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Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams Boissonnat, Jean-Daniel Rouxel-Labbé, Mael Wintraecken, Mathijs Riemannian Geometry Voronoi diagram Delaunay triangulation 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 R^2 and on surfaces embedded in 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 P in a domain Omega 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 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jean-Daniel Boissonnat and Mael Rouxel-Labbé and Mathijs Wintraecken 2017 Is Part Of LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2017.19 urn:nbn:de:0030-drops-72060 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.19 eng https://creativecommons.org/licenses/by/3.0/legalcode