Visualizing fortune's sweepline algorithm for planar Voronoi diagrams

Abstract

A visualization of Fortune's sweepline algorithm for planar Voronoi diagrams is presented. The algorithm sweeps upward through the set of input sites, maintaining a parabolic wavefront comprised of points equidistant from the sweepline and the sites. Voronoi edges arc traced wherever adjacent parabolic arcs intersect; Voronoi vertices are left behind wherever an arc is overtaken by its neighbors. In fact, the algorithm computes a transformation of the diagram, in which the wavefront maps to the sweepline itself, and all events (i.e., are disappearances) occur above the sweepline. Some applications of the Voronoi diagram are demonstrated, including nearest neighbor-finding, Delaunay triangulation, and planar convex hull. Finally, an intriguing connection between d-dimensional Voronoi diagrams and (d + 1)-dimensional convex hulls is depicted (for d = 2).