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 ℝ d . 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 ℝ d 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.

[1]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[2]  D. Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[3]  Kevin Q. Brown,et al.  Voronoi Diagrams from Convex Hulls , 1979, Inf. Process. Lett..

[4]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[5]  R. B. Simpson,et al.  On optimal interpolation triangle incidences , 1989 .

[6]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[7]  N. S. Barnett,et al.  Private communication , 1969 .

[8]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[9]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1985, SCG '85.

[10]  L. Nackman,et al.  Automatic mesh generation using the symmetric axis transformation of polygonal domains , 1992, Proc. IEEE.

[11]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[12]  R. Cottle.  Symmetric dual quadratic programs , 1963 .

[13]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[14]  Herbert Edelsbrunner,et al.  An acyclicity theorem for cell complexes in d dimensions , 1989, SCG '89.

[15]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .