TY - JOUR

T1 - On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

AU - Kedem, Klara

AU - Livne, Ron

AU - Pach, János

AU - Sharir, Micha

PY - 1986/12

Y1 - 1986/12

N2 - Let γ1,..., γm be m simple Jordan curves in the plane, and let K1,..., 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 of K=∩i=1 mKi 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,..., 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 corners of the Ai's.

AB - Let γ1,..., γm be m simple Jordan curves in the plane, and let K1,..., 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 of K=∩i=1 mKi 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,..., 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 corners of the Ai's.

UR - http://www.scopus.com/inward/record.url?scp=7544245726&partnerID=8YFLogxK

U2 - 10.1007/BF02187683

DO - 10.1007/BF02187683

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AN - SCOPUS:7544245726

SN - 0179-5376

VL - 1

SP - 59

EP - 71

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -