Discrete & Computational Geometry

, Volume 9, Issue 3, pp 267–291 | Cite as

The upper envelope of voronoi surfaces and its applications

  • Daniel P. Huttenlocher
  • Klara Kedem
  • Micha Sharir


Given a setS ofsources (points or segments) in ℜ211C;d, we consider the surface in ℜ211C;d+1 that is the graph of the functiond(x)=min pεS ρ(x, p) for some metricρ. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metricρ. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.


Line Segment Voronoi Diagram Computational Geometry Hausdorff Distance Combinatorial Complexity 
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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Daniel P. Huttenlocher
    • 1
  • Klara Kedem
    • 1
    • 2
  • Micha Sharir
    • 2
    • 3
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA
  2. 2.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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