Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Continuous fields and discrete samples: reconstruction through delaunay tessellations

Abstract: Here we introduce the Delaunay Density Estimator Method. Its purpose is rendering a fully volume-covering reconstruction of a density field from a set of discrete data points sampling this field. Reconstructing density or intensity fields from a set of irregularly sampled data is a recurring key issue in operations on astronomical data sets, both in an observational context as well as in the context of numerical simulations. Our technique is based upon the stochastic geometric concept of the Delaunay tessellation generated by the point set. We shortly describe the method, and illustrate its virtues by means of an application to an N-body simulation of cosmic structure formation. The presented technique is a fully adaptive method: automatically it probes high density regions at maximum possible resolution, while low density regions are recovered as moderately varying regions devoid of the often irritating shot-noise effects. Of equal importance is its capability to sharply and undilutedly recover anisotropic density features like filaments and walls. The prominence of such features at a range of resolution levels within a hierarchical clustering scenario as the example of the standard CDM scenario is shown to be impressively recovered by our scheme.

Properties of the Delaunay triangulation

Abstract: 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.

Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

Abstract: The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each type of face. The representative of a face type is specified by a decoration of the corresponding Coxeter-Dynkin diagram. The rules of domain description are uniform for root lattices of simple Lie groups of all types. An explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple Lie groups. The Coxeter-Dynkin diagrams required here are the diagrams extended by the highest short root. Each diagram is partitioned into two subdiagrams, one describing completely a d-face of the Voronoi domain, its complement completely describing the dual of the d-face. The applicability of the authors' classification method to generalized kaleidoscopes is explained.

On deletion in delaunay triangulations

Abstract: This paper presents how the space of spheres and shelling may be used to delete a point from a d-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller's algorithm,1,2 which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.

Using longest‐side bisection techniques for the automatic refinement of delaunay triangulations

Abstract: In this paper we discuss, study and compare two linear algorithms for the triangulation refinement problem: the known longest-side (triangle bisection) refinement algorithm, as well as a new algorithm that uses longest side bisection techniques for refining Delaunay triangulations. We show that the automatic point insertion criterion, taken from the fractal property of optimal (linear) longest-side bisection algorithms, assures the construction of good quality Delaunay triangulations in linear time. Numerical evidence, showing that the practical behaviour of the new algorithm is in complete agreement with the theory, is included. © 1997 by John Wiley & Sons, Ltd.