Exact Construction of Minimum-Width Annulus of Disks in the Plane

The construction of a minimum-width annulus of a set of objects in the plane has useful applications in diverse fields, such as tolerancing metrology and facility location. We present a novel implementation of an algorithm for obtaining a minimum-width annulus containing a given set of disks in the plane, in case one exists. The algorithm extends previously known methods for constructing minimum-width annuli of sets of points. The algorithm for disks requires the construction of two Voronoi diagrams of different types, one of which we call the “farthest-point farthest-site” Voronoi diagram and appears not to have been investigated before. The vertices of the overlay of these two diagrams are candidates for the annulus’ center. The implementation employs an asymptotically nearoptimal randomized divide-and-conquer algorithm for constructing two-dimensional Voronoi diagrams. Our software utilizes components from Cgal, the Computational Geometry Algorithms Library, and follows the exact computation paradigm. We do not assume general position. Namely, we handle degenerate input and produce exact results.