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Voronoi Diagrams and Delaunay Triangulations

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

Years and Authors of Summarized Original Work

  • 1975; Shamos, Hoey

  • 1987; Fortune

Problem Definition

Suppose there is some set of objects p called sites that exert influence over their surrounding space, M. For each site p, we consider the set of all points z in M for which the influence of p is strongest.

Such decompositions have already been considered by R. Descartes [5] for the fixed stars in solar space. In mathematics and computer science, they are called Voronoi diagrams, honoring work by G.F. Voronoi on quadratic forms. Other sciences know them as domains of action, Johnson-Mehl model, Thiessen polygons, Wigner-Seitz zones, or medial axis transform.

In the case most frequently studied, the space M is the real plane, the sites are n points, and influence corresponds to proximity in the Euclidean metric, so that the points most strongly influenced by site p are those for which p is the nearest neighbor among all sites. They form a convex region called the Voronoi region of p. The...

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Recommended Reading

  1. Aurenhammer F, Klein R, Lee DT (2013) Voronoi diagrams and delaunay triangulations. World Scientific, Singapore

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  2. Bentley J, Ottman T (1979) Algorithms for reporting and counting geometric intersections. IEEE Trans Comput C-28:643–647

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  3. Brown KQ (1979) Voronoi diagrams from Convex Hulls. Inf Process Lett 9:223–228

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  4. Cole R (1989) as reported by ÓDúnlaing, oral communication

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  5. Descartes R (1644) Principia philosophiae. Ludovicus Elsevirius, Amsterdam

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  6. Edelsbrunner H, Seidel R (1986) Voronoi diagrams and arrangements. Discret Comput Geom 1:25–44

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  7. Fortune S (1978) A sweepline algorithm for Voronoi diagrams. Algorithmica 2:153–174

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  8. Green PJ, Sibson RR (1978) Computing dirichlet tessellations in the plane. Comput J 21: 168–173

    Article  MathSciNet  MATH  Google Scholar 

  9. Okabe A, Boots B, Sugihara K, Chiu SN (2000) Spatial tessellations: concepts and applications of Voronoi diagrams. Wiley, Chichester

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  10. Seidel R (1988) Constrained delaunay triangulations and Voronoi diagrams with obstacles. Technical report 260, TU Graz

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  11. Shamos MI, Hoey D (1975) Closest-point problems. In: Proceedings 16th annual IEEE symposium on foundations of computer science, Berkeley, pp 151–162

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Author information

Authors and Affiliations

  1. Institute for Computer Science, University of Bonn, Bonn, Germany

    Rolf Klein

Authors
  1. Rolf Klein
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Correspondence to Rolf Klein .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Klein, R. (2016). Voronoi Diagrams and Delaunay Triangulations. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_507

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