Discrete & Computational Geometry

, Volume 12, Issue 2, pp 189–202 | Cite as

Optimality of the Delaunay triangulation in ℝd

  • V. T. Rajan


In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝd. These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝd is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.


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  1. 1.
    F. Aurenhammer, Voronoi diagrams—a survey of a fundamental geometric data structure,ACM Comput. Surveys 23 (1991), 345–405.CrossRefGoogle Scholar
  2. 2.
    M. Bern and D. Eppstein, Mesh generation and optimal triangulation, inComputing in Euclidean Geometry (F. K. Hwang and D. Z. Du, eds.), World Scientific, Singapore, 1992, pp. 23–90.CrossRefGoogle Scholar
  3. 3.
    K. Q. Brown, Voronoi diagrams from convex hulls,Inform. Process. Lett. 9 (1970), 223–228.CrossRefGoogle Scholar
  4. 4.
    R. W. Cottle, Symmetric dual quadratic programs,Quart. Appl. Math. 21 (1963), 237–243.MATHMathSciNetGoogle Scholar
  5. 5.
    E. F. D’Azevedo and R. B. Simpson, On optimal interpolation triangle incidences,SIAM J. Sci. Statist. Comput. 10 (1989), 1063–1075.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    W. S. Dorn, Duality in quadratic programming,Quart. Appl. Math. 18 (1960), 155–162.MATHMathSciNetGoogle Scholar
  7. 7.
    H. Edelsbrunner and R. Seidel, Voronoi diagrams and arrangements,Discrete Comput. Geom. 1 (1986), 25–44.MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Edelsbrunner,Algorithms in Combinatorial Geometry, Springer-Verlag, New York, 1987.MATHCrossRefGoogle Scholar
  9. 9.
    H. Edelsbrunner and E. P. Mucke, Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms,ACM Trans. Graphics 9 (1990), 66–104.MATHCrossRefGoogle Scholar
  10. 10.
    S. Fortune, A sweepline algorithm for Voronoi diagrams,Algorithmica 2(2) (1987), 153–174.MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    L. Guibas and J. Stolfi, Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams.ACM Trans. Graphics 4 (1985), 74–123.MATHCrossRefGoogle Scholar
  12. 12.
    M. S. Karasick, D. Lieber, L. R. Nackman, and V. T. Rajan, Fast Visualization of Three-Dimensional Delaunay Meshes, RC 18347, IBM T. J. Watson Research Center, Yorktown Heights, NY, 1992.Google Scholar
  13. 13.
    C. L. Lawson, Generation of a Triangular Grid with Applications to Contour Plotting, Internal Technical Memorandum No. 299, Jet Propulsion Laboratory, Pasadena, CA, 1972.Google Scholar
  14. 14.
    S. Meshkat, J. Ruppert, and H. Li, Three-Dimensional automatic unstructured grid generation based on Delaunay tetrahedralization,Proc. Internat. Conf. on Numerical grid Generation, 1991, pp. 841–851.Google Scholar
  15. 15.
    F. P. Preparata, and M. I. Shamos,Computational Geometry—An Introduction, Springer-Verlag, New York, 1985.Google Scholar
  16. 16.
    V. T. Rajan. Optimality of the Delaunay triangulation in ℝd,Proc. 7th Ann. Symp. on Computational Geometry, 1991, pp. 357–363.Google Scholar
  17. 17.
    M. I. Shamos and D. Hoey, Closest-point problems,Proc. 16th Ann. IEEE Symp. on Foundations of Computer Science, 1975, pp. 151–162.Google Scholar
  18. 18.
    V. Srinivasan, L. R. Nackman, J. Tang, and S. N. Meshkat, Automatic mesh generation using the symmetric axis transformation of polygonal domains,Proc. IEEE,80(9) (1992), 1485–1501.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1994

Authors and Affiliations

  • V. T. Rajan
    • 1
  1. 1.Manufacturing Research Department, IBM Research DivisionT. J. Watson Research CenterYorktown HeightsUSA

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