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Animating a continuous family of two-site Voronoi diagrams (and a proof of a bound on the number of regions)

Authors:

Matthew T. Dickerson,

David EppsteinAuthors Info & Claims

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

Pages 92 - 93

https://doi.org/10.1145/1542362.1542380

Published: 08 June 2009 Publication History

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Abstract

A two-site distance function defines a "distance" measure from a point to a pair of points; mathematically, it is a mapping D:R2×(R2×R2)R+. A Voronoi diagram for a two-site distance function D and a set S of planar point sites has a region V (p, q) for each pair of sites p,q-S, where V(p,q) is defined as the set of all points in the plane "closer" to (p, q)"under distance function D"than to any other pair of sites in S. Two-site distance functions and their Voronoi diagrams have been explored by Barequet et al. (2002) and animated by Barequet et al. (2001), who give

the complexity of the Voronoi diagram for the two-site sum function (among others), and leave as an open question the complexity of the diagram for the two-site perimeter function. In this video, we introduce and animate a new continuous family of two-site distance functions Dc defined for any constant ce-1. This family includes both the sum and perimeter distance functions, providing a unifying model. We also present and animate in this video a new proof that the perimeter function Voronoi diagram has O(n) non-empty regions. The proof generalizes to any function in the Dc family when ce0. The animation also shows how the various functions in the family relate to one another.

References

[1]

F. Aurenhammer. Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv., 23(3):345--405, Sept. 1991.

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[2]

G. Barequet, M. Dickerson, and R. Drysdale. 2-point site voronoi diagrams. Discrete Applied Mathematics, 122:37--54, 2002.

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[3]

G. Barequet, M. Dickerson, R. Drysdale, and D. Guertin. 2-point site voronoi diagrams. In Video Review at the 17th Ann. ACM Symp. on Computational Geometry, 323--324, 2001.

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M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997.

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A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, UK, 1992.

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Cited By

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Chen DHuang ZLiu YXu J(2013)On Clustering Induced Voronoi DiagramsProceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2013.49(390-399)Online publication date: 26-Oct-2013
https://dl.acm.org/doi/10.1109/FOCS.2013.49
Barequet GDickerson MEppstein DHodorkovsky DVyatkina K(2013)On 2-Site Voronoi Diagrams Under Geometric Distance FunctionsJournal of Computer Science and Technology10.1007/s11390-013-1328-228:2(267-277)Online publication date: 12-Mar-2013
https://doi.org/10.1007/s11390-013-1328-2
Vyatkina KBarequet G(2011)On multiplicatively weighted Voronoi diagrams for lines in the planeTransactions on computational science XIII10.5555/2028176.2028180(44-71)Online publication date: 1-Jan-2011
https://dl.acm.org/doi/10.5555/2028176.2028180
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Index Terms

Animating a continuous family of two-site Voronoi diagrams (and a proof of a bound on the number of regions)
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Published In

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

June 2009

426 pages

ISBN:9781605585017

DOI:10.1145/1542362

Program Chairs:
John Hershberger
Mentor Graphics Corp.
,
Efi Fogel
Tel Aviv University

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 08 June 2009

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SoCG '09

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SoCG '09: 25th Annual Symposium on Computational Geometry

June 8 - 10, 2009

Aarhus, Denmark

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Chen DHuang ZLiu YXu J(2013)On Clustering Induced Voronoi DiagramsProceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science10.1109/FOCS.2013.49(390-399)Online publication date: 26-Oct-2013
https://dl.acm.org/doi/10.1109/FOCS.2013.49
Barequet GDickerson MEppstein DHodorkovsky DVyatkina K(2013)On 2-Site Voronoi Diagrams Under Geometric Distance FunctionsJournal of Computer Science and Technology10.1007/s11390-013-1328-228:2(267-277)Online publication date: 12-Mar-2013
https://doi.org/10.1007/s11390-013-1328-2
Vyatkina KBarequet G(2011)On multiplicatively weighted Voronoi diagrams for lines in the planeTransactions on computational science XIII10.5555/2028176.2028180(44-71)Online publication date: 1-Jan-2011
https://dl.acm.org/doi/10.5555/2028176.2028180
Barequet GDickerson MEppstein DHodorkovsky DVyatkina K(2011)On 2-Site Voronoi Diagrams under Geometric Distance FunctionsProceedings of the 2011 Eighth International Symposium on Voronoi Diagrams in Science and Engineering10.1109/ISVD.2011.13(31-38)Online publication date: 28-Jun-2011
https://dl.acm.org/doi/10.1109/ISVD.2011.13
Hanniel IBarequet G(2010)On the triangle-perimeter two-site Voronoi diagramTransactions on computational science IX10.5555/1986573.1986576(54-75)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1986573.1986576
Hanniel IBarequet G(2010)On the triangle-perimeter two-site Voronoi diagramTransactions on computational science IX10.5555/1980587.1980590(54-75)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1980587.1980590
Hanniel IBarequet G(2010)On the Triangle-Perimeter Two-Site Voronoi DiagramTransactions on Computational Science IX10.1007/978-3-642-16007-3_3(54-75)Online publication date: 2010
https://doi.org/10.1007/978-3-642-16007-3_3
Hanniel IBarequet G(2009)On the Triangle-Perimeter Two-Site Voronoi DiagramProceedings of the 2009 Sixth International Symposium on Voronoi Diagrams10.1109/ISVD.2009.12(129-136)Online publication date: 23-Jun-2009
https://dl.acm.org/doi/10.1109/ISVD.2009.12

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Round-trip voronoi diagrams and doubling density in geographic networks