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| [[Image:Euclidean Voronoi Diagram.png|thumb|right|width=200|20 points and their Voronoi cells (larger version [[#Illustration|below]]).]]
| | So, here is one of the easiest fat reduction plans which has no limited food options like a fad diet, nor any tedious techniques. But, a set of easy rules plus strategies that will assist you reduce weight in ten days. You should remember which because 1 does not gain fat overnight, neither may it be lost overnight. These tips usually assist you to lower fat faster than a normal weight loss regime.<br><br>Then, in a quest tolose weight fast, would we want to feel hungry all of the time; needing to taste the bitter, sour bile inside the vomit; having hair which becomes thin and brittle; enduring heart, kidney, muscles and blood problems; having ugly dry skin; and lookin like a bag of bones and eventually die? These are the consequences of anorexia nervosa. Of course not, enjoying the food plus feeling wonderful about our healthy bodies is what we wish to achieve. So why not do it the healthy, secure, efficient means. And these techniques are truly frequently much easier plus quicker than you think.<br><br>Chew the food thoroughly - If you chew a food thoroughly, you'd not only enjoy the food better, nevertheless it could also enable we lose weight. This is because doing which would prolong a time of eating, that would offer your body a chance to distinguish being full. In brief, when you eat slower, you'd be consuming lesser amounts of foods, than eating quicker.<br><br>First of all, many people mistakenly think which HCG is simply another fancy designer drug. In fact, HCG is a hormone that occurs naturally in every of our bodies. When females are pregnant, they have a very big supply of this hormone. Because of this, whenever a girl takes a pregnancy test, when a certain concentration of HCG is found in the circulation, then the girl is considered positive for the test.<br><br>Eat more often. This really is probably the number one tip of the "7 ways [http://safedietplansforwomen.com/how-to-lose-weight-fast how to lose weight fast for women] and easy". By eating little and often throughout the day, you'll naturally accelerate your metabolism, meaning you'll burn calories more effectively. Needless to say, it doesn't indicate eat a 3-course meal, 6 occasions a day?snacks such as raisins, an apple, celery or carrot sticks are advantageous options and because you may be eating continuously, we are less probably to stray towards foods you need to be limiting.<br><br>Cardio is an aerobic exercise which we do for hours at slow - moderate pace with an intensity level where you are able to nevertheless talk. It does burn fat however only during the exercise bout. Once we stop, the fat-burning impact stops.<br><br>Use Behavioral Change to implement the changes that we require in the brand-new healthy eating practices. This can insure which you are on a fresh healthy eating program till a reach a goals and forever if you want to remain healthy plus fit. |
| In [[mathematics]], a '''Voronoi diagram''' is a way of dividing space into a number of regions. A set of points (called seeds, sites, or generators) is specified beforehand and for each seed there will be a corresponding region consisting of all points [[distance|closer]] to that seed than to any other. The regions are called Voronoi cells. It is [[Duality (mathematics)|dual]] to the [[Delaunay triangulation]].
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| It is named after [[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 diagrams can be found in a large number of fields in [[science]] and [[technology]], even in [[art]], and they have found numerous practical and theoretical applications.<ref>Franz Aurenhammer (1991). ''Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure''. ACM Computing Surveys, 23(3):345–405, 1991</ref><ref>Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). ''Spatial Tessellations – Concepts and Applications of Voronoi Diagrams''. 2nd edition. John Wiley, 2000, 671 pages ISBN 0-471-98635-6</ref>
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| ==The simplest case==
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| In the simplest and most familiar case (shown in the first picture), we are given a finite set of points {''p''<sub>1</sub>, …, ''p''<sub>''n''</sub>} in the [[Euclidean plane]]. In this case each site ''p''<sub>''k''</sub> is simply a point and its corresponding Voronoi cell (also called Voronoi region or Dirichlet cell) ''R''<sub>''k''</sub> consisting of every point whose distance to ''p''<sub>''k''</sub> is less than or equal to its distance to any other site. Each such cell is obtained from the intersection of half-spaces, and hence it is a [[convex polygon]]. The segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices ([[Node (graph theory)|node]]s) are the points equidistant to three (or more) sites.
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| ==Formal definition==
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| Let <math>\scriptstyle X</math> be a space (a nonempty [[Set (mathematics)|set]]) endowed with a distance function <math>\scriptstyle d</math>. Let <math>\scriptstyle K</math> be a set of indices and let <math>\scriptstyle (P_k)_{k \in K}</math> be a [[tuple]] (ordered collection) of nonempty [[subsets]] (the sites) in the space <math>\scriptstyle X</math>. The Voronoi cell, or Voronoi region, <math>\scriptstyle R_k</math>, associated with the site <math>\scriptstyle P_k</math> is the set of all points in <math>\scriptstyle X</math> whose distance to <math>\scriptstyle P_k</math> is not greater than their distance to the other sites <math>\scriptstyle P_j</math>, where <math>\scriptstyle j</math> is any index different from <math>\scriptstyle k</math>. In other words, if <math>\scriptstyle d(x,\, A) \;=\; \inf\{d(x,\, a) \;|\; a \,\in\, A\}</math> denotes the distance between the point <math>\scriptstyle x</math> and the subset <math>\scriptstyle A</math>, then
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| :<math> R_k = \{x \in X \;|\; d(x, P_k) \leq d(x, P_j)\; \text{for all}\; j \neq k\}</math>
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| The Voronoi diagram is simply the [[tuple]] of cells <math>\scriptstyle (R_k)_{k \in K} </math>. In principle some of the sites can intersect and even coincide (an application is described below for sites representing shops), but usually they are assumed to be disjoint. In addition, infinitely many sites are allowed in the definition (this setting has applications in [[geometry of numbers]] and [[crystallography]]), but again, in many cases only finitely many sites are considered.
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| In the particular case where the space is a [[finite dimensional]] [[Euclidean space]], each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are [[convex polytope]]s and they can be represented in a combinatorial way using their vertices, sides, 2-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. However, in general the Voronoi cells may not be convex or even connected.
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| ==Illustration==
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| As a simple illustration, consider a group of shops in a flat city. Suppose we want to estimate the number of customers of a given shop. With all else being equal (price, products, quality of service, etc.), it is reasonable to assume that customers choose their preferred shop simply by distance considerations: they will go to the shop located nearest to them. In this case the Voronoi cell <math>\scriptstyle R_k</math> of a given shop <math>\scriptstyle P_k</math> can be used for giving a rough estimate on the number of potential customers going to this shop (which is modeled by a point in our flat city).
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| So far it was assumed that the distance between points in the city is measured using the standard distance, the familiar
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| [[Euclidean distance]]: <math>\ell_2 = d\left[\left(a_1, a_2\right), \left(b_1, b_2\right)\right] = \sqrt{\left(a_1 - b_1\right)^2 + \left(a_2 - b_2\right)^2}</math>
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| However, if we consider the case where customers only go to the shops by a vehicle and the traffic paths are parallel to the <math> x</math> and <math> y </math> axes, as in [[Manhattan distance | Manhattan]], then a more realistic distance function will be the [[Taxicab geometry|<math>\ell_1</math> distance]], namely <math>d\left[\left(a_1, a_2\right), \left(b_1, b_2\right)\right] = \left|a_1 - b_1\right| + \left|a_2 - b_2\right|</math>.
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| {{multiple image
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| | align = center
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| | direction = horizontal
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| | width = 382
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| | header = Voronoi diagrams of 20 points under two different metrics
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| | header_align = center
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| | image1 = Euclidean Voronoi Diagram.png
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| | alt1 = Voronoi diagram under Euclidean distance
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| | caption1 = [[Euclidean distance]]
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| | image2 = Manhattan Voronoi Diagram.png
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| | alt2 = Voronoi diagram under Manhattan distance
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| | caption2 = [[Taxicab geometry|Manhattan distance]]
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| }}
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| ==Properties==
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| * The [[dual graph]] for a Voronoi diagram (in the case of a [[Euclidean space]] with point sites) corresponds to the [[Delaunay triangulation]] for the same set of points.
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| * The [[closest pair of points]] corresponds to two adjacent cells in the Voronoi diagram.
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| * Assume the setting is the [[Euclidean plane]] and a group of different points are given. Then two points are adjacent on the [[convex hull]] if and only if their Voronoi cells share an infinitely long side.
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| * If the space is a [[normed space]] and the distance to each site is attained (e.g., when a site is a [[compact set]] or a closed ball), then each Voronoi cell can be represented as a union of line segments emanating from the sites.<ref name=Reem_alg>Daniel Reem, ''An algorithm for computing Voronoi diagrams of general generators in general normed spaces, [http://dx.doi.org/10.1109/ISVD.2009.23 In Proceedings of the sixth International Symposium on Voronoi Diagrams in science and engineering (ISVD 2009), 2009, pp. 144–152]</ref> As shown there, this property does not necessarily hold when the distance is not attained.
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| * Under relatively general conditions (the space is a possibly infinite dimensional [[uniformly convex space]], there can be infinitely many sites of a general form, etc.) Voronoi cells enjoy a certain stability property: a small change in the shapes of the sites, e.g., a change caused by some translation or distortion, yields a small change in the shape of the Voronoi cells. This is the geometric stability of Voronoi diagrams.<ref name=Reem_geo_stable>Daniel Reem, ''The geometric stability of Voronoi diagrams with respect to small changes of the sites'', Full version: [http://arxiv.org/abs/1103.4125 arXiv 1103.4125 (2011)], Extended abstract [http://dx.doi.org/10.1145/1998196.1998234 in Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG 2011), pp. 254–263]</ref> As shown there, this property does not hold in general, even if the space is two-dimensional (but non-uniformly convex, and, in particular, non-Euclidean) and the sites are points.
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| ==History and research==
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| Informal use of Voronoi diagrams can be traced back to [[Descartes]] in 1644. [[Peter Gustav Lejeune Dirichlet]] used 2-dimensional and 3-dimensional Voronoi diagrams in his study of quadratic forms in 1850.
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| British physician [[John Snow (physician)|John Snow]] used a Voronoi diagram in 1854 to illustrate how the majority of people who died in the [[1854 Broad Street cholera outbreak|Soho cholera epidemic]] lived closer to the infected [[Soho#Broad_Street_pump|Broad Street pump]] than to any other water pump.
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| Voronoi diagrams are named after Ukrainian mathematician [[Georgy Voronoy|Georgy Fedosievych Voronyi]] (or ''Voronoy'') who defined and studied the general ''n''-dimensional case in 1908. Voronoi diagrams that are used in [[geophysics]] and [[meteorology]] to analyse spatially distributed data (such as rainfall measurements) are called Thiessen polygons after American meteorologist [[Alfred H. Thiessen]]. In [[condensed matter physics]], such tessellations are also known as [[Wigner–Seitz unit cell]]s. Voronoi tessellations of the [[reciprocal lattice]] of [[momentum|momenta]] are called [[Brillouin zone]]s. For general lattices in [[Lie group]]s, the cells are simply called [[fundamental domain]]s. In the case of general [[metric space]]s, the cells are often called metric [[fundamental polygon]]s.
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| Other equivalent names for this concept (or particular important cases of it) : [[Voronoi polyhedra]], [[Voronoi polygon]]s, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), [[Dirichlet tessellation]](s).
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| == Examples ==
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| [[Image:Coloured Voronoi 3D slice.svg|right|thumb|This is a slice of the Voronoi diagram of a random set of points in a 3D box. In general a cross section of a 3D Voronoi tessellation is not a 2D Voronoi tessellation itself. (The cells are all [[Convex set|convex]] [[polyhedron|polyhedra]].)]]
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| Voronoi tessellations of regular [[Lattice (group)|lattice]]s of points in two or three dimensions give rise to many familiar tessellations.
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| * 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 (geometry)|square]] lattice gives the regular tessellation of squares; note that the rectangles and the squares can also be generated by other lattices (for example the lattice defined by the vectors (1,0) and (1/2,1/2) gives squares). See [http://mbostock.github.com/d3/ex/voronoi.html here] for a dynamic visual example.
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| * A [[simple cubic lattice]] gives the [[cubic honeycomb]].
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| * A [[hexagonal close-packed]] lattice gives a tessellation of space with [[trapezo-rhombic dodecahedron|trapezo-rhombic dodecahedra]].
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| * A [[face-centred cubic]] lattice gives a tessellation of space with [[rhombic dodecahedron|rhombic dodecahedra]].
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| * A [[body-centred cubic]] lattice gives a tessellation of space with [[truncated octahedron|truncated octahedra]].
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| * Parallel planes with regular triangular lattices aligned with each other's centers give the [[hexagonal prismatic honeycomb]].
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| * Certain body centered tetragonal lattices give a tessellation of space with [[rhombo-hexagonal dodecahedron|rhombo-hexagonal dodecahedra]].
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| For the set of points (''x'', ''y'') with ''x'' in a discrete set ''X'' and ''y'' in a discrete set ''Y'', we get rectangular tiles with the points not necessarily at their centers.
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| == Higher-order Voronoi diagrams ==
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| {{Expand section|date=June 2009}}
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| {{cleanup|section|date=August 2009}}
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| Although a normal Voronoi cell is defined as the set of points closest to a single point in ''S'', an ''n''th-order Voronoi cell is defined as the set of points having a particular set of ''n'' points in ''S'' as its ''n'' nearest neighbors. Higher-order Voronoi diagrams also subdivide space.
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| Higher-order Voronoi diagrams can be generated recursively. To generate the ''n''<sup>th</sup>-order Voronoi diagram from set ''S'', start with the (''n'' − 1)<sup>th</sup>-order diagram and replace each cell generated by ''X'' = {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''−1</sub>} with a Voronoi diagram generated on the set ''S'' − ''X''.
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| ===Farthest-point Voronoi diagram===
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| For a set of ''n'' points the (''n''−1)<sup>th</sup>-order Voronoi diagram is called a farthest-point Voronoi diagram.
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| For a given set of points ''S'' = {''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''</sub>} the farthest-point Voronoi diagram divides the plane into cells in which the same point of ''P'' is the farthest point. Note that a point of ''P'' has a cell in the farthest-point Voronoi diagram if and only if it is a vertex of the [[convex hull]] of ''P''. Thus, let ''H'' = {''h''<sub>1</sub>, ''h''<sub>2</sub>, ..., ''h''<sub>''k''</sub>} be the convex hull of ''P'' we define the farthest-point Voronoi diagram as the subdivision of the plane into ''k'' cells, one for each point in ''H'', with the property that a point ''q'' lies in the cell corresponding to a site ''h''<sub>''i''</sub> if and only if dist(''q'', ''h''<sub>''i''</sub>) > dist(''q'', ''p''<sub>''j''</sub>) for each ''p''<sub>''j''</sub> ∈ ''S'' with ''h''<sub>''i''</sub> ≠ ''p''<sub>''j''</sub>. Where dist(''p'', ''q'') is the [[Euclidean distance]] between two points ''p'' and ''q''.<ref name="berg2008">{{cite book|author=[[Mark de Berg]], [[Marc van Kreveld]], [[Mark Overmars]], and [[Otfried Schwarzkopf]]|year=2008|title=Computational Geometry|publisher=[[Springer-Verlag]]|edition=Third edition}} 7.4 Farthest-Point Voronoi Diagrams. Includes a description of the algorithm.</ref>
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| <ref>{{cite journal|first=Sven|last=Skyum|first2=S.|title=A simple algorithm for computing the smallest enclosing circle|journal=Information Processing Letters 37(1991)121–125|year=1991}}, contains a simple algorithm to compute the farthest-point Voronoi diagram.</ref>
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| == Generalizations and variations ==
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| As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean (such as the [[Mahalanobis distance|Mahalanobis]] or [[Manhattan distance|Manhattan]]) distances. However in these cases the boundaries of the Voronoi cells may be more complicated than in the Euclidean case, since the equidistant locus for two points may fail to be subspace of codimension 1, even in the 2-dimensional case.
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| A [[weighted Voronoi diagram]] is the one in which the function of a pair of points to define a Voronoi cell is a distance function modified by multiplicative or additive weights assigned to generator points. In contrast to the case of Voronoi cells defined using a distance which is a [[metric (mathematics)|metric]], in this case some of the Voronoi cells may be empty. A [[power diagram]] is a type of Voronoi diagram defined from a set of circles using the [[Power of a point|power distance]]; it can also be thought of as a weighted Voronoi diagram in which a weight defined from the radius of each circle is added to the [[quadrance|squared distance]] from the circle's center.<ref>{{citation|last=Edelsbrunner|first=Herbert|author-link=Herbert Edelsbrunner|contribution=13.6 Power Diagrams|pages=327–328|publisher=Springer-Verlag|series=EATCS Monographs on Theoretical Computer Science|title=Algorithms in Combinatorial Geometry|volume=10|year=1987}}.</ref>
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| [[Image:ApproximateVoronoiDiagram.png|right|thumb|Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells.]]
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| The Voronoi diagram of ''n'' points in ''d''-dimensional space requires <math>\scriptstyle O\left(n^{\left\lceil \frac{1}{2}d \right\rceil}\right)</math> storage space. Therefore, Voronoi diagrams are often not feasible for ''d'' > 2. An alternative is to use [[approximate Voronoi diagram]]s, where the Voronoi cells have a fuzzy boundary, which can be approximated.<ref>S. Arya, T. Malamatos, and [[David Mount|D. M. Mount]],
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| [http://www.cs.ust.hk/faculty/arya/pub/stoc02.pdf Space-Efficient Approximate Voronoi Diagrams], Proc. 34th ACM Symp. on Theory of Computing (STOC 2002), pp. 721–730.</ref> Another alternative is when any site is a fuzzy circle and as a result the cells become fuzzy too.<ref>{{Cite journal|first1=Mohammadreza|last1=Jooyandeh|first2=Ali|last2=Mohades|first3=Maryam|last3=Mirzakhah|title=Uncertain Voronoi Diagram|journal=Information Processing Letters|volume=109|issue=13|pages=709–712|doi=10.1016/j.ipl.2009.03.007|url=http://jooyandeh.info/Publications/UncertainVoronoiDiagram.pdf|year=2009|publisher=Elsevier}}</ref>
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| Voronoi diagrams are also related to other geometric structures such as the [[medial axis]] (which has found applications in image segmentation, [[optical character recognition]], and other computational applications), [[straight skeleton]], and [[zone diagrams]].
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| ==Applications==
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| *In [[epidemiology]], Voronoi diagrams can be used to correlate sources of infections in epidemics. One of the early applications of Voronoi diagrams was implemented by [[John Snow (physician)|John Snow]] to study the [[1854 Broad Street cholera outbreak]] in Soho, England. He showed the correlation between areas on the map of London using a particular water pump, and the areas with most deaths due to the outbreak.
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| *A [[point location]] data structure can be built on top of the Voronoi diagram in order to answer [[nearest neighbor search|nearest neighbor]] queries, where one wants to find the object that is closest to a given query point. Nearest neighbor queries have numerous applications. For example, one might want to find the nearest hospital, or the most similar object in a [[database]]. A large application is [[vector quantization]], commonly used in [[data compression]].
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| *In [[geometry]], Voronoi diagrams can be used to find the [[Largest empty sphere|largest empty circle]] amid a set of points, and in an enclosing polygon; e.g. to build a new supermarket as far as possible from all the existing ones, lying in a certain city.
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| *Voronoi diagrams together with farthest-point Voronoi diagrams are used for efficient algorithms to compute the [[Roundness (object)|roundness]] of a set of points.<ref name="berg2008">{{cite book|author=[[Mark de Berg]], [[Marc van Kreveld]], [[Mark Overmars]], and [[Otfried Schwarzkopf]]|year=2008|title=Computational Geometry |isbn=978-3-540-77974-2 |publisher=[[Springer-Verlag]]|edition=Third edition}} 7.4 Farthest-Point Voronoi Diagrams. Includes a description of the algorithm.</ref>
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| *The Voronoi approach is also put to good use in the evaluation of circularity/[[roundness (object)|roundness]] while assessing the dataset from a [[coordinate-measuring machine]].
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| *In networking, Voronoi diagrams can be used in derivations of the capacity of a [[wireless network]].
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| *In [[climatology]], Voronoi diagrams are used to calculate the rainfall of an area, based on a series of point measurements. In this usage, they are generally referred to as Thiessen polygons.
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| *In [[ecology]], Voronoi diagrams are used to study the growth patterns of forests and forest canopies, and may also be helpful in developing predictive models for forest fires.
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| *In [[computational chemistry]], Voronoi cells defined by the positions of the nuclei in a molecule are used to compute [[partial charge|atomic charge]]s. This is done using the [[Voronoi deformation density]] method.
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| *In polymer physics, Voronoi diagrams can be used to represent free volumes of polymers.
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| *In [[materials science]], polycrystalline microstructures in metallic alloys are commonly represented using Voronoi tessellations. In solid state physics, the [[Wigner-Seitz cell]] is the Voronoi tessellation of a solid, and the [[Brillouin zone]] is the Voronoi tessellation of reciprocal (wave number) space of crystals which have the symmetry of a space group.
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| *In [[mining]], Voronoi polygons are used to estimate the reserves of valuable materials, minerals, or other resources. Exploratory drillholes are used as the set of points in the Voronoi polygons.
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| *In computer graphics, Voronoi diagrams are used to procedurally generate organic or lava-looking textures.
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| * In autonomous [[robot navigation]], Voronoi diagrams are used to find clear routes. If the points are obstacles, then the edges of the graph will be the routes furthest from obstacles (and theoretically any collisions).
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| *In [[machine learning]], Voronoi diagrams are used to do [[k-nearest neighbor algorithm|1-NN]] classifications.<ref>{{cite book|title=Machine Learning|author=Tom M. Mitchell|year=1997|publisher=McGraw-Hill|edition=International Edition 1997|isbn= 0-07-042807-7|page=233}}</ref>
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| ==See also==
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| ;Algorithms
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| *[[Bowyer–Watson algorithm]], an algorithm for generating a Voronoi diagram in any number of dimensions.
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| *[[Fortune's algorithm]], an [[big O notation|O]](''n'' log(''n'')) algorithm for generating a Voronoi diagram from a set of points in a plane.
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| *[[Lloyd's algorithm]], aka [[k-means]] clustering, produces a Voronoi tessellation in a space of arbitrary dimensions
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| ;Related subjects
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| *[[Centroidal Voronoi tessellation]]
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| *[[Computational geometry]]
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| *[[Delaunay triangulation]]
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| *[[Mathematical diagram]]
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| *[[Natural neighbor| Natural neighbor interpolation ]]
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| *[[Nearest neighbor search]]
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| *[[Nearest-neighbor interpolation]]
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| *[[Voronoi pole]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal
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| |author=[[Peter Gustav Lejeune Dirichlet|G. Lejeune Dirichlet]]
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| |year=1850
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| |title=Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen
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| |journal=Journal für die Reine und Angewandte Mathematik
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| |volume=40
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| |pages=209–227
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| }}
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| * {{cite journal
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| |first1=Georgy
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| |last1= Voronoi
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| |year=1908
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| |title=Nouvelles applications des paramètres continus à la théorie des formes quadratiques
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| |journal=Journal für die Reine und Angewandte Mathematik
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| |volume=133
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| |doi=10.1515/crll.1908.133.97
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| |pages=97–178
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| }}
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| * Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). ''Spatial Tessellations – Concepts and Applications of Voronoi Diagrams''. 2nd edition. John Wiley, 2000, 671 pages ISBN 0-471-98635-6
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| * {{cite journal
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| |first1=Franz
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| |last1= Aurenhammer
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| |year=1991
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| |title=Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure
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| |journal=ACM Computing Surveys
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| |volume= 23
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| |doi=10.1145/116873.116880
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| |number=3
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| |pages=345–405
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| }}
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| * {{cite doi|10.1093/comjnl/24.2.162}}
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| * {{Cite news
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| |first1=Daniel
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| |last1=Reem
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| |year=2009
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| |title=An algorithm for computing Voronoi diagrams of general generators in general normed spaces
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| |doi=10.1109/ISVD.2009.23
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| |journal=Proceedings of the sixth International Symposium on Voronoi Diagrams in science and engineering (ISVD 2009)
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| |pages=144—152
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| }}
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| * Daniel Reem (2011). ''The geometric stability of Voronoi diagrams with respect to small changes of the sites''. Full version: [http://arxiv.org/abs/1103.4125 arXiv 1103.4125 (2011)], Extended abstract: [http://dx.doi.org/10.1145/1998196.1998234 in Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG 2011), pp. 254–263].
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| * {{cite doi|10.1093/comjnl/24.2.167}}
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| * {{cite book|author = Mark de Berg, Marc van Kreveld, [[Mark Overmars]], and Otfried Schwarzkopf | year = 2000 | title = Computational Geometry | publisher = [[Springer-Verlag]] | edition = 2nd revised edition | isbn = 3-540-65620-0}} Chapter 7: Voronoi Diagrams: pp. 147–163. Includes a description of Fortune's algorithm.
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| * {{cite book|author = Rolf Klein | year = 1989 | title = Abstract voronoi diagrams and their applications |series=[[Lecture Notes in Computer Science]]|volume=333|
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| pages=148—157
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| |doi= 10.1007/3-540-50335-8_31| publisher = [[Springer-Verlag]] | edition = | isbn = 3-540-52055-4}}
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| ==External links==
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| {{Commons category|Voronoi diagrams}}
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| * [http://www.pi6.fernuni-hagen.de/GeomLab/VoroGlide/index.html.en Real time interactive Voronoi / Delaunay diagrams with draggable points, Java 1.0.2, 1996–1997]
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| * [http://www.cs.cornell.edu/Info/People/chew/Delaunay.html Real time interactive Voronoi and Delaunay diagrams with source code]
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| * [http://alexbeutel.com/webgl/voronoi.html Interactive Voronoi diagrams with Natural Neighbor Interpolation visualization (in WebGL)]
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| * [http://www.nirarebakun.com/eng.html Demo for various metrics]
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| * [http://mathworld.wolfram.com/VoronoiDiagram.html Mathworld on Voronoi diagrams]
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| * [http://www.qhull.org Qhull for computing the Voronoi diagram in 2-d, 3-d, etc.]
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| * [http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html Voronoi Diagrams: Applications from Archaeology to Zoology]
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| * [http://www.cgal.org/Part/VoronoiDiagrams Voronoi Diagrams] in [[CGAL]], the Computational Geometry Algorithms Library
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| * [http://www.voronoi.com/ Voronoi Web Site : using Voronoi diagrams for spatial analysis]
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| * [http://www.math.psu.edu/qdu/Res/Pic/gallery3.html More discussions and picture gallery on centroidal Voronoi tessellations]
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| * [http://demonstrations.wolfram.com/VoronoiDiagrams/ Voronoi Diagrams] by [[Ed Pegg, Jr.]], Jeff Bryant, and [[Theodore Gray]], [[Wolfram Demonstrations Project]].
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| * [http://www.diku.dk/hjemmesider/studerende/duff/Fortune/ Nice explanation of voronoi diagram and visual implementation of fortune's algorithm]
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| * [http://www.preschern.org/pdf/WorldMap.pdf A Voronoi diagram on a sphere]
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| * [http://datavoreconsulting.com/programming-tips/voronoi-diagrams-in-mathematica/ Plot a Voronoi diagram with Mathematica]
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| {{DEFAULTSORT:Voronoi Diagram}}
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