The Voronoi diagram is a fundamental geometric structure that encodes proximity information. Given a set of geometric objects, called sites, their Voronoi diagram is a subdivision of the underlying space into maximal regions, such that all points within one region have the same nearest site. Problems in diverse application domains (such as VLSI CAD, robotics, facility location, etc.) demand var...

Abstract. We study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular, the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the error of the ...

In this paper an on-line distributed algorithm, namely, Voronoi Partition-based Coverage (VPC), is proposed to solve the problem of complete coverage of a geographical region with known boundary and unknown obstacles using multiple robots. The region is decomposed into Voronoi cells based on the initial positions of robots. Each robot is responsible for covering its Voronoi cell. It is shown th...

We define the spatial constraints and objective function for weight-proportional partitioning of information spaces. We evaluate existing methods in light of these definitions and find that none performs as desired. We then formulate an alternative approach based on an adaptive version of the multiplicatively weighted Voronoi diagram; i.e., the diagram’s weights are computed based on a set of p...

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) a...

G. Leibon and D. Letscher showed that for general and sufficiently dense point set its Delaunay triangulation and Voronoi diagram in Riemannian manifold exist. They also proposed an algorithm to construct them for a given set. In this paper we estimate the necessary number of points for computing the Voronoi diagram in the manifold by using sectional curvature of the manifold. Moreover, we show...

Consider a dense sampling S of the smooth boundary of a planar shape O. We show that the medial axis of the union of Voronoi balls centered at Voronoi vertices inside O has a particularly simple structure: it is the union of all Voronoi vertices inside O and the Voronoi edges connecting them. Therefore, the medial axis of the union of these inner balls can be computed more efficiently and robus...

Despite of many important applications in various disciplines from sciences and engineering, Voronoi diagram for spheres in a 3-dimensional Euclidean distance has not been studied as much as it deserves. In this paper, we present an edge-tracing algorithm to compute the Euclidean Voronoi diagram of 3-dimensional spheres in O( ) in the worst-case, where is the number of edges of Voronoi diagram ...

In this paper, we introduce the fuzzy Voronoi diagram as an extension of the Voronoi diagram. We assume Voronoi sites to be fuzzy points and then define the Voronoi diagram for this kind of sites, then we provide an algorithm for computing this diagram based on Fortune’s algorithm which costs O(n log n) time. Also we introduce the fuzzy Voronoi diagram for a set of fuzzy circles, rather than fu...

We study a probabilistic algorithm for the computation of the centroidal Voronoi tessellation which is a Voronoi tessellation of a given set such that the associated generating points are centroids (centers of mass) of the corresponding Voronoi regions. We discuss various issues related to the implementation of the algorithm and provide numerical results. Some measures to improve the performanc...