Anisotropic triangulations via discrete Riemannian Voronoi diagrams

J.-D. Boissonnat, M. Rouxel-Labbé, M. Wintraecken, SIAM Journal on Computing 48 (2019) 1046–1097.


Journal Article | Published | English
Author
Boissonnat, Jean-Daniel; Rouxel-Labbé, Mael; Wintraecken, MathijsIST Austria
Abstract
The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in $\mathbb{R}^2$ and on surfaces embedded in $\mathbb{R}^3$ as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points $\mathcal{P}$ in a domain $\Omega$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $\mathcal{P}$ to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in $\Omega$ under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened.
Publishing Year
Date Published
2019-05-21
Journal Title
SIAM Journal on Computing
Volume
48
Issue
3
Page
1046-1097
ISSN
eISSN
IST-REx-ID

Cite this

Boissonnat J-D, Rouxel-Labbé M, Wintraecken M. Anisotropic triangulations via discrete Riemannian Voronoi diagrams. SIAM Journal on Computing. 2019;48(3):1046-1097. doi:10.1137/17m1152292
Boissonnat, J.-D., Rouxel-Labbé, M., & Wintraecken, M. (2019). Anisotropic triangulations via discrete Riemannian Voronoi diagrams. SIAM Journal on Computing, 48(3), 1046–1097. https://doi.org/10.1137/17m1152292
Boissonnat, Jean-Daniel, Mael Rouxel-Labbé, and Mathijs Wintraecken. “Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams.” SIAM Journal on Computing 48, no. 3 (2019): 1046–97. https://doi.org/10.1137/17m1152292.
J.-D. Boissonnat, M. Rouxel-Labbé, and M. Wintraecken, “Anisotropic triangulations via discrete Riemannian Voronoi diagrams,” SIAM Journal on Computing, vol. 48, no. 3, pp. 1046–1097, 2019.
Boissonnat J-D, Rouxel-Labbé M, Wintraecken M. 2019. Anisotropic triangulations via discrete Riemannian Voronoi diagrams. SIAM Journal on Computing. 48(3), 1046–1097.
Boissonnat, Jean-Daniel, et al. “Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams.” SIAM Journal on Computing, vol. 48, no. 3, Society for Industrial & Applied Mathematics (SIAM), 2019, pp. 1046–97, doi:10.1137/17m1152292.
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