A Subdivision Approach to Weighted Voronoi Diagrams

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 some postulated capability. In R, (1) might say that the bisector of two objects is a simple infinite curve, and (2) might assume the ability to compute the intersection of any pair of bisectors. For example, see [2] in R and [1] for semi-algebraic convex objects in R. On the other hand, if we want to implement these algorithms, these abstract algorithms often pose immense barriers. An example of this phenomenon is the fact that there is currently no implementable algorithm for the Euclidean Voronoi diagram of a set S of polyhedral objects. See [8] for a discussion of the issues. This is a strong motivation for exploring models of computation other than the Real RAM. This paper continues our exploration of approximate numerical models: we use a specific form of this model, based on the subdivision paradigm [7]. The first subdivision algorithm for the Voronoi diagram of polyhedral solids is from [4]. Other subdivision algorithms for Voronoi diagrams are surveyed in [8]. A key observation about such models is that they are relatively easy to implement and very flexible. That is in marked constrast with exact algorithms which often require completely different algorithms when the setting is slightly generalized. This phenomenon is demonstrated in the present paper. In [8], we described and implemented an algorithm for the Euclidean Voronoi diagram for a set of pairwise disjoint set of polygons, assuming suitable non-degeneracy conditions. In this paper, we show how to extend such an algorithm is handle weighted polygons. This generalizes previously studied classes of weighted Voronoi diagrams: multiplicative or additive weights [5] and anisotropic weights [3]. Such Voronoi diagrams have quartic curves. But if we drop the additive weights, the resulting Voronoi diagram has only conic curves, and moreover all three types of conics (parabola, ellipse and hyperbola) do appear. These Voronoi diagrams appear to be new, and offer a flexible modeling tool in applications. 1.1 Overview