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

Let γ₁,..., γ_m be m simple Jordan curves in the plane, and let K₁,..., K_m be their respective interior regions. It is shown that if each pair of curves γ_i, γ_j, i ≠j, intersect one another in at most two points, then the boundary of K=∩_i^{=1 m}K_i 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,..., A_m. 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 A_i's.