An optimal algorithm for the intersection radius of a set of convex polygons

Abstract

The intersection radius of a finite set of geometrical objects in the plane is the radius of the smallest closed disk that intersects all the objects in the set. Bhattacharya et. al. showed how the intersection radius can be found in linear time for a set of line segments in the plane by combining the prune-and-search strategy of Megiddo [8] with the strategy of replacing line segments by lines or points [2]. In this paper, we enlarge the scope of this technique by showing that it can also be used to find the intersection radius of a set of convex polygons in linear time. Moreover, it is immaterial if the set also contains other types of geometric objects like points, lines, rays, line segments and wedges. In fact, we will show how to handle such a mixed set of objects in a unified way; and this is the other important contribution of this paper.