TY - GEN

T1 - The upper envelope of Voronoi surfaces and its applications

AU - Huttenlocher, Daniel P.

AU - Kedem, Klara

AU - Sharir, Micha

N1 - Publisher Copyright:
© 1991 ACM.

PY - 1991/6/1

Y1 - 1991/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=2342478039&partnerID=8YFLogxK

U2 - 10.1145/109648.109670

DO - 10.1145/109648.109670

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AN - SCOPUS:2342478039

SN - 0897914260

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 194

EP - 203

BT - Proceedings of the Annual Symposium on Computational Geometry

PB - Association for Computing Machinery

T2 - 7th Annual Symposium on Computational Geometry, SCG 1991

Y2 - 10 June 1991 through 12 June 1991

ER -