External polygon containment problems

Micha Sharir*, Sivan Toledo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Given a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k26(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon.

Original languageEnglish
Pages (from-to)99-118
Number of pages20
JournalComputational Geometry: Theory and Applications
Volume4
Issue number2
DOIs
StatePublished - Jun 1994

Keywords

  • Motion planning
  • Parametric searching
  • Polygon containment

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