Matthew T. Dickerson
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.
- F. Aurenhammer. Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv., 23(3):345--405, Sept. 1991. Google Scholar
Digital Library
- G. Barequet, M. Dickerson, and R. Drysdale. 2-point site voronoi diagrams. Discrete Applied Mathematics, 122:37--54, 2002. Google ScholarDigital Library
- 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. Google ScholarDigital Library
- M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 1997. Google ScholarDigital Library
- A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, UK, 1992. Google ScholarDigital Library
Index Terms
- Animating a continuous family of two-site Voronoi diagrams (and a proof of a bound on the number of regions)
Recommendations
Voronoi diagrams with overlapping regions
Voronoi diagrams are a tessellation of the plane based on a given set of points (called Voronoi points) into polygons so that all the points inside a polygon are closest to the Voronoi point in that polygon. All the regions of existing Voronoi diagrams ...
Sorting helps for voronoi diagrams
It is well known that, using standard models of computation, Ω(n logn) time is required to build a Voronoi diagram forn point sites. This follows from the fact that a Voronoi diagram algorithm can be used to sort. However, if the sites are sorted before ...
Round-trip voronoi diagrams and doubling density in geographic networks
Transactions on Computational Science XIVGiven a geographic network G (e.g. road network, utility distribution grid) and a set of sites (e.g. post offices, fire stations), a two-site Voronoi diagram labels each vertex v∈G with the pair of sites that minimizes some distance function. The sum ...
Comments