Voronoi diagrams and arrangements
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
On the construction of abstract Voronoi diagrams
Discrete & Computational Geometry
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
Randomized incremental construction of abstract Voronoi diagrams
Computational Geometry: Theory and Applications
Randomized incremental construction of simple abstract Voronoi diagrams in 3-space
Computational Geometry: Theory and Applications
On non-smooth convex distance functions
Information Processing Letters
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first annual symposium on Computational geometry
Convex Distance Functions In 3-Space Are Different
Fundamenta Informaticae
Abstract Voronoi diagrams revisited
Computational Geometry: Theory and Applications
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We propose a class of abstract Voronoi diagrams in 3-space that generalizes the planar abstract Voronoi diagram of Klein. Our class of abstract Voronoi diagrams includes the Voronoi diagram of point sites in general position under any convex distance function. To characterize the abstract Voronoi diagram in 3- space, we introduce the notion of intersection characteristic. We determine the intersection characteristic for the simplex, the L∞, and the Lp distance function. We find that the intersection characteristic in case of the simplex distance function is similar to that of the usual Euclidean distance. This enables us to give a randomized incremental algorithm for computing the Voronoi diagram under the simplex distance function in quadratic expected time.