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# Topic: Dirichlet tessellation

###### In the News (Wed 22 May 13)

 Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia His family hailed from the town of Richelet in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet" = "the young chap from Richelet") was derived, and that was where his grandfather lived. Dirichlet was born in Düren, where his father was the postmaster. He married Rebecka Mendelssohn Bartholdy, who came from a distinguished family of converts from Judaism to Christianity; she was a granddaughter of the philosopher Moses Mendelssohn, daughter of Abraham Mendelssohn Bartholdy and a sister of the composer Felix Mendelssohn Bartholdy. en.wikipedia.org /wiki/Dirichlet   (312 words)

 Voronoi diagram - Wikipedia, the free encyclopedia In mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation (after Lejeune Dirichlet), is special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. A 2D lattice gives an irregular honeycomb tessellation, with equal hexagons with point symmetry; in the case of a regular triangular lattice it is regular; in the case of a rectangular lattice the hexagons reduce to rectangles in rows and columns; a square lattice gives the regular tessellation of squares. However in these cases the Voronoi tessellation is not guaranteed to exist (or to be a "true" tessellation), since the equidistant locus for two points may fail to be subspace of codimension 1, even in the 2-dimensional case. en.wikipedia.org /wiki/Dirichlet_tessellation   (880 words)

 Dirichlet tessellation of bark beetle spatial attack points This tessellation was proposed in 1850 by Dirichlet (Upton and Fingleton 1985) and a formal mathematical definition is given by Green and Sibson (1978). Dirichlet tessellations can be thought of as representing the areas of territorial animals, allelochemic-producing plants, or the packing of cells in a tissue. Dirichlet tessellation (irregular pentagon) composed of perpendicular bisectors (solid lines) between center point and six surrounding points. vinsonlab.tamu.edu /former/john/papers/jae92a.htm   (6012 words)

 How Tessellations Works   (Site not responding. Last check: 2007-10-22) A more direct method of evaluating whether particles are ordered, clustered, or randomly distributed is Dirichlet tessellations, named for the man who described their mathematical properties, in 1850*. The tessellation cells are formed by expanding the perimeters of all particles until the expanding regions meet their nearest neighbors. By using Dirichlet tessellations, we can construct a quantitative measure of clustering that can be used to relate materials processes with physical and mechanical properties. www.pgt.com /Tessel1.html   (412 words)

 Correct calculation of Dirichlet polygon areas The Dirichlet cell, first proposed in 1850, has been useful in many scientific disciplines and thus is known under a variety of names including Voronoi, 1909, Thiessen, 1911, Wigner-Seitz, 1933, cell model, 1953, and S-mosaic, 1977 (Rogers 1964; Mead 1971; Rhynsburger 1973; Upton & Fingleton 1985; David 1988). More recently, a Dirichlet tessellation algorithm was developed to define colonization territories of bark beetles (Coleoptera: Scolytidae) under the bark of host trees (Byers 1992). However, calculations of the area of a Dirichlet cell are undervalued by about 10 to 30 percent. chemical-ecology.net /z65-abs.htm   (974 words)

 A geometrical description of particle distributions in materials Extension of the tessellation technique by specifying a distance over which second-phase particles are assumed to interact mechanically (particle interaction distance) facilitates quantitative characterization of particle clustering by allowing evaluation of characteristics such as the fraction of particles clustered, and the number, size, and spatial distribution of clusters. The tessellation and clustering characteristics of several types of computer-generated particle arrays (random, clustered, and hexagonal cellular) are presented. Tessellation and clustering characteristics of the random particle arrays are insensitive to plane-strain deformation. stacks.iop.org /0965-0393/1/275   (289 words)

 Elastic moduli of model random three-dimensional closed-cell cellular solids - Voronoi tessellations The most common models of cellular solids are generated by Voronoi tessellation of distributions of 'seed-points' in space. The cells of the Kelvin foam are uniformly shaped, fill space, and satisfy Plateau's law of foam equilibrium (three faces meet at angles of 120º, and four struts join at 109.5º). Data at the two lowest densities were obtained for 3 realizations of the 26 cell model, and the remainder were obtained for 5 realizations of the 122 cell model. ciks.cbt.nist.gov /~garbocz/closedcell/node5.html   (579 words)

 Elastic properties of model random three-dimensional open-cell solids-Voronoi tessellations An open-cell Voronoi tessellation results if only the edges where two cell-walls intersect are defined as solid. It is worth noting that the tessellation of the BCC array (the tetrakaidecahedral cell model discussed above) is a reasonable approximation to the foam introduced by Lord Kelvin [Weaire and Fortes, 1994,Warren and Kraynik, 1997,Grenestedt, 1999]. The open-cell Voronoi tessellations of these distributions showed a polydisperse size distribution since the seed points may now be bunched in regions (see Figure 7). ciks.cbt.nist.gov /~garbocz/opencell/node4.html   (1125 words)

 About "What is Dirichlet Tessellation?" But the Delaunay triangulation also has a mathematical dual called the Dirichlet tessellation. This Dirichlet tessellation is also known as a Voronoi diagram or a collection of Theissen regions. The Math Forum is a research and educational enterprise of the Drexel School of Education. mathforum.org /library/view/4811.html   (83 words)

 [No title] The Dirichlet tessellation and its dual Delaunay network are used to connect the neighboring particles. A discrete homogenization method is used to estimate the linear effective properties of the material. The validity and limitations of the model have been discussed through comparisons with finite element simulations on regular periodic unit cells in the limit case of rigid inclusions. www.enpc.fr /lami/equipe/home/perso/lachihab/abstract.doc   (239 words)

 National Federation of Badger Groups   (Site not responding. Last check: 2007-10-22) The Independent Scientific Group recommends that boundaries of badger territories are established using the 'Dirichlet tessellations' method backed by field signs. It assumes that each social group has only one main sett, that neighbouring territories are contiguous, that the boundary of neighbouring social groups is positioned mid-way between each main sett and that territory configurations approximate to polygons. The Dirichlet tessellation method also assumes that territory boundaries do not change between years. www.badger.org.uk /tb/reason-2-4.html   (333 words)

 Voronoi tessellation   (Site not responding. Last check: 2007-10-22) Voronoi tessellation is a geometric dual of Delaunay triangulation and one can be derived from the other. Given a set of N points in a plane, Voronoi tessellation divides the domain in a set of polygonal regions, the boundaries of which are the perpendicular bisectors of the lines joining the points (Figure 4.1). Except in degenerate cases, the vertices of Voronoi tessellation occur where three tiles meet. www.anirudh.net /btp/main/node19.html   (225 words)

 [No title] However, as early as in 1850 another mathematician, G. Dirichlet, studied the problem. Accordingly the Voronoi diagram is sometimes named Dirichlet tessellation. The Voronoi diagram is a partition (tessellation) of R www.mlahanas.de /CompGeom/vor1.htm   (319 words)

 Voronoi diagrams   (Site not responding. Last check: 2007-10-22) Different names particular to the respective field have been used, such as medial axis transform in biology and physiology, Wigner-Seitz zones in chemistry and physics, domains of action in crystallography, and Thiessen polygons in metereology and geography. The mathematicians Dirichlet and Voronoi were the first to formally introduce this concept. The resulting structure has been called Dirichlet tessellation or Voronoi diagram, which has become its standard name today. www.igi.tugraz.at /Abstracts/ak-vd-00   (325 words)

 Voronoi Diagrams in Biology Byers, J. "Dirichlet tessellation of bark beetle spatial attack points." Journal of Animal Ecology. Use of such properties to determine the Dirichlet center when the regions are known. Sibson, R.; "The Dirichlet Tessellation as an Aid in Data Analysis"; Scandanavian Journal of Statistics, 7, pp. www.beloit.edu /~biology/zdravko/vor_reference.html   (2914 words)

 Dirichlet tesselation - Abstract (1) Algorithms for Dirichlet tessellation of spatial points are developed and implemented on personal computer. (2) The program also calculates Dirichlet cell areas and their coefficient of variation (CV) as well as the average nearest neighbour distance between points. All three species exhibited spacing between attack sites, in agreement with known behavioural mechanisms that are proposed for avoiding intraspecific competition for food resources. vinsonlab.tamu.edu /former/john/abstracts/dir-abs.htm   (195 words)

 [No title]   (Site not responding. Last check: 2007-10-22) } \item{col}{ the colour numbers for plotting the triangulation, the tesselation, the data points, the dummy points, and the point numbers, in that order; defaults to c(1,1,1,1,1). As well, the edges of the Delaunay triangles and/or of the Dirichlet tiles are plotted. By default the real points are plotted as circles (pch=1) and the dummy points are plotted as triangles (pch=2). www.stat.umn.edu /R/library/deldir/man/deldir.Rd   (1312 words)

 [No title]   (Site not responding. Last check: 2007-10-22) In this section each row has three entries: the number of sides --- within the rectangular window --- of the Dirichlet tile surrounding a given point, the number of points in which the Dirichlet tile intersects the boundary of the rectangular window, and the area of the Dirichlet tile surrounding the point. (The given point'' is the same as that in the corresponding row of the Delaunay summary''.) Note that the total area of the Dirichlet tessellation is equal to the sum of the last column of the Dirichlet summary''. 'dirseg.r' subroutine dirseg(dirsgs,ndir,nadj,madj,x,y,npd,ntot,rw,eps,ind,nerror) # Output the endpoints of the segments of boundary of Dirichlet # tiles. sunsite.univie.ac.at /statlib/general/delaunay   (6301 words)

 A Dirichlet Tessellation-based Sampling Scheme for Measuring Whole-body Exposure -- WHEELER and WARREN 46 (2): 209 -- ...   (Site not responding. Last check: 2007-10-22) A Dirichlet Tessellation-based Sampling Scheme for Measuring Whole-body Exposure -- WHEELER and WARREN 46 (2): 209 -- Annals of Occupational Hygiene Panel cut from overall covering left arm and upper left body showing the position of the PXRF measurements and their Dirichlet tessellation (a colour version of this figure is available with the online version of this paper). Correlation between exposure estimates from acid digestion of patches and the Dirichlet PXRF method with whole suit acid digestion. annhyg.oxfordjournals.org /cgi/content/full/46/2/209   (4520 words)

 Johann Peter Gustav Lejeune Dirichlet - ExampleProblems.com Biography of Dirichlet found at Fermat's Last Theorem Blogde:Peter Gustav Lejeune Dirichlet es:Peter Gustav Lejeune Dirichlet fr:Johann Peter Gustav Lejeune Dirichlet ko:페터 구스타프 르죈 디리클레 id:Peter Gustav Lejeune Dirichlet it:Peter Gustav Lejeune Dirichlet nl:Johann Dirichlet pl:Peter Gustav Lejeune Dirichlet pt:Johann Peter Gustav Lejeune Dirichlet ru:Лежён-Дирихле, Петер Густав sk:Peter Gustav Lejeune Dirichlet sl:Johann Peter Gustav Lejeune Dirichlet sv:Johann Peter Gustav Lejeune Dirichlet zh:約翰·彼得·狄利克雷 he:דיריכלה This page was last modified 14:03, 27 July 2006. www.exampleproblems.com /wiki/index.php/Dirichlet   (332 words)

 R: Compute Quadrature Weights Based on Dirichlet Tessellation   (Site not responding. Last check: 2007-10-22) Computes quadrature weights for a given set of points, using the areas of tiles in the Dirichlet tessellation. The weights are computed using the Dirichlet tessellation. the Dirichlet tessellation is computed exactly by the Lee-Schachter algorithm using the package pbil.univ-lyon1.fr /library/spatstat/html/dirichlet.weights.html   (175 words)

 TILE   (Site not responding. Last check: 2007-10-22) Subroutines for creating and manipulating the Dirichlet (Voronoi) tessellation of a set of points in the plane, for natural neighbour interpolation based on the tessellation, and for plotting perspective block views of surfaces. Based on 'Computing Dirichlet tessellations in the plane' by Peter Green and Robin Sibson, Computer Journal 21, 168-173 (1978). This software was first developed while the originators were at the University of Bath. www.stats.bris.ac.uk /~peter/Tile.html   (206 words)

 Voronoi polygons   (Site not responding. Last check: 2007-10-22) Tessellation of the plane such that any given location is assigned to a tile according to the minimum distance between it and a single, previously sampled point Thiessen polygons is that they can be easily used with qualitative data like vegetation classes or land use if all you need is a choropleth map and do not mind the strange geometrical pattern of the boundaries Dirichlet tessellation produces, as one may expect, values Y j at the pseudo design points X j which are highly discontinuous along the borders of the Dirichlet tiles www.uem.es /web/fil/alumnos/proyectos/glosarios/teledeteccion/19EN.htm   (158 words)

 [No title] In first part students have to demonstrate they can construct a tessellation; and in second part, turn it into a triangulation. The diagram below shows the Dirichlet tessellation of three points, A, B, C, in a plane. Rather than a set of lines dividing the space into regions ‘closest’ to a point, we now have a set of planes, again constructed as perpendicular bisectors (3 marks). www.comp.leeds.ac.uk /kwb/VIS/vis_03_solutions.doc   (1933 words)

 On Finding p-th Nearest Neighbours of Scattered Points in Two Dimensions for Small p (ResearchIndex) If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies. Abstract: Given a large set of scattered points in the plane, we describe a new and efficient algorithm to find, for each point, the subset of p closest points, using the Dirichlet tessellation of the set of points, for small values of p. 40 Computing Dirichlet tessellations in the plane (context) - Green, Sibson - 1978 citeseer.ist.psu.edu /643267.html   (300 words)

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