It is well known that the complexity, i.e. the number of vertices, edges and faces, of the 3-dimensional Voronoi diagram of n points can be as bad as (n2). It is also known that if the points are chosen Independently Identically Distributed uniformly from a 3-dimensional region such as a cube or sphere, then the expected complexity falls to O(n). In this paper we introduce the problem of analyz...

In this paper we will extend the concept of Voronoi diagrams to parameterized surfuces where distance between two points is defined as infimum over the lengths of surface paths connecting these points. We will present a method to compute Voronoi diagrams on these surfuces.

Voronoi Diagrams Revisited Rolf Klein∗ Elmar Langetepe∗ Zahra Nilforoushan† Abstract Abstract Voronoi diagrams [21] were designed as a unifying concept that should include as many concrete types of diagrams as possible. To ensure that abstract Voronoi diagrams, built from given sets of bisecting curves, are finite graphs, it was required that any two bisecting curves intersect only finitely oft...

This paper considers the interpolation for multi-dimensional data using Voronoi diagrams. Sibson's interpolant is well-known as an interpolation method using Voronoi diagrams for discretely distributed data, and it is extended to continuously distributed data by Gross. On the other hand, the authors studied another interpolation method using Voronoi diagrams recently. This paper outlines the au...

Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphan-free (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its dual have properties that parallel those of ordinary Voronoi diagrams: the primal’s vertices, edg...

The ordinary point Voronoi diagram is a partition of the plane, in the way that each object (point) partitions the euclidean plane into a region, that is the locus of points which are closer from that object than from any other object (see Preparata and Shamos [7]). The Voronoi diagram has many applications in a variety of disciplines, and has been widely treated in the literature (see Okabe an...

In this paper, we concentrate on the problem of computing a Voronoi diagram using Hypercube model of computation. The main contribution of this work is the O(log n) parallel algorithm for computing Voronoi diagram on the Euclidean plane. Our technique parallelizes the wellknown seemingly inherent sequential technique of Shamos and Hoey, and makes use of a number of special properties of the div...

The Centroidal Voronoi Diagram (CVD) is a very versatile structure, well studied in Computational Geometry. It is used as the basis for a number of applications. This paper presents a deterministic algorithm, entirely computed using graphics hardware resources, based on Lloyd’s Method for computing CVDs. While the computation of the ordinary Voronoi diagram on GPU is a well explored topic, its ...

The Voronoi diagram of n distinct generating points divides the plane into cells, each of which consists of points most close to one particular generator. After introducing ‘limit Voronoi diagrams’ by analyzing diagrams of moving and coinciding points, we define compactifications of the configuration space of n distinct, labeled points. On elements of these compactifications we define Voronoi d...