Calculating Voronoi Diagrams using Convex Sweep Curves

The Voronoi diagram is an important data structure in computational geometry. Given n sites in the plane, the Voronoi diagram partitions the plane into n regions. The region of a site p consists of all those points that lie closer to p than to any of the other sites. For a survey on Voronoi diagrams and their applications we refer to Aurenhammer [1]. A generalization of the sweep line method of Fortune [2] was developed in [4] which will be presented here. To achieve this generalization, the sweep line was replaced by a general sweep curve. In order to look for sweep curves that are feasable for such an algorithm, equidistant curves were introduced. It turned out that there are two useful forms of sweep curves for the Euclidian plane, lines and circles. As a further result, a sweep algorithm based on sweep circles was implemented; this algorithms runs with O(n · log n) time and O(n) space requirements, which is optimal.