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Topic: Voronoi tessellation

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	Vaisman et al, Computational Geometry of Macromolecular Structure
		The method, including the design and implementation of practical algorithms, was further developed by Finney for the case of Voronoi tessellation (Finney 1970, 1977).
		The topological difference between these objects is that the Voronoi polyhedron represents the environment of individual atoms whereas the Delaunay simplex represents the ensemble of neighboring atoms.
		Whereas the Voronoi polyhedra may differ topologically (i.e., they may have different numbers of faces and edges), the Delaunay simplices are always topologically equivalent (i.e., in three-dimensional space they are always tetrahedra).
	www.unc.edu /~ivaisman/delaunay/tessell.htm (391 words)

	Geometry in Action: Voronoi Diagrams
		The Voronoi diagram of a collection of geometric objects is a partition of space into cells, each of which consists of the points closer to one particular object than to any others.
		Voronoi diagrams tend to be involved in situations where a space should be partitioned into "spheres of influence", including models of crystal and cell growth as well as protein molecule volume analysis.
		Elias Kalaitzis of Edinburgh uses 3d Voronoi diagrams in an iterated parallel procedure for approximating a geometric transformation aligning a pair of shapes.
	www.ics.uci.edu /~eppstein/gina/voronoi.html (1071 words)

	Voronoi (Site not responding. Last check: )
		Example of Voronoi tessellation of a two dimensional face-space.
		The coloured areas are Voronoi cells defined by a nearest neighbour rule.
		The circles and lines demonstrate how a simple neural network can produce a Voronoi tessellation of identity fields (those are the area over which that identity produces the strongest response of that identity).
	www.cf.ac.uk /psych/LewisMB/vorono.shtml (137 words)

	Science Fair Projects - Voronoi diagram
		In mathematics, a Voronoi diagram, also called a Voronoi tessellation or Voronoi decomposition, named after Georgy Voronoi, also called 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.
		Voronoi diagrams are named after Russian mathematician Georgy Fedoseevich Voronoi (or Voronoy) who defined and studied the general n-dimensional case in 1908.
		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.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Voronoi_diagram (809 words)

	Voronoi Diagrams in Biology
		However, Voronoi diagrams are used much more widely today because another German mathematician M. Voronoi in 1908 studied the concept and defined it for a more general n-dimensional case.
		On Figure 7 all the numbers of the coordinates from Table 1 are marked- the Voronoi centers (entrance points) with fl dots and numbers while the circumcenters are marked with gray dots and numbers on a white background.
		Voronoi diagrams are marked with fl capital letters on white background.
	www.beloit.edu /~biology/zdravko/vor_paper.html (1783 words)

	Transform - Voronoi Operators
		Voronoi diagrams are very important for dividing drawings into regions associated with points.
		All of the Voronoi operators except Voronoi Lines will transfer column data from source to target (created) objects using whatever transfer rules are in force for the data attribute columns.
		Voronoi later published a generalization of this concept that would apply to higher dimensions and so introduced the concept in its modern form.
	exchange.manifold.net /manifold/manuals/5_userman/mfd50Transform__Voronoi_Operators.htm (556 words)

	Centroidal Voronoi Tessellations
		A centroidal Voronoi tessellation is a Voronoi tessellation of a given set such that the associated generating points are centroids (centers of mass with respect to a given density function) of the corresponding Voronoi regions.
		Such tessellations are useful, in among other contexts, in data compression, optimal quadrature rules, optimal representation and quantization, finite difference schemes, optimal distribution of resources, cellular biology, and the territorial behavior of animals.
		A centroidal Voronoi tessellation is a Voronoi tessellation of a given set such that the associated generating points are centroids (centers of mass) of the corresponding Voronoi regions.
	people.scs.fsu.edu /~gunzburg/voronoi.html (1969 words)

	Voronoi tessellation - references
		Hinde, A.L. and R.E. Miles (1980) Monte Carlo estimates of the distribution of the random polygons of the Voronoi tessellation with respect to a Poisson process.
		Voronoi, G. (1908) Nouvelles applications des parametres continus a la théorie des formes quadratiques, deuxieme memoire, recherches sur les parallelloedres primitifs.
		Zuyev, S.A. (1992) Estimates of the Voronoi polygon's geometric characteristics.
	fyzika.ft.utb.cz /voronoi/references.htm (1278 words)

	Using the Voronoi Tessellation for Grouping Words and Multi-part Symbols in Documents
		We present a number of benefits to using the Voronoi neighborhood definition; however, we argue that definitions based upon the point Voronoi diagrams are insufficient in the general case (e.g.
		We give the definition of a generalized (Euclidean distance measure, two dimensional Cartesian space, and an area based generator set) Voronoi tessellation and then present our algorithm for approximating this generalized tessellation.
		The algorithm is constructed from a normal point Voronoi tessellation algorithm.
	www.computing.armstrong.edu /FacNStaff/burge/publications/papers/visgeo-95 (220 words)

	Voronoi diagram Information
		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.
		The set of such polytopes tessellates the whole space, and is the Voronoi tessellation corresponding to the set S.
		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.
	www.bookrags.com /wiki/Voronoi_diagram (857 words)

	Fracture Animation
		Domain discretization is based on a Voronoi diagram on an irregular set of points, which serve as computational nodes for both the elasticity and diffusion analyses.
		By definition, the Voronoi polyhedron (or cell) associated with node i is the set of points closer to node i than all other nodes in the domain (Okabe et al., 1992).
		Here, the Voronoi diagram is constructed from its dual, the Delaunay tessellation, since the latter is generally easier to construct and a robust program was available for doing so (Taniguchi et al., 2002).
	cee.engr.ucdavis.edu /faculty/bolander/voronoi.html (553 words)

	RANDOM VORONOI CELLS OF HIGHER DIMENSIONS
		Tessellation of space into convex polyhedra is often an interesting theme of the research of symmetry.
		As the characteristics of Poisson Voronoi polygons, Hinde and Miles (1980) reported the number of edges N, the area A, the perimeter S and so on, and then fitted their histograms to the three parameter generalized gamma density.
		In case of three dimensional Poisson Voronoi cells, Tanemura (1988) reported the distributions of the volume V and the number of faces F based on a hundred thousand samples of Voronoi polyhedra.
	www.mi.sanu.ac.yu /vismath/proceedings/tanemura.htm (1041 words)

	Voronoi Diagrams
		The Voronoi diagram of a collection of geometric objects is a partition of space into cells, each of which consists of the points closer to one particular object than to any others.
		Voronoi diagrams tend to be involved in situations where a space should be partitioned into "spheres of influence", including models of crystal and cell growth as well as protein molecule volume analysis.
		US Patent 5564004 uses Voronoi diagrams as part of a user interface that highlights the icon nearest the cursor in a windowing system.
	www.labvis.unam.mx /elio/Geom/local/voronoi.html (882 words)

	Voronoi diagram - Term Explanation on IndexSuche.Com
		or Dirichlet tessellation 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.
		Hence ''Voronoi diagram'' is a less restrictive term to describe the partition of the space by Voronoi polytopes.
		Extensions of the Voronoi tesselation for the case when distances to areas are compared rather than to points are of use in image segmentation, optical_character_recognition and other computational applications.
	www.indexsuche.com /Voronoi_diagram.html (387 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)

	ETHZ - Computer Vision Lab: Publications
		The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image.
		The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms.
		Consequently, in Computer Vision applications, the computation of the Voronoi tessellation represents an attractive alternative to raster-based techniques in terms of both computational complexity and quality of data structures.
	www.vision.ee.ethz.ch /publications/get_abstract.cgi?articles=50&mode=&lang=en (192 words)

	Voronoi diagram at AllExperts
		In general a cross section of a 3D Voronoi tesselation is not a 2D Voronoi tesselation itself.
		* 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.
		In materials science, polycrystalline microstructures in metallic alloys are commonly represented using Voronoi tessellations.
	en.allexperts.com /e/v/vo/voronoi_diagram.htm (939 words)

	Voronoi diagrams (Site not responding. Last check: )
		Voronoi was the first to consider the dual of this structure, where any two point sites are connected whose regions have a boundary in common.
		Moreover, these structures are very appealing, and a lot of research has been devoted to their study (about one out of 16 papers in computational geometry), ever since Shamos and Hoey introduced them to the field.
		Instead, we are trying to highlight the intrinsic potential of Voronoi diagrams, that lies in its structural properties, in the existence of efficient algorithms for its construction, and in its adaptability.
	www.igi.tugraz.at /Abstracts/ak-vd-00 (325 words)

	8. Voronoi Tessellation & Percolation (Site not responding. Last check: )
		The application of a non parametric percolation to the tessellation cells exceeding this noise level leads directly to a source list which is free of any assumptions about the source geometry.
		Apart from the Voronoi cells in high-flux regions around the sources we are actually looking for, fluctuations in the random background will of course also be found in the upper end of the flux distribution.
		Knowing the total area covered by each source from the Voronoi tessellations, a background correction can be applied by simply subtracting the number of background photons statistically expected in the same area.
	ledas-cxc.star.le.ac.uk /udocs/docs/swdocs/detect/html/node16.html (3205 words)

	Introduction to the Theory of Design (Site not responding. Last check: )
		A simple example is the point distribution with regular triangle configuration, which leads to a Voronoi tessellation composed of equal regular hexagons (see the right figure).
		An important property of the Voronoi tessellation is that any point on the Voronoi network is separated equally from the two points on both sides of the network line (see the right figure) This property seems to give a mechnism how borders of countries are determined.
		It is an element of a 3D Voronoi tessellation for a point distribution called body-centered cubic lattice, which is a periodic arrangement made of the cubic lattice and one point at the center of each cube (see figure below).
	www.kobe-du.ac.jp /gsdr/gsdr/kiso04/05-e.html (1511 words)

	Computational Geometry
		Defining these Voronoi regions for all the points in the set S results in a tessellation of the plane which is called the Voronoi tessellation[1,2].
		The Voronoi tessellation and the polygonal region assigned to each dot in this representation lead to a natural and intuitive definition of neighborhood of a dot.
		Because the Voronoi tessellation is adaptive to variations in dot distribution, such properties of the Voronoi polygons as the area, shape, etc. also vary with changes of scale, density variations, etc., thus reflecting the differences in spatial distributions of dots.
	www.cs.iupui.edu /~tuceryan/research/ComputerVision/computational-geometry.html (594 words)

	Voronoi diagram
		In mathematics, a Voronoi diagram or Dirichlet tessellation 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.
		However in these cases the Voronoi tessellation is not guaranteed to exist (or to be a "true" tessellation), since the equidistant for two points may fail to be subspace of codimension 1, even in the 2-dimensional case.
		Spatial Tessellations - Concepts and Applications of Voronoi Diagrams.
	www.xasa.com /wiki/en/wikipedia/v/vo/voronoi_diagram.html (592 words)

	The large-scale structure of the universe and quasi-Voronoi tessellation of shock fronts in forced Burgers turbulence ...
		Burgers turbulence is an accepted formalism for the adhesion model of the large-scale distribution of matter in the universe.
		The paper uses variational methods to establish evolution of quasi-Voronoi (curved boundaries) tessellation structure of shock fronts for solutions of the inviscid nonhomogeneous Burgers equation in $R^d$ in the presence of random forcing due to a degenerate potential.
		The mean rate of growth of the quasi-Voronoi cells is calculated and a scaled limit random tessellation structure is found.
	www.projecteuclid.org /Dienst/UI/1.0/Display/euclid.aoap/1034625260 (160 words)

	Voronoi tessellation
		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)

	The Voronoi Diagram Package
		The Voronoi diagram is a simple mathematical construct that has proved useful in fields as diverse as environmental studies, cell biology, crystallography, transportation planning, and communications theory.
		A Voronoi diagram may be generated by any finite set of points in a plane.
		A related concept is the Delaunay triangulation, which is obtained from a Voronoi diagram by drawing straight lines between each polling place and polling places in neighboring districts.
	users.rcn.com /peterash/voronoi.html (832 words)

	Voronoi Tessellation of Points with Integer Coordinates: Time-Efficient Implementation and Online Edge-List Generation ...
		Abstract: The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image.
		The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms.
		29 Hierarchic Voronoi skeletons - Ogniewicz, Kubler - 1995
	citeseer.ist.psu.edu /ogniewicz93voronoi.html (495 words)

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