Planning a Purely Translational Motion for a Convex Object in Two-Dimensional Space Using Generalized Voronoi Diagrams

Generalized Voronoi diagrams of objects are di cult to compute in a robust way, especially in higher dimensions. For a number of applications an approximation of the real diagram within some predetermined precision is su cient. In this paper we study the computation of such approximate Voronoi diagrams. The emphasis is on practical applicability, therefore we are mainly concerned with fast (in terms of running time) computation, generality, robustness, and easy implementation, rather than optimal combinatorial and computational complexity. Given a set of disjoint convex sites in any dimension we describe a general algorithm that approximates their Voronoi diagram with arbitrary precision; the only primitive operation that is required is the computation of the distance from a point to a site. The method is illustrated by its application to motion planning using retraction. To justify our claims on practical applicability, we provide experimental results obtained with implementations of the method in two and three dimensions.