Properties of the Delaunay triangulation

Some of the most well-known names in Computational Geometry are those of two prominent Russian mathematicians: Georgy F. Voronoi (1868 – 1908) and Boris N. Delaunay (1890 1980). Their considerable contribution to the Number Theory and Geometry is well known to the specialists in these fields. Surprisingly, their names (their works remained unread and later re-discovered) became the most popular not among “pure” mathematician, but among the researchers who used geometric applications. Such terms as “ Voronoi diagram” and “ Delaunay triangulation” are very important not only for Computational Geometry, but also for Geometric Modeling, Image Processing, CAD, GIS etc. Delaunay triangulation is used in numerous applications. It is widely used in plane and 3D case. A natural question may arise: why th~ triangulation is better than the others. Usually the advantages of Delaunay triangulation are rationalized by the max-min angle criterion and other properties [1,2,5,10,11,12]. The max-min angle criterion requires that the diagonal of every convex quadrilateral occurring in the triangulation “should be well chosen” [12], in the sense that replacement of the chosen diagonal by the alternative one must not increase the minimum oft he six angles in the two triangles making up the quadrilateral. Thus the Delaunay triangulation of a planar point set maximizes the minimum angle in any triangle. More specifically, the sequence of triangle angles, sorted from sharpest to leaat sharp, is lexicographlcally maximized over all such sequences constructed from triangulation of S. We defined several functional on the set of all triangulations of the finite system of sites in Rd attaining global minimum on the Delaunay triangulation (DT). First we consider a so called “parabolic” functional and prove that it attains its minimum on DT in all dimensions. It could be used as an equivalent definition for DT. Secondly we treat “mean radius” functiorral(the mean of circumradii of triangles) for planar triangulations. Thirdly we treat a so called “harmonic” functional. For a triangle this functional equals the ration of the sum of squaresof sides over area. Finally, we consider a discrete anidogue of the Dirichlet functional. Actually in all these cases the optimality of DT in 2D directly follow from flipping (swapping) aIgorithm: after each flip the corresponding functional decrease until Delaunay triangulation is reached. In 2D case all of these functional on triagles are Iexicographically minimised over all such sequences constructed from triangulation of S like for the max-min angle criterion. If d >2 then Delaunay triangulation is not optimal for the functional “mean radius”, “harmonic” and “ Dirichlet”. ~l?rom this point of view the usage of DT in dimensions d >2 may be nonappropriate. Thus the problem of finding” good” triangulations for this functional in higher dimensions is opened and more detailed consideration is necessary.