Voronoi Tessellation
Voronoi or Voronoy is a Slavic masculine surname; its feminine counterpart is Voronaya. It may refer to *Georgy Voronoy (1868–1908), Russian and Ukrainian mathematician **Voronoi diagram **Weighted Voronoi diagram ** Voronoi deformation density **Voronoi formula **Voronoi pole **Centroidal Voronoi tessellation In geometry, a centroidal Voronoi tessellation (CVT) is a special type of Voronoi tessellation in which the generating point of each Voronoi cell is also its centroid (center of mass). It can be viewed as an optimal partition corresponding to an ...

Georgy Voronoy
Georgy Feodosevich Voronoy (russian: Георгий Феодосьевич Вороной; ukr, Георгій Феодосійович Вороний; 28 April 1868 – 20 November 1908) was an Russian Empire, Imperial Russian mathematician of Ukraine, Ukrainian descent noted for defining the Voronoi diagram. Biography Voronoy was born in the village of Zhuravka, Pyriatyn, in the Poltava Governorate, which was a part of the Russian Empire at that time and is in Varva Raion, Chernihiv Oblast, Ukraine. Beginning in 1889, Voronoy studied at Saint Petersburg State University, Saint Petersburg University, where he was a student of Andrey Markov. In 1894 he defended his master's thesis ''On algebraic integers depending on the roots of an equation of third degree''. In the same year, Voronoy became a professor at the University of Warsaw, where he worked on continued fractions. In 1897, he defended his doctoral thesis ''On a generalisation of a continuous fraction''. He was an Invite ...
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Voronoi Diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. The simplest case In the simplest case, shown in the first picture, we are given a finite set of points in the Euclid ...
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Weighted Voronoi Diagram
In mathematics, a weighted Voronoi diagram in ''n'' dimensions is a generalization of a Voronoi diagram. The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The distance function may specify the usual Euclidean distance, or may be some other, special distance function. In weighted Voronoi diagrams, each site has a weight that influences the distance computation. The idea is that larger weights indicate more important sites, and such sites will get bigger Voronoi cells. In a multiplicatively weighted Voronoi diagram, the distance between a point and a site is divided by the (positive) weight of the site."Dictionary of distances", by Elena Deza and Michel Dezabr>pp. 255, 256/ref> In the plane under the ordinary Euclidean distance, the multiplicatively weighted Voronoi diagram is also called circular Dirichlet tessellation and its edges are circular arcs and straight line segments. A Voronoi cell may be non-convex, disconnected and may hav ...
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Voronoi Formula
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups. The Voronoi (summation) formula for GL(2) has long been a standard tool for studying analytic properties of automorphic forms and their ''L''-functions. There have been numerous results coming out the Voronoi formula on GL(2). The concept is named after Georgy Voronoy. Classical application To Voronoy and his contemporaries, the formula appeared tailor-made to evaluate certain finite sums. That seemed significant because several important questions in number theory involve finite sums of arithmetic quantities. In this connection, let us mention two classical examples, Dirichlet’s divisor problem and the Gauss’ circle problem. The former estimates the size of ''d''(''n''), the number of positive divisors of an integer ''n''. Di ...
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Voronoi Pole
In geometry, the positive and negative Voronoi poles of a cell in a Voronoi diagram are certain vertices of the diagram. Definition Let V be the Voronoi diagram for a set of sites P, and let V_p be the Voronoi cell of V corresponding to a site p\in P. If V_p is bounded, then its positive pole is the vertex of the boundary of V_p that has maximal distance to the point p. If the cell is unbounded, then a positive pole is not defined. Furthermore, let \bar be the vector from p to the positive pole, or, if the cell is unbounded, let \bar be a vector in the average direction of all unbounded Voronoi edges of the cell. The negative pole is then the Voronoi vertex v in V_p with the largest distance to p such that the vector \bar and the vector from p to v make an angle larger than \tfrac. References * {{Cite book , last = Boissonnat , first = Jean-Daniel , title = Effective Computational Geometry for Curves and Surfaces , publisher = Springer Springer or springers may r ...
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