, Volume 1, Issue 1, pp 59-71

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

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Abstract

Let γ1,..., γ m bem simple Jordan curves in the plane, and letK 1,...,K m be their respective interior regions. It is shown that if each pair of curves γ i , γ j ,ij, intersect one another in at most two points, then the boundary ofK=∩ i =1m K i contains at most max(2,6m − 12) intersection points of the curvesγ 1, 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 obstaclesA 1,...,A m . Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2 n), wheren is the total number of corners of theA i 's.

Work on this paper by the second author has been supported in part by a grant from the Bat-Sheva Fund at Israel. Work by the fourth author has been supported in part by a grant from the U.S.-Israeli Binational Science Foundation.