A Voronoi diagram of a set of n sites partitions a finite set of m points into regions of different areas, called the capacities of the sites. In this paper we are interested in Voronoi diagrams in which the capacities are given as constraints. We compute such capacity-constrained Voronoi diagrams in finite spaces by starting with an arbitrary partition that fulfills the capacity constraint wit...

Voronoi diagrams are fundamental geometric structures that have been both deeply and widely investigated since their inception in disguise by René Descartes for analyzing the gravitational influence of stars, in the 17 century. Voronoi diagrams find countless applications in science and engineering beyond graphics as attested by the long representative, and yet non-exhaustive, list of applicati...

We consider the question under which circumstances the straight skeleton and the Voronoi diagram of a given input shape coincide. More precisely, we investigate convex distance functions that stem from centrally symmetric convex polyhedra as unit balls and derive sufficient and necessary conditions for input shapes in order to obtain identical straight skeletons and Voronoi diagrams with respec...

In this paper, two-dimensional manifolds in threedimensional Euclidian space are considered in order to single out surfaces whose skeletons fulfill the grass-fire concept. Among such surfaces we distinguish those ones that can be covered by finite number of adjacent patches. We name them multi-ribbon surfaces. We aim to calculate a multi-ribbon surface skeleton by constructing every patch Voron...

Given a set of sites in a simple polygon, a geodesic Voronoi diagram partitions the polygon into regions based on distances to sites under the geodesic metric. We present algorithms for computing the geodesic nearest-point, higher-order and farthest-point Voronoi diagrams of m point sites in a simple n-gon, which improve the best known ones form ≤ n/polylogn. Moreover, the algorithms for the ne...

Voronoi diagrams are a central topic in Computational Geometry. They can be generalized in many ways. Suppose S is a finite set of geometric objects and there is a notion of Voronoi diagram V or(S) that we want to compute. One approach to designing algorithms for V or(S) is to (1) assume some general properties of S, and (2) describe an abstract algorithm based on the Real RAM model that has so...

Dit proefschrift werd mede mogelijk gemaakt met financiële steun van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek.

Given a family of k disjoint connected polygonal sites in general position and of total complexity n, we consider the farthest-site Voronoi diagram of these sites, where the distance to a site is the distance to a closest point on it. We show that the complexity of this diagram is O(n), and give an O(n log n) time algorithm to compute it. We also prove a number of structural properties of this ...

In Euclidean geometry, it is well-known that the k-order Voronoi diagram in R can be computed from the vertical projection of the k-level of an arrangement of hyperplanes tangent to a convex potential function in R: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a concave potential function: the northern hemisphere. Furthermore, we also show how to bui...

In typical 2D Voronoi diagrams, the distance from a site to a point in the plane is unaffected by the existance of other sites. In 2D pursuit/evasion Voronoi diagrams, the distance from an evader to a point is the length of the shortest path to that point that avoids all pursuers. Since pursuers can move, the paths that evaders follow to reach certain points in the plane can be quite complicate...