The upper envelope of Voronoi surfaces and its applications

Daniel P. Huttenlocher, Klara Kedem, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

31 Scopus citations


Given a set S of sources (points or segments), we consider the surface that is the graph of the function d(x) - minp∈s ρ(x,p), for some metric ρ. This surface is closely related to the Voronoi diagram, Vor(S), of S under the metric ρ. The upper envelope of a set of these Voronoi surfaces, each defined for a different set of sources, can be used to solve a number of problems, including finding the minimum Hausdorff distance between two sets of points or segments under translation, and determining the optimal placement of a site with respect to sets of utilities. We derive bounds on the number of vertices on the upper envelope of m Voronoi surfaces, provide efficient algorithms to calculate these vertices, and discuss applications to the aforementioned problems.

Original languageEnglish
Title of host publicationProceedings of the Annual Symposium on Computational Geometry
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)0897914260
StatePublished - 1 Jun 1991
Externally publishedYes
Event7th Annual Symposium on Computational Geometry, SCG 1991 - North Conway, United States
Duration: 10 Jun 199112 Jun 1991


Conference7th Annual Symposium on Computational Geometry, SCG 1991
Country/TerritoryUnited States
CityNorth Conway

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics


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