Optimality of the Delaunay triangulation in ℝd
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.
Unable to display preview. Download preview PDF.
- 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.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.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.F. P. Preparata, and M. I. Shamos,Computational Geometry—An Introduction, Springer-Verlag, New York, 1985.Google Scholar
- 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.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