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The construction of anisotropic triangulations is desirable for variousapplications, such as the numerical solving of partial differential equationsand the representation of surfaces in graphics. To solve this notoriouslydifficult problem in a practical way, we introduce the discrete RiemannianVoronoi diagram, a discrete structure that approximates the Riemannian Voronoidiagram. This structure has been implemented and was shown to lead to goodtriangulations in $\mathbb{R}^2$ and on surfaces embedded in $\mathbb{R}^3$ asdetailed in our experimental companion paper.  In this paper, we study theoretical aspects of our structure. Given a finiteset of points $\cal P$ in a domain $\Omega$ equipped with a Riemannian metric,we compare the discrete Riemannian Voronoi diagram of $\cal P$ to itsRiemannian Voronoi diagram. Both diagrams have dual structures called thediscrete Riemannian Delaunay and the Riemannian Delaunay complex. We provideconditions that guarantee that these dual structures are identical. It thenfollows from previous results that the discrete Riemannian Delaunay complex canbe embedded in $\Omega$ under sufficient conditions, leading to an anisotropictriangulation with curved simplices. Furthermore, we show that, under similarconditions, the simplices of this triangulation can be straightened.

Anisotropic triangulations via discrete Riemannian Voronoi diagrams