TY - GEN

T1 - The upper envelope of Voronoi surfaces and its applications

AU - Huttenlocher, Daniel P.

AU - Kedem, Klara

AU - Sharir, Micha

N1 - Funding Information:
diagrams are a central construct in compu-geometry and serve as an important tool in fields ([PS], [Ed], [Ya], [AU], [LS]). In this *The first author was supported in part by NSF grant IRI-9057928 and matching funds from Kodak Corporation. The second and third authors were supported by the Fund for Basic Research administered by the Israeli Academy of Sciences. The second author was also supported by a fellowship from the Pikkowski-Vrdazzi Fund and by the Eshkol grant 04601-90. The third author was also supported by Office of Navaf Research Grants by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, and the G .I.F., the German-Israeli Foundation for Scientific Research and Development.
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

M3 - Conference contribution

AN - SCOPUS:2342478039

SN - 0897914260

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 -