Discrete & Computational Geometry

, Volume 1, Issue 1, pp 59–71

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

  • Klara Kedem
  • Ron Livne
  • János Pach
  • Micha Sharir
Article

DOI: 10.1007/BF02187683

Cite this article as:
Kedem, K., Livne, R., Pach, J. et al. Discrete Comput Geom (1986) 1: 59. doi:10.1007/BF02187683

Abstract

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,ij, 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.

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Klara Kedem
    • 1
  • Ron Livne
    • 1
  • János Pach
    • 3
  • Micha Sharir
    • 1
    • 2
  1. 1.School of Mathematical SciencesTel Aviv UniversityIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Mathematical Institute of the Hungarian Academy of SciencesHungary

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