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

Let γ1,..., γm bem simple Jordan curves in the plane, and letK1,...,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 ofK=∩i=1mKi contains at most max(2,6m − 12) intersection points of the curvesγ1, 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 ofm 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 polygonB amidst several (convex) polygonal obstaclesA1,...,Am. Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theAi's.