ISAAC 1996: Algorithms and Computation pp 126-135 | Cite as

Computing the maximum overlap of two convex polygons under translations

  • Mark de Berg
  • Olivier Devillers
  • Marc van Kreveld
  • Otfried Schwarzkopf
  • Monique Teillaud
Session 4a: Invited Presentation
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1178)

Abstract

Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O(n2+m2+min(nm2+n2m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n+m) log(n+m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q.

We also prove that the placement of Q that makes the centroids of Q and P coincide realizes an overlap of at least 9/25 of the maximum possible overlap. As an upper bound, we show an example where the overlap in this placement is 4/9 of the maximum possible overlap.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Mark de Berg
    • 1
  • Olivier Devillers
    • 2
  • Marc van Kreveld
    • 1
  • Otfried Schwarzkopf
    • 3
  • Monique Teillaud
    • 2
  1. 1.Dept. of Computer ScienceUtrecht UniversityTB UtrechtThe Netherlands
  2. 2.INRIASophia-Antipolis cedex
  3. 3.Dept. of Computer SciencePohang University of Science and TechnologyPohangSouth Korea

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