Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
SCG '01 Proceedings of the seventeenth annual symposium on Computational geometry
Discrete Applied Mathematics
On the triangle-perimeter two-site Voronoi diagram
Transactions on computational science IX
On the triangle-perimeter two-site Voronoi diagram
Transactions on computational science IX
On multiplicatively weighted Voronoi diagrams for lines in the plane
Transactions on computational science XIII
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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.