Constrained Construction of Planar Delaunay Triangulations without Flipping

Abstract

The construction of Voronoi diagrams and Delaunay triangulations finds wide application in many branches of science. Delaunay triangulations have properties which make them more desirable than other triangulations for the same node set. Delaunay has characterized his triangulations by the empty circle property. The partitioning and flipping methods which have been developed for digital construction of Voronoi diagrams and Delaunay triangulations only make indirect use of this property. A novel method of construction is proposed, which is based directly on the empty circle property of Delaunay. The geometry of the steps of the algorithm is simple and can be grasped intuitively. The method can be applied to constrained triangulations, in which a triangulation domain and some of the edges are prescribed. A data structure for triangulations of concave and multiply-connected domains is presented which permits convenient specification of the constraints and the triangulation. The method is readily implemented, efficient and robust.Keywords: Delaunay, Voronoi, empty circle, shortest diagonal, triangulation theorem, partitioning, flipping, constrained, half edge, dihedral cycle Voronoi Diagrams and Delaunay TriangulationsOver centuries, outstanding scientists have considered the following problem: Given a set of sites in a plane, partition the plane into regions in such a way that all points of a region are at least as near to a particular site as to any other site of the set. This problem has arisen in many applications [1] [2] [3]. Johannes Keppler (1571-1639) encountered Voronoi diagrams when he considered the densest packing of spheres, René Descartes (1596-1650) when he investigated the distribution of matter relative to fixed stars, Johann Dirichlet (1805-1859) when he constructed integer lattices, John Snow (1813-1853) when he related the outbreak of cholera in London to the location of water pumps, Georges Voronoi (1868Voronoi ( -1908 when he extended the Dirichlet tessellations to higher dimensions and Boris Delaunay when he generalized Voronoi diagrams and their duals to irregularly spaced sites in ddimensional space. Many additional examples of Voronoi diagrams have been reported, for example in gold mining, crystallography, metallurgy and meteorology. Today diagrams which relate sets of nearest points to sites in a plane are known as Voronoi diagrams, sometimes also as Dirichlet tessellations. Extensions of the nearness concept to weighted tessellations

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“…The oscillation frequency w 0 is defined by the spring constants c 1 and c 2 . Performance function is defined by Equations ( 53)- (55) in Equation (56). The surrogation problem stated to find a model m ai that is capable of representing the nonlinear oscillator performance (Equation ( 56)).…”

“…Plant Evaluation: the plant evaluation will be performed using Equation (56). Model and Hyperparameter constraints: the search of the surrogate model will be constrained to the algorithms and parameters proposed in Listing 2.…”

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