Anisotropic triangulations via discrete Riemannian Voronoi diagrams

Jean-Daniel Boissonnat 1 Maël Rouxel-Labbé 2 Mathijs Wintraecken 1
1 DATASHAPE - Understanding the Shape of Data
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
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 R2 and on surfaces embedded in R3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of 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 Ω 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.
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Jean-Daniel Boissonnat, Maël Rouxel-Labbé, Mathijs Wintraecken. Anisotropic triangulations via discrete Riemannian Voronoi diagrams. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 2019. ⟨hal-02419460⟩

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