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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 have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi AAAAA14
{{Disambiguation, surname ...
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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. 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, after Alfred H. Thiessen. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. Simplest case In the simplest case, shown in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's ''Elements'', distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. The connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elena Deza
Elena Ivanovna Deza (, née Panteleeva; born 23 August 1961) is a French and Russian mathematician known for her books on metric spaces and figurate numbers. Education and career Deza was born on 23 August 1961 in Volgograd Volgograd,. formerly Tsaritsyn. (1589–1925) and Stalingrad. (1925–1961), is the largest city and the administrative centre of Volgograd Oblast, Russia. The city lies on the western bank of the Volga, covering an area of , with a population ..., and is a French and Russian citizen. She earned a diploma in mathematics in 1983, a candidate's degree (doctorate) in mathematics and physics in 1993, and a docent's certificate in number theory in 1995, all from Moscow State Pedagogical University. From 1983 to 1988, Deza was an assistant professor of mathematics at Moscow State Forest University. In 1988 she moved to Moscow State Pedagogical University; she became a lecturer there in 1993, a reader in 1994, and a full professor in 2006. Books As well as m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Deza
Michel Marie Deza (27 April 1939. – 23 November 2016) was a Soviet and French mathematician, specializing in combinatorics, discrete geometry and graph theory. He was the retired director of research at the French National Centre for Scientific Research (CNRS), the vice president of the European Academy of Sciences, a research professor at the Japan Advanced Institute of Science and Technology, and one of the three founding editors-in-chief of the European Journal of Combinatorics. Deza graduated from Moscow University in 1961, after which he worked at the Soviet Academy of Sciences until emigrating to France in 1972. In France, he worked at CNRS from 1973 until his 2005 retirement. He has written eight books and about 280 academic papers with 75 different co-authors, including four papers with Paul Erdős, giving him an Erdős number of 1. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometriae Dedicata
''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands.. It is published by Springer Netherlands. The Editor-in-Chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Richard Alan Wentworth. References External links Springer site Geometry journals Springer Science+Business Media academic journals Algebra journals Dynamical systems journals Topology journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Tessellation
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, he advanced the theory of Fourier series and was one of the first to give the modern formal definition of a function (mathematics), function. In mathematical physics, he studied potential theory, boundary-value problem, boundary-value problems, and heat diffusion, and hydrodynamics. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him. Biography Early life (1805–1822) Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Growth
Crystal growth is a major stage of a crystallization, crystallization process, and consists of the addition of new atoms, ions, or polymer strings into the characteristic arrangement of the crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) nucleation, unless a "seed" crystal, purposely added to start the growth, was already present. The action of crystal growth yields a crystalline solid whose atoms or molecules are close packed, with fixed positions in space relative to each other. The crystalline states of matter, state of matter is characterized by a distinct structural rigidity and very high resistance to Plastic deformation in solids, deformation (i.e. changes of shape and/or volume). Most crystalline solids have high values both of Young's modulus and of the shear modulus of elasticity (physics), elasticity. This contrasts with most liquids or fluids, which have a low shear modulus, and typically exh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Diagram
In computational geometry, a power diagram, also called a Laguerre–Voronoi diagram, Dirichlet cell complex, radical Voronoi tesselation or a sectional Dirichlet tesselation, is a partition of the Euclidean plane into polygonal cells defined from a set of circles. The cell for a given circle ''C'' consists of all the points for which the Power of a point, power distance to ''C'' is smaller than the power distance to the other circles. The power diagram is a form of generalized Voronoi diagram, and coincides with the Voronoi diagram of the circle centers in the case that all the circles have equal radii.... Definition If ''C'' is a circle and ''P'' is a point outside ''C'', then the Power of a point, power of ''P'' with respect to ''C'' is the square of the length of a line segment from ''P'' to a point ''T'' of tangency with ''C''. Equivalently, if ''P'' has distance ''d'' from the center of the circle, and the circle has radius ''r'', then (by the Pythagorean theorem) the power ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Of A Point
In elementary plane geometry, the power of a point is a real number that reflects the relative distance of a given point from a given circle. It was introduced by Jakob Steiner in 1826. Specifically, the power \Pi(P) of a point P with respect to a circle c with center O and radius r is defined by : \Pi(P)=, PO, ^2 - r^2. If P is ''outside'' the circle, then \Pi(P)>0, if P is ''on'' the circle, then \Pi(P)=0 and if P is ''inside'' the circle, then \Pi(P)<0. Due to the Pythagorean theorem the number has the simple geometric meanings shown in the diagram: For a point outside the circle is the squared tangential distance of point to the circle . Points with equal power, isolines of , are circles concentric to circle . Steiner used the power of a point for proofs of several statements on circles, for example: * Determination of a circle, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Geometry
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algorithms
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. Automated planning Combinatorial algorithms General combinatorial algorithms * Brent's algorithm: finds a cycle in function value iterations using only two iterators * Floyd's cycle-finding algorithm: finds a cycle in function value iterations * Gale–Shapley algorithm: solves the stable matching problem * Pseudorandom number generators (uniformly dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
