Aperture angle optimization problems

Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P, the aperture angle of x with respect to Q is deened as the angle subtended by the cone that contains Q, has apex at x, and has its two rays emanating from x tangent to Q. We present algorithms with complexities O(n logm) and O(n+m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P. To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. In fact, this is optimal as we show that (maxfm; ng) is a lower bound for the minimization problem. Finally, we establish an (n) time lower bound for the maximization problem.