertices and generalized intersection angles at edges, or by specifying a certain way of dividing the edges. This geometric structure gives rise to a discrete Laplacian operator acting on functions on the vertices. We study these geometric structures in some detail, considering when dual volumes are nondegenerate, which corresponds to weighted Delaunay triangulations in dimension 2, and how one might find such nondegenerate weighted triangulations. Finally, we talk briefly about the possibilities of discrete Riemannian manifolds.

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来源期刊
CiteScore
1.60
自引率
16.70%
发文量
43
审稿时长
>12 weeks
期刊介绍: Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.
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