The Seven Dimensional Perfect Delaunay Polytopes and Delaunay Simplices

For a lattice L of ℝn, a sphere S(c, r) of center c and radius r is called empty if for any v ∈ L we have. Then the set S(c, r) ∩ L is the vertex set of a Delaunay polytope P = conv(S(c, r) ∩ L). A Delaunay polytope is called perfect if any aõne transformation ø such that ø(P) is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.Perfect Delaunay polytopes are remarkable structures that exist only if n = 1 or n ≥ 6, and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension seven, which allows us to find that there are only two perfect Delaunay polytopes: 321, which is a Delaunay polytope in the root lattice E7, and the Erdahl Rybnikov polytope.We then use this classification in order to get the list of all types of Delaunay simplices in dimension seven and found that there are eleven types.

Using Voronoi Tessellation and Delaunay Triangulation to Evaluate Spatial Uniformity of Particle Distribution

Evaluation methods for the spatial distribution uniformity of a large number of particles have important applications in a range of scientific and industrial problems. We report a quantitative image-processing method to evaluate the spatial distribution uniformity of particles using the Voronoi tessellation and Delaunay triangulation. Given the geometric particle centroids, we constructed the Voronoi tessellation and Delaunay triangulation. For each of the Voronoi cell, the volume, facet area, and number of neighbors (facets) were calculated. For the Delaunay simplices, the volume and the distance between neighbors were calculated and the distributive characteristics were compared between different particle patterns. Simulation results demonstrated that distributions of the above metrics were numerically related to the uniformity of the particle spatial displacement. More studies are needed to further validate the results and refine the method.

THE STABILITY OF DELAUNAY TRIANGULATIONS

We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of δ-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation remains unchanged.