On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

Abstract

Let γ1,..., γm be m simple Jordan curves in the plane, and let K1,..., Km be their respective interior regions. It is shown that if each pair of curves γi, γj, i ≠j, intersect one another in at most two points, then the boundary of K=∩i=1 mKi contains at most max(2,6 m - 12) intersection points of the curves γ1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union of m Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygon B amidst several (convex) polygonal obstacles A1,..., Am. Assuming that the number of corners of B is fixed, the algorithm presented here runs in time O (n log2n), where n is the total number of corners of the Ai's. © 1986 Springer-Verlag New York Inc.