Anisotropic Triangulations via Discrete Riemannian Voronoi Diagrams

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 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 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.

Learning Proofs of Motion Planning Infeasibility

“…This section provides a brief introduction of the permutahedral representation [21] used in the manifold tracing algorithm with Coxeter triangulation. Coxeter triangulation is a way to triangulate any n-dimensional Euclidean space R n , and is defined using root systems [39].…”

“…The entire triangulation and collision checking step runs on GPU, which ensures minimum data transfer between CPU and GPU. We design the triangulation algorithm based on manifold tracing with Coxeter triangulation [21] to fit the computation architecture of GPUs. To handle the significant memory requirements within the constraints of limited GPU memory during triangulation, we design the algorithm to support batch triangulation.…”

Computation of Compact Distributions of Discrete Elements

“…The Voronoi diagram over curved surfaces [27,28] has also been researched. Some researchers even conducted work on the Voronoi diagram in non-Euclidean space metrics (like Riemannian geometry metric) [29,30].…”

“…To generate a Voronoi diagram, the iteration process must be repeated and every iteration process needs to loop through all the distribution points, which is quite time-consuming. Hoff et al [30] proposed a method of converting the Voronoi generation process into a graphical rendering process. First, the distribution of the initial points is determined, and then a circular cone with height equal to the radius is constructed at each initial point location.…”

The reach of subsets of manifolds

“…One particular motivation is the search for anisotropic triangulations for numerical partial differential equations, see e.g. [24][25][26][27][28][29][30][31], based on Riemannian metrics as in [32] but with interfaces or boundaries. To triangulate these complicated spaces one needs to understand the geometry of the interfaces and boundaries with respect to the Riemannian metric.…”