On the union of Jordan regions and collisionfree translational motion amidst polygonal obstacles
 Klara Kedem,
 Ron Livne,
 János Pach,
 Micha Sharir
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Abstract
Let γ_{1},..., 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 },i ≠j, 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 collisionfree 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 log^{2} n), wheren is the total number of corners of theA _{ i }'s.
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 Title
 On the union of Jordan regions and collisionfree translational motion amidst polygonal obstacles
 Journal

Discrete & Computational Geometry
Volume 1, Issue 1 , pp 5971
 Cover Date
 19861201
 DOI
 10.1007/BF02187683
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Klara Kedem ^{(1)}
 Ron Livne ^{(1)}
 János Pach ^{(2)}
 Micha Sharir ^{(1)} ^{(3)}
 Author Affiliations

 1. School of Mathematical Sciences, Tel Aviv University, Israel
 2. Mathematical Institute of the Hungarian Academy of Sciences, Hungary
 3. Courant Institute of Mathematical Sciences, New York University, New York, New York, USA