An Optimal Algorithm for Finding Minimal Enclosing Triangles

The problems of circumscribing and inscribing convex polygons with, respectively, minimum and maximum area k-gons have been studied extensively in the recent past because of their applications to robotics and collision avoidance problems [CY, DA]. In particular, Klee and Laskowski [KL] have given an O(n log%) algorithm for finding all local minima (with respect to area) among the triangles that contain a given convex n-gon P. (A triangle T is a local minimum if there exists some E > 0 such that area of T’ 2 area of T for each triangle T’ that is at a Hausdorff distance less than E from T.) The strength of their paper lies in establishing an elegant geometric characterization of these minima, which permits the avoidance of brute-force optimization. They show that although there may be infinitely many local minima, these fall into at most n equivalence classes, each of which is a “segment” of triangles having the same area. Their algorithm computes all the local minima in O(n log2n) time. Selecting the global minima from these can then be accomplished in an additional O(n) time. Klee and Laskowski find each local minimum afresh, without using any information gleaned from finding previous local minima. Inspired by Toussaint’s “rotating caliper” algorithms [T], we show that it is possible to move from one local minimum to the “next” in an orderly fashion,