George Voronoi (1908, 1909) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with Ltype domains, often called domains of Delaunay type. The first method is important in studies of dense lattice packings of spheres. The second method provides the key tools for finding the least dense lattice coverings with equal spheres ...

Voronoi diagrams are fundamental data structures in computational geometry with several applications on different fields inside and outside computer science. This paper shows a CUDA algorithm to compute Voronoi diagrams on a 2D image where the distance between points cannot be directly computed in the euclidean plane. The proposed method extends an existing Dijkstra-based GPU algorithm to treat...

We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation. We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges. The clipping itself i...

We present novel exploration algorithms and control law that enable the construction of Voronoi diagram in an initially unknown area using one vehicle equipped with range scanners. Our control law and exploration algorithms are provably complete, i.e., convergence is guaranteed. The control laws use range measurements to make the vehicle track Voronoi edges between two obstacles. The exploratio...

A packing of congruent balls of unit radius in three-dimensional Euclidean space is determined by and is identified with the set Λ of centers of the balls. A packing Λ determines a region called the Voronoi cell around each ball. The Voronoi cell Ω(Λ, v) around a ball at v ∈ Λ consists of points of space that are closer to v than to any other w ∈ Λ. The Voronoi cell is a convex polyhedron conta...

In this review paper, we shall discuss some recent results concerning several models of random geometric graphs, including the Gilbert disc model Gr , the k-nearest neighbour model G nn k and the Voronoi model GP . Many of the results concern finite versions of these models. In passing, we shall mention some of the applications to engineering and biology.

One of most famous theorems in computational geometry is the duality between Voronoi diagram and Delaunay triangulation in Euclidean space. This paper proposes an extension of that theorem to the Voronoi diagram and Delaunay-type triangulation in dually at space. In that space, the Voronoi diagram and the triangulation can be computed e ciently by using potential functions. We also propose high...

Voronoi diagrams are widely used to represent geographical distributions of information, but they are not readily stacked in a hierarchical fashion. We propose a simple mechanism whereby each index Voronoi cell contains the generators of several Voronoi cells in the next lower level. This allows various processes of indexing, paging, visualization and generalization to be performed on various t...

In this paper, we study the problem of multi-robot coverage when the evolution of the robots’ positions is uncertain. More precisely, we use the popular Voronoi coverage control approach [1], which is based on the iterative construction of a centroidal Voronoi tessellation, and ask what happens to the control law and the Voronoi tessellation under actuation and measurement noise. We present a f...

Voronoi diagram and its geometric dual, the Delaunay triangulation, both are practical geometric constructions which have been applied extensively in spatial analysis. Considering the low efficiency of the algorithm of indirectly building Voronoi diagram, this paper proposes an improved Voronoi diagram generation algorithm based on Delaunay triangulation of randomly distributed points in the Eu...