# Anisotropic triangulations via discrete Riemannian Voronoi diagrams

## TL;DR

This paper introduces a discrete Riemannian Voronoi diagram to create anisotropic triangulations, providing theoretical guarantees for their properties and embedding conditions, advancing computational geometry and mesh generation.

## Contribution

It offers a theoretical analysis of the discrete Riemannian Voronoi diagram, establishing conditions for the equivalence of dual structures and triangulation properties.

## Key findings

- Discrete Riemannian Voronoi diagram approximates the Riemannian Voronoi diagram.
- Conditions are provided for the dual structures to be identical.
- Triangulations can be embedded and straightened under certain conditions.

## 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 $\cal P$ in a domain $\Omega$ equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of $\cal P$ 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.