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Register a new userDelaunay Hodge Star
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Added by
Anil Hirani
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.
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We study metrics that assess how close a triangulation is to being a Delaunay triangulation, for use in contexts where a good triangulation is desired but constraints (e.g., maximum degree) prevent the use of the Delaunay triangulation itself. Our near-Delaunay metrics derive from common Delaunay properties and satisfy a basic set of design criteria, such as being invariant under similarity transformations. We compare the metrics, showing that each can make different judgments as to which triangulation is closer to Delaunay. We also present a preliminary experiment, showing how optimizing for these metrics under different constraints gives similar, but not necessarily identical results, on random and constructed small point sets.
We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced provided the transition functions are bi-Lipschitz with a constant close to 1, and the sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the ouput complex is also a Delaunay triangulation of its vertices with respect to this piecewise flat metric.
We present an algorithm that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions. There is also a guarantee on the quality of the simplices: they cannot be too flat. The algorithm provides an alternative tool to the weighting or refinement methods to remove poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation.
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