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It is shown that if each pair of curves 7i, ~'j, i ~ j, intersect one another in at most two points, then the boundary of K =1.)~"= 1Ki contains at most max(2, 6m-12) intersection points of the curves y,, 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 rn 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 A 1 ..... 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 comers of the Ai's.

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