Discrete & Computational Geometry

, Volume 1, Issue 1, pp 59–71 | Cite as

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

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


Let γ1,..., γ m bem simple Jordan curves in the plane, and letK1,...,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 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,...,A m . 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 theA i 's.


Voronoi Diagram Jordan Curve Simple Polygon Convex Corner Motion Planning Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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