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It can be classified also as a (/wiki/Tessellation) (Tessellation) 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 (/wiki/Region_(mathematics)) (Region (mathematics)) 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 (/wiki/Duality_(mathematics)) (Duality (mathematics)) dual to that set's (/wiki/Delaunay_triangulation) (Delaunay triangulation) Delaunay triangulation . (/wiki/File:Euclidean_Voronoi_diagram.svg) 20 points and their Voronoi cells (larger version below ) The Voronoi diagram is named after mathematician (/wiki/Georgy_Voronoy) (Georgy Voronoy) Georgy Voronoy , and is also called a Voronoi tessellation , a Voronoi decomposition , a Voronoi partition , or a Dirichlet tessellation (after (/wiki/Peter_Gustav_Lejeune_Dirichlet) (Peter Gustav Lejeune Dirichlet) Peter Gustav Lejeune Dirichlet ). Voronoi cells are also known as Thiessen polygons , after (/wiki/Alfred_H._Thiessen) (Alfred H. Thiessen) Alfred H. Thiessen .[ 1] [ 2] [ 3] Voronoi diagrams have practical and theoretical applications in many fields, mainly in (/wiki/Science) (Science) science and (/wiki/Technology) (Technology) technology , but also in (/wiki/Visual_art) (Visual art) visual art .[ 4] [ 5] Contents 1 Simplest case 2 Formal definition 3 Illustration 4 Properties 5 History and research 6 Examples 7 Higher-order Voronoi diagrams 7.1 Farthest-point Voronoi diagram 8 Generalizations and variations 9 Applications 9.1 Meteorology/Hydrology 9.2 Humanities and social sciences 9.3 Natural sciences 9.4 Health 9.5 Engineering 9.6 Mathematics 9.7 Informatics 10 Algorithms 11 Voronoi in 3D 12 See also 13 Notes 14 References 15 External links Simplest case (/w/index.php?title=Voronoi_diagram&action=edit§ion=1) (Edit section: Simplest case) edit In the simplest case, shown in the first picture, we are given a finite set of points { p 1 , \u2026 p n } {\\displaystyle \\{p_{1},\\dots p_{n}\\}} ({\\displaystyle \\{p_{1},\\dots p_{n}\\}}) in the (/wiki/Euclidean_plane) (Euclidean plane) Euclidean plane . In this case, each point p k {\\displaystyle p_{k}} ({\\displaystyle p_{k}}) has a corresponding cell R k {\\displaystyle R_{k}} ({\\displaystyle R_{k}}) consisting of the points in the Euclidean plane for which p k {\\displaystyle p_{k}} ({\\displaystyle p_{k}}) is the nearest site: the distance to p k {\\displaystyle p_{k}} ({\\displaystyle p_{k}}) is less than or equal to the minimum distance to any other site p j {\\displaystyle p_{j}} ({\\displaystyle p_{j}}) . For one other site p j {\\displaystyle p_{j}} ({\\displaystyle p_{j}}) , the points that are closer to p k {\\displaystyle p_{k}} ({\\displaystyle p_{k}}) than to p j {\\displaystyle p_{j}} ({\\displaystyle p_{j}}) , or equally distant, form a (/wiki/Half-space_(geometry)) (Half-space (geometry)) closed half-space , whose boundary is the (/wiki/Perpendicular_bisector) (Perpendicular bisector) perpendicular bisector of line segment p j p k {\\displaystyle p_{j}p_{k}} ({\\displaystyle p_{j}p_{k}}) . Cell R k {\\displaystyle R_{k}} ({\\displaystyle R_{k}}) is the intersection of all of these n \u2212 1 {\\displaystyle n-1} ({\\displaystyle n-1}) half-spaces, and hence it is a (/wiki/Convex_polygon) (Convex polygon) convex polygon .[ 6] When two cells in the Voronoi diagram share a boundary, it is a (/wiki/Line_segment) (Line segment) line segment , (/wiki/Ray_(geometry)) (Ray (geometry)) ray , or line, consisting of all the points in the plane that are equidistant to their two nearest sites. The (/wiki/Vertex_(geometry)) (Vertex (geometry)) vertices of the diagram, where three or more of these boundaries meet, are the points that have three or more equally distant nearest sites. Formal definition (/w/index.php?title=Voronoi_diagram&action=edit§ion=2) (Edit section: Formal definition) edit Let X {\\textstyle X} ({\\textstyle X}) be a (/wiki/Metric_space) (Metric space) metric space with distance function d {\\textstyle d} ({\\textstyle d}) . Let K {\\textstyle K} ({\\textstyle K}) be a set of indices and let ( P k ) k \u2208 K {\\textstyle (P_{k})_{k\\in K}} ({\\textstyle (P_{k})_{k\\in K}}) be a (/wiki/Tuple) (Tuple) tuple (indexed collection) of nonempty (/wiki/Subsets) (Subsets) subsets (the sites) in the space X {\\textstyle X} ({\\textstyle X}) . The Voronoi cell, or Voronoi region, R k {\\textstyle R_{k}} ({\\textstyle R_{k}}) , associated with the site P k {\\textstyle P_{k}} ({\\textstyle P_{k}}) is the set of all points in X {\\textstyle X} ({\\textstyle X}) whose distance to P k {\\textstyle P_{k}} ({\\textstyle P_{k}}) is not greater than their distance to the other sites P j {\\textstyle P_{j}} ({\\textstyle P_{j}}) , where j {\\textstyle j} ({\\textstyle j}) is any index different from k {\\textstyle k} ({\\textstyle k}) . In other words, if d ( x , A ) = inf { d ( x , a ) \u2223 a \u2208 A } {\\textstyle d(x,\\,A)=\\inf\\{d(x,\\,a)\\mid a\\in A\\}} ({\\textstyle d(x,\\,A)=\\inf\\{d(x,\\,a)\\mid a\\in A\\}}) denotes the distance between the point x {\\textstyle x} ({\\textstyle x}) and the subset A {\\textstyle A} ({\\textstyle A}) , then R k = { x \u2208 X \u2223 d ( x , P k ) \u2264 d ( x , P j ) for all j \u2260 k } {\\displaystyle R_{k}=\\{x\\in X\\mid d(x,P_{k})\\leq d(x,P_{j})\\;{\\text{for all}}\\;j\\neq k\\}} ({\\displaystyle R_{k}=\\{x\\in X\\mid d(x,P_{k})\\leq d(x,P_{j})\\;{\\text{for all}}\\;j\\neq k\\}}) The Voronoi diagram is simply the (/wiki/Tuple) (Tuple) tuple of cells ( R k ) k \u2208 K {\\textstyle (R_{k})_{k\\in K}} ({\\textstyle (R_{k})_{k\\in K}}) . 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 (/wiki/Geometry_of_numbers) (Geometry of numbers) geometry of numbers and (/wiki/Crystallography) (Crystallography) crystallography ), but again, in many cases only finitely many sites are considered. In the particular case where the space is a (/wiki/Finite-dimensional) (Finite-dimensional) finite-dimensional (/wiki/Euclidean_space) (Euclidean space) Euclidean space , each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are (/wiki/Convex_polytope) (Convex polytope) convex polytopes and they can be represented in a combinatorial way using their vertices, sides, two-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however, the Voronoi cells may not be convex or even connected. In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon R k {\\textstyle R_{k}} ({\\textstyle R_{k}}) is associated with a generator point P k {\\textstyle P_{k}} ({\\textstyle P_{k}}) .\nLet X {\\textstyle X} ({\\textstyle X}) be the set of all points in the Euclidean space. Let P 1 {\\textstyle P_{1}} ({\\textstyle P_{1}}) be a point that generates its Voronoi region R 1 {\\textstyle R_{1}} ({\\textstyle R_{1}}) , P 2 {\\textstyle P_{2}} ({\\textstyle P_{2}}) that generates R 2 {\\textstyle R_{2}} ({\\textstyle R_{2}}) , and P 3 {\\textstyle P_{3}} ({\\textstyle P_{3}}) that generates R 3 {\\textstyle R_{3}} ({\\textstyle R_{3}}) , and so on. Then, as expressed by Tran et al ,[ 7] \"all locations in the Voronoi polygon are closer to the generator point of that polygon than any other generator point in the Voronoi diagram in Euclidean plane\". Illustration (/w/index.php?title=Voronoi_diagram&action=edit§ion=3) (Edit section: Illustration) edit As a simple illustration, consider a group of shops in a 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 R k {\\displaystyle R_{k}} ({\\displaystyle R_{k}}) of a given shop P k {\\displaystyle P_{k}} ({\\displaystyle P_{k}}) 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 city). For most cities, the distance between points can be measured using the familiar (/wiki/Euclidean_distance) (Euclidean distance) Euclidean distance : \u2113 2 = d [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] = ( a 1 \u2212 b 1 ) 2 + ( a 2 \u2212 b 2 ) 2 {\\displaystyle \\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}}}} ({\\displaystyle \\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}}}}) or the (/wiki/Manhattan_distance) (Manhattan distance) Manhattan distance : d [ ( a 1 , a 2 ) , ( b 1 , b 2 ) ] = | a 1 \u2212 b 1 | + | a 2 \u2212 b 2 | {\\displaystyle 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|} ({\\displaystyle 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|}) . The corresponding Voronoi diagrams look different for different distance metrics. Voronoi diagrams of 20 points under two different metrics (/wiki/File:Euclidean_Voronoi_diagram.svg) (Voronoi diagram under Euclidean distance) (/wiki/Euclidean_distance) (Euclidean distance) Euclidean distance (/wiki/File:Manhattan_Voronoi_Diagram.svg) (Voronoi diagram under Manhattan distance) (/wiki/Manhattan_distance) (Manhattan distance) Manhattan distance Properties (/w/index.php?title=Voronoi_diagram&action=edit§ion=4) (Edit section: Properties) edit The (/wiki/Dual_graph) (Dual graph) dual graph for a Voronoi diagram (in the case of a (/wiki/Euclidean_space) (Euclidean space) Euclidean space with point sites) corresponds to the (/wiki/Delaunay_triangulation) (Delaunay triangulation) Delaunay triangulation for the same set of points. The (/wiki/Closest_pair_of_points) (Closest pair of points) closest pair of points corresponds to two adjacent cells in the Voronoi diagram. If the setting is the (/wiki/Euclidean_plane) (Euclidean plane) Euclidean plane and a discrete set of points is given, then two points of the set are adjacent on the (/wiki/Convex_hull) (Convex hull) convex hull if and only if their Voronoi cells share an infinitely long side. If the space is a (/wiki/Normed_space) (Normed space) normed space and the distance to each site is attained (e.g., when a site is a (/wiki/Compact_set) (Compact set) compact set or a closed ball), then each Voronoi cell can be represented as a union of line segments emanating from the sites.[ 8] As shown there, this property does not necessarily hold when the distance is not attained. Under relatively general conditions (the space is a possibly infinite-dimensional (/wiki/Uniformly_convex_space) (Uniformly convex space) 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.[ 9] 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. History and research (/w/index.php?title=Voronoi_diagram&action=edit§ion=5) (Edit section: History and research) edit Informal use of Voronoi diagrams can be traced back to (/wiki/Descartes) (Descartes) Descartes in 1644.[ 10] (/wiki/Peter_Gustav_Lejeune_Dirichlet) (Peter Gustav Lejeune Dirichlet) Peter Gustav Lejeune Dirichlet used two-dimensional and three-dimensional Voronoi diagrams in his study of quadratic forms in 1850.\nBritish physician (/wiki/John_Snow_(physician)) (John Snow (physician)) John Snow used a Voronoi-like diagram in 1854 to illustrate how the majority of people who died in the (/wiki/1854_Broad_Street_cholera_outbreak) (1854 Broad Street cholera outbreak) Broad Street cholera outbreak lived closer to the infected (/wiki/Soho#Broad_Street_pump) (Soho) Broad Street pump than to any other water pump. Voronoi diagrams are named after (/wiki/Georgy_Voronoy) (Georgy Voronoy) Georgy Feodosievych Voronoy who defined and studied the general n -dimensional case in 1908.[ 11] Voronoi diagrams that are used in (/wiki/Geophysics) (Geophysics) geophysics and (/wiki/Meteorology) (Meteorology) meteorology to analyse spatially distributed data are called Thiessen polygons after American meteorologist (/wiki/Alfred_H._Thiessen) (Alfred H. Thiessen) Alfred H. Thiessen , who used them to estimate rainfall from scattered measurements in 1911. Other equivalent names for this concept (or particular important cases of it): Voronoi polyhedra, Voronoi polygons, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), Dirichlet tessellation(s). Examples (/w/index.php?title=Voronoi_diagram&action=edit§ion=6) (Edit section: Examples) edit (/wiki/File:Coloured_Voronoi_3D_slice.svg) 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 a (/wiki/Power_diagram) (Power diagram) power diagram , a weighted form of a 2d Voronoi diagram, rather than being an unweighted Voronoi diagram. Voronoi tessellations of regular (/wiki/Lattice_(group)) (Lattice (group)) lattices of points in two or three dimensions give rise to many familiar tessellations. 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 (/wiki/Square_(geometry)) (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). A (/wiki/Simple_cubic_lattice) (Simple cubic lattice) simple cubic lattice gives the (/wiki/Cubic_honeycomb) (Cubic honeycomb) cubic honeycomb . A (/wiki/Hexagonal_close-packed) (Hexagonal close-packed) hexagonal close-packed lattice gives a tessellation of space with (/wiki/Trapezo-rhombic_dodecahedron) (Trapezo-rhombic dodecahedron) trapezo-rhombic dodecahedra . A (/wiki/Face-centred_cubic) (Face-centred cubic) face-centred cubic lattice gives a tessellation of space with (/wiki/Rhombic_dodecahedron) (Rhombic dodecahedron) rhombic dodecahedra . A (/wiki/Body-centred_cubic) (Body-centred cubic) body-centred cubic lattice gives a tessellation of space with (/wiki/Truncated_octahedron) (Truncated octahedron) truncated octahedra . Parallel planes with regular triangular lattices aligned with each other's centers give the (/wiki/Hexagonal_prismatic_honeycomb) (Hexagonal prismatic honeycomb) hexagonal prismatic honeycomb . Certain body-centered tetragonal lattices give a tessellation of space with (/wiki/Rhombo-hexagonal_dodecahedron) (Rhombo-hexagonal dodecahedron) rhombo-hexagonal dodecahedra . Certain body-centered tetragonal lattices give a tessellation of space with (/wiki/Rhombo-hexagonal_dodecahedron) (Rhombo-hexagonal dodecahedron) rhombo-hexagonal dodecahedra . 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. Higher-order Voronoi diagrams (/w/index.php?title=Voronoi_diagram&action=edit§ion=7) (Edit section: Higher-order Voronoi diagrams) edit 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. Higher-order Voronoi diagrams can be generated recursively. To generate the n th -order Voronoi diagram from set S , start with the (n \u2212\u00a01)th -order diagram and replace each cell generated by X =\u00a0{x 1 , x 2 ,\u00a0..., x n \u22121 } with a Voronoi diagram generated on the set S \u2212 X . Farthest-point Voronoi diagram (/w/index.php?title=Voronoi_diagram&action=edit§ion=8) (Edit section: Farthest-point Voronoi diagram) edit For a set of n points, the (n \u2212\u00a01)th -order Voronoi diagram is called a farthest-point Voronoi diagram. For a given set of points S =\u00a0{p 1 , p 2 ,\u00a0..., p n }, the farthest-point Voronoi diagram divides the plane into cells in which the same point of P is the farthest point. A point of P has a cell in the farthest-point Voronoi diagram if and only if it is a vertex of the (/wiki/Convex_hull) (Convex hull) convex hull of P . Let H =\u00a0{h 1 , h 2 ,\u00a0..., h k } be the convex hull of P ; then the farthest-point Voronoi diagram is a 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 i if and only if d(q , h i ) > d(q , p j ) for each p j \u2208 S with h i \u2260 p j , where d(p , q ) is the (/wiki/Euclidean_distance) (Euclidean distance) Euclidean distance between two points p and q .[ 12] [ 13] The boundaries of the cells in the farthest-point Voronoi diagram have the structure of a (/wiki/Real_tree) (Real tree) topological tree , with infinite (/wiki/Ray_(mathematics)) (Ray (mathematics)) rays as its leaves. Every finite tree is isomorphic to the tree formed in this way from a farthest-point Voronoi diagram.[ 14] Generalizations and variations (/w/index.php?title=Voronoi_diagram&action=edit§ion=9) (Edit section: Generalizations and variations) edit As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as the (/wiki/Mahalanobis_distance) (Mahalanobis distance) Mahalanobis distance or (/wiki/Manhattan_distance) (Manhattan distance) Manhattan distance . 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 two-dimensional case. (/wiki/File:Approximate_Voronoi_Diagram.svg) Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells. A (/wiki/Weighted_Voronoi_diagram) (Weighted Voronoi diagram) 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 (/wiki/Metric_(mathematics)) (Metric (mathematics)) metric , in this case some of the Voronoi cells may be empty. A (/wiki/Power_diagram) (Power diagram) power diagram is a type of Voronoi diagram defined from a set of circles using the (/wiki/Power_of_a_point) (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 (/wiki/Squared_Euclidean_distance) (Squared Euclidean distance) squared Euclidean distance from the circle's center.[ 15] The Voronoi diagram of n {\\displaystyle n} ({\\displaystyle n}) points in d {\\displaystyle d} ({\\displaystyle d}) -dimensional space can have O ( n \u2308 d / 2 \u2309 ) {\\textstyle O(n^{\\lceil d/2\\rceil })} ({\\textstyle O(n^{\\lceil d/2\\rceil })}) vertices, requiring the same bound for the amount of memory needed to store an explicit description of it. Therefore, Voronoi diagrams are often not feasible for moderate or high dimensions. A more space-efficient alternative is to use approximate Voronoi diagrams.[ 16] Voronoi diagrams are also related to other geometric structures such as the (/wiki/Medial_axis) (Medial axis) medial axis (which has found applications in image segmentation, (/wiki/Optical_character_recognition) (Optical character recognition) optical character recognition , and other computational applications), (/wiki/Straight_skeleton) (Straight skeleton) straight skeleton , and (/wiki/Zone_diagram) (Zone diagram) zone diagrams . Applications (/w/index.php?title=Voronoi_diagram&action=edit§ion=10) (Edit section: Applications) edit Meteorology/Hydrology (/w/index.php?title=Voronoi_diagram&action=edit§ion=11) (Edit section: Meteorology/Hydrology) edit It is used in meteorology and engineering hydrology to find the weights for precipitation data of stations over an area (watershed). The points generating the polygons are the various station that record precipitation data. Perpendicular bisectors are drawn to the line joining any two stations. This results in the formation of polygons around the stations. The area ( A i ) {\\displaystyle (A_{i})} ({\\displaystyle (A_{i})}) touching station point is known as influence area of the station. The average precipitation is calculated by the formula P \u00af = \u2211 A i P i \u2211 A i {\\displaystyle {\\bar {P}}={\\frac {\\sum A_{i}P_{i}}{\\sum A_{i}}}} ({\\displaystyle {\\bar {P}}={\\frac {\\sum A_{i}P_{i}}{\\sum A_{i}}}}) See also: (/wiki/Delaunay_triangulation#Applications) (Delaunay triangulation) Delaunay triangulation \u00a7\u00a0Applications Humanities and social sciences (/w/index.php?title=Voronoi_diagram&action=edit§ion=12) (Edit section: Humanities and social sciences) edit In (/wiki/Classical_archaeology) (Classical archaeology) classical archaeology , specifically (/wiki/Art_history) (Art history) art history , the symmetry of (/wiki/Statue) (Statue) statue heads is analyzed to determine the type of statue a severed head may have belonged to. An example of this that made use of Voronoi cells was the identification of the (/wiki/Sabouroff_head) (Sabouroff head) Sabouroff head , which made use of a high-resolution (/wiki/Polygon_mesh) (Polygon mesh) polygon mesh .[ 17] [ 18] In (/wiki/Dialectometry) (Dialectometry) dialectometry , Voronoi cells are used to indicate a supposed linguistic continuity between survey points. In (/wiki/Political_science) (Political science) political science , Voronoi diagrams have been used to study multi-dimensional, multi-party competition.[ 19] Natural sciences (/w/index.php?title=Voronoi_diagram&action=edit§ion=13) (Edit section: Natural sciences) edit (/wiki/File:Voronoi_growth_euclidean.gif) A Voronoi tessellation emerges by radial growth from seeds outward. In (/wiki/Biology) (Biology) biology , Voronoi diagrams are used to model a number of different biological structures, including (/wiki/Cell_(biology)) (Cell (biology)) cells [ 20] and (/wiki/Cancellous_bone) (Cancellous bone) bone microarchitecture. [ 21] Indeed, Voronoi tessellations work as a geometrical tool to understand the physical constraints that drive the organization of biological tissues.[ 22] In (/wiki/Hydrology) (Hydrology) hydrology , 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. In (/wiki/Ecology) (Ecology) 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. In (/wiki/Ethology) (Ethology) ethology , Voronoi diagrams are used to model domains of danger in the (/wiki/Selfish_herd_theory) (Selfish herd theory) selfish herd theory . In (/wiki/Computational_chemistry) (Computational chemistry) computational chemistry , ligand-binding sites are transformed into Voronoi diagrams for (/wiki/Machine_learning) (Machine learning) machine learning applications (e.g., to classify binding pockets in proteins).[ 23] In other applications, Voronoi cells defined by the positions of the nuclei in a molecule are used to compute (/wiki/Partial_charge) (Partial charge) atomic charges . This is done using the (/wiki/Voronoi_deformation_density) (Voronoi deformation density) Voronoi deformation density method. In (/wiki/Astrophysics) (Astrophysics) astrophysics , Voronoi diagrams are used to generate adaptative smoothing zones on images, adding signal fluxes on each one. The main objective of these procedures is to maintain a relatively constant (/wiki/Signal-to-noise_ratio) (Signal-to-noise ratio) signal-to-noise ratio on all the images. In (/wiki/Computational_fluid_dynamics) (Computational fluid dynamics) computational fluid dynamics , the Voronoi tessellation of a set of points can be used to define the computational domains used in (/wiki/Finite_volume) (Finite volume) finite volume methods, e.g. as in the moving-mesh cosmology code AREPO.[ 24] In (/wiki/Computational_physics) (Computational physics) computational physics , Voronoi diagrams are used to calculate profiles of an object with (/wiki/Shadowgraph) (Shadowgraph) Shadowgraph and proton radiography in (/wiki/High_energy_density_physics) (High energy density physics) High energy density physics .[ 25] Health (/w/index.php?title=Voronoi_diagram&action=edit§ion=14) (Edit section: Health) edit In (/wiki/Medical_diagnosis) (Medical diagnosis) medical diagnosis , models of muscle tissue, based on Voronoi diagrams, can be used to detect neuromuscular diseases.[ 22] In (/wiki/Epidemiology) (Epidemiology) epidemiology , Voronoi diagrams can be used to correlate sources of infections in epidemics. One of the early applications of Voronoi diagrams was implemented by (/wiki/John_Snow_(physician)) (John Snow (physician)) John Snow to study the (/wiki/1854_Broad_Street_cholera_outbreak) (1854 Broad Street cholera outbreak) 1854 Broad Street cholera outbreak in Soho, England. He showed the correlation between residential areas on the map of Central London whose residents had been using a specific water pump, and the areas with the most deaths due to the outbreak.[ 26] Engineering (/w/index.php?title=Voronoi_diagram&action=edit§ion=15) (Edit section: Engineering) edit In (/wiki/Polymer_physics) (Polymer physics) polymer physics , Voronoi diagrams can be used to represent free volumes of polymers. In (/wiki/Materials_science) (Materials science) materials science , polycrystalline microstructures in metallic alloys are commonly represented using Voronoi tessellations. In island growth, the Voronoi diagram is used to estimate the growth rate of individual islands.[ 27] [ 28] [ 29] [ 30] [ 31] In (/wiki/Solid-state_physics) (Solid-state physics) solid-state physics , the (/wiki/Wigner-Seitz_cell) (Wigner-Seitz cell) Wigner-Seitz cell is the Voronoi tessellation of a solid, and the (/wiki/Brillouin_zone) (Brillouin zone) Brillouin zone is the Voronoi tessellation of reciprocal ((/wiki/Wavenumber) (Wavenumber) wavenumber ) space of crystals which have the symmetry of a space group. In (/wiki/Aviation) (Aviation) aviation , Voronoi diagrams are superimposed on oceanic plotting charts to identify the nearest airfield for in-flight diversion (see (/wiki/ETOPS) (ETOPS) ETOPS ), as an aircraft progresses through its flight plan. In (/wiki/Architecture) (Architecture) architecture , Voronoi patterns were the basis for the winning entry for the redevelopment of (/wiki/The_Arts_Centre_Gold_Coast) (The Arts Centre Gold Coast) The Arts Centre Gold Coast .[ 32] In (/wiki/Urban_planning) (Urban planning) urban planning , Voronoi diagrams can be used to evaluate the Freight Loading Zone system.[ 33] In (/wiki/Mining) (Mining) 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. In (/wiki/Surface_metrology) (Surface metrology) surface metrology , Voronoi tessellation can be used for (/wiki/Surface_roughness) (Surface roughness) surface roughness modeling.[ 34] In (/wiki/Robotics) (Robotics) robotics , some of the control strategies and path planning algorithms[ 35] of (/wiki/Multi-agent_system) (Multi-agent system) multi-robot systems are based on the Voronoi partitioning of the environment.[ 36] [ 37] Mathematics (/w/index.php?title=Voronoi_diagram&action=edit§ion=16) (Edit section: Mathematics) edit A (/wiki/Point_location) (Point location) point location data structure can be built on top of the Voronoi diagram in order to answer (/wiki/Nearest_neighbor_search) (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 (/wiki/Database) (Database) database . A large application is (/wiki/Vector_quantization) (Vector quantization) vector quantization , commonly used in (/wiki/Data_compression) (Data compression) data compression . In (/wiki/Geometry) (Geometry) geometry , Voronoi diagrams can be used to find the (/wiki/Largest_empty_sphere) (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. Voronoi diagrams together with farthest-point Voronoi diagrams are used for efficient algorithms to compute the (/wiki/Roundness_(object)) (Roundness (object)) roundness of a set of points.[ 12] The Voronoi approach is also put to use in the evaluation of circularity/(/wiki/Roundness_(object)) (Roundness (object)) roundness while assessing the dataset from a (/wiki/Coordinate-measuring_machine) (Coordinate-measuring machine) coordinate-measuring machine . Zeroes of iterated derivatives of a rational function on the complex plane accumulate on the edges of the Voronoi diagam of the set of the poles ((/w/index.php?title=P%C3%B3lya%27s_shires_theorem&action=edit&redlink=1) (P\u00f3lya's shires theorem (page does not exist)) P\u00f3lya's shires theorem [ 38] ). Informatics (/w/index.php?title=Voronoi_diagram&action=edit§ion=17) (Edit section: Informatics) edit In (/wiki/Computer_network) (Computer network) networking , Voronoi diagrams can be used in derivations of the capacity of a (/wiki/Wireless_network) (Wireless network) wireless network . In (/wiki/Computer_graphics) (Computer graphics) computer graphics , Voronoi diagrams are used to calculate 3D shattering / fracturing geometry patterns. It is also used to (/wiki/Procedural_generation) (Procedural generation) procedurally generate organic or lava-looking textures. In autonomous (/wiki/Robot_navigation) (Robot navigation) 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). In (/wiki/Machine_learning) (Machine learning) machine learning , Voronoi diagrams are used to do (/wiki/K-nearest_neighbor_algorithm) (K-nearest neighbor algorithm) 1-NN classifications.[ 39] In global scene reconstruction, including with random sensor sites and unsteady wake flow, geophysical data, and 3D turbulence data, Voronoi tesselations are used with (/wiki/Deep_learning) (Deep learning) deep learning .[ 40] In (/wiki/User_interface) (User interface) user interface development, Voronoi patterns can be used to compute the best hover state for a given point.[ 41] Algorithms (/w/index.php?title=Voronoi_diagram&action=edit§ion=18) (Edit section: Algorithms) edit Several efficient algorithms are known for constructing Voronoi diagrams, either directly (as the diagram itself) or indirectly by starting with a (/wiki/Delaunay_triangulation) (Delaunay triangulation) Delaunay triangulation and then obtaining its dual.\nDirect algorithms include (/wiki/Fortune%27s_algorithm) (Fortune's algorithm) Fortune's algorithm , an (/wiki/Big_O_notation) (Big O notation) O (n log(n )) algorithm for generating a Voronoi diagram from a set of points in a plane. (/wiki/Bowyer%E2%80%93Watson_algorithm) (Bowyer\u2013Watson algorithm) Bowyer\u2013Watson algorithm , an (/wiki/Big_O_notation) (Big O notation) O (n log(n )) to (/wiki/Big_O_notation) (Big O notation) O (n 2 ) algorithm for generating a Delaunay triangulation in any number of dimensions, can be used in an indirect algorithm for the Voronoi diagram. The (/wiki/Jump_flooding_algorithm) (Jump flooding algorithm) Jump Flooding Algorithm can generate approximate Voronoi diagrams in constant time and is suited for use on commodity graphics hardware.[ 42] [ 43] (/wiki/Lloyd%27s_algorithm) (Lloyd's algorithm) Lloyd's algorithm and its generalization via the (/wiki/Linde%E2%80%93Buzo%E2%80%93Gray_algorithm) (Linde\u2013Buzo\u2013Gray algorithm) Linde\u2013Buzo\u2013Gray algorithm (aka (/wiki/K-means_clustering) (K-means clustering) k-means clustering ) use the construction of Voronoi diagrams as a subroutine.\nThese methods alternate between steps in which one constructs the Voronoi diagram for a set of seed points, and steps in which the seed points are moved to new locations that are more central within their cells. These methods can be used in spaces of arbitrary dimension to iteratively converge towards a specialized form of the Voronoi diagram, called a (/wiki/Centroidal_Voronoi_tessellation) (Centroidal Voronoi tessellation) Centroidal Voronoi tessellation , where the sites have been moved to points that are also the geometric centers of their cells. Voronoi in 3D (/w/index.php?title=Voronoi_diagram&action=edit§ion=19) (Edit section: Voronoi in 3D) edit Voronoi meshes can also be generated in 3D. (/wiki/File:Random_points_in_3D_for_forming_a_3D_Voronoi_partition.svg) (Random points in 3D for forming a 3D Voronoi partition) (Random points in 3D for forming a 3D Voronoi partition) Random points in 3D for forming a 3D Voronoi partition (/wiki/File:3D_Voronoi_mesh_of_25_random_points.svg) (3D Voronoi mesh of 25 random points) (3D Voronoi mesh of 25 random points) 3D Voronoi mesh of 25 random points (/wiki/File:3D_Voronoi_mesh_of_25_random_points_with_0.3_opacity_and_points.svg) (3D Voronoi mesh of 25 random points with 0.3 opacity and points) (3D Voronoi mesh of 25 random points with 0.3 opacity and points) 3D Voronoi mesh of 25 random points with 0.3 opacity and points (/wiki/File:3D_Voronoi_mesh_of_25_random_points_convex_polyhedra_pieces.svg) (3D Voronoi mesh of 25 random points convex polyhedra pieces) (3D Voronoi mesh of 25 random points convex polyhedra pieces) 3D Voronoi mesh of 25 random points convex polyhedra pieces See also (/w/index.php?title=Voronoi_diagram&action=edit§ion=20) (Edit section: See also) edit (/wiki/Delaunay_triangulation) (Delaunay triangulation) Delaunay triangulation (/wiki/Map_segmentation) (Map segmentation) Map segmentation (/wiki/Natural_element_method) (Natural element method) Natural element method (/wiki/Natural_neighbor_interpolation) (Natural neighbor interpolation) Natural neighbor interpolation (/wiki/Nearest-neighbor_interpolation) (Nearest-neighbor interpolation) Nearest-neighbor interpolation (/wiki/Power_diagram) (Power diagram) Power diagram (/wiki/Voronoi_pole) (Voronoi pole) Voronoi pole Notes (/w/index.php?title=Voronoi_diagram&action=edit§ion=21) (Edit section: Notes) edit ^ Burrough, Peter A.; McDonnell, Rachael; McDonnell, Rachael A.; Lloyd, Christopher D. (2015). (https://books.google.com/books?id=kvoJCAAAQBAJ&pg=PA160) \"8.11 Nearest neighbours: Thiessen (Dirichlet/Voroni) polygons\" . Principles of Geographical Information Systems . Oxford University Press. pp.\u00a0160\u2013. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-0-19-874284-5) (Special:BookSources/978-0-19-874284-5) 978-0-19-874284-5 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=8.11+Nearest+neighbours%3A+Thiessen+%28Dirichlet%2FVoroni%29+polygons&rft.btitle=Principles+of+Geographical+Information+Systems&rft.pages=160-&rft.pub=Oxford+University+Press&rft.date=2015&rft.isbn=978-0-19-874284-5&rft.aulast=Burrough&rft.aufirst=Peter+A.&rft.au=McDonnell%2C+Rachael&rft.au=McDonnell%2C+Rachael+A.&rft.au=Lloyd%2C+Christopher+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkvoJCAAAQBAJ%26pg%3DPA160&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) Longley, Paul A.; Goodchild, Michael F.; Maguire, David J.; Rhind, David W. (2005). (https://books.google.com/books?id=-FbVI-2tSuYC&pg=PA333) \"14.4.4.1 Thiessen polygons\" . Geographic Information Systems and Science . Wiley. pp.\u00a0333\u2013. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-0-470-87001-3) (Special:BookSources/978-0-470-87001-3) 978-0-470-87001-3 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=14.4.4.1+Thiessen+polygons&rft.btitle=Geographic+Information+Systems+and+Science&rft.pages=333-&rft.pub=Wiley&rft.date=2005&rft.isbn=978-0-470-87001-3&rft.aulast=Longley&rft.aufirst=Paul+A.&rft.au=Goodchild%2C+Michael+F.&rft.au=Maguire%2C+David+J.&rft.au=Rhind%2C+David+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-FbVI-2tSuYC%26pg%3DPA333&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) Sen, Zekai (2016). (https://books.google.com/books?id=6N0yDQAAQBAJ&pg=PA57) \"2.8.1 Delaney, Varoni, and Thiessen Polygons\" . Spatial Modeling Principles in Earth Sciences . Springer. pp.\u00a057\u2013. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-3-319-41758-5) (Special:BookSources/978-3-319-41758-5) 978-3-319-41758-5 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.8.1+Delaney%2C+Varoni%2C+and+Thiessen+Polygons&rft.btitle=Spatial+Modeling+Principles+in+Earth+Sciences&rft.pages=57-&rft.pub=Springer&rft.date=2016&rft.isbn=978-3-319-41758-5&rft.aulast=Sen&rft.aufirst=Zekai&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6N0yDQAAQBAJ%26pg%3DPA57&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) (/wiki/Franz_Aurenhammer) (Franz Aurenhammer) Aurenhammer, Franz (1991). \"Voronoi Diagrams \u2013 A Survey of a Fundamental Geometric Data Structure\". ACM Computing Surveys . 23 (3): 345\u2013 405. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1145%2F116873.116880) 10.1145/116873.116880 . (/wiki/S2CID_(identifier)) (S2CID (identifier)) S2CID (https://api.semanticscholar.org/CorpusID:4613674) 4613674 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+Computing+Surveys&rft.atitle=Voronoi+Diagrams+%E2%80%93+A+Survey+of+a+Fundamental+Geometric+Data+Structure&rft.volume=23&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E345-%3C%2Fspan%3E405&rft.date=1991&rft_id=info%3Adoi%2F10.1145%2F116873.116880&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4613674%23id-name%3DS2CID&rft.aulast=Aurenhammer&rft.aufirst=Franz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) Okabe, Atsuyuki; Boots, Barry; Sugihara, Kokichi; Chiu, Sung Nok (2000). Spatial Tessellations \u2013 Concepts and Applications of Voronoi Diagrams (2nd\u00a0ed.). John Wiley. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-0-471-98635-5) (Special:BookSources/978-0-471-98635-5) 978-0-471-98635-5 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spatial+Tessellations+%E2%80%93+Concepts+and+Applications+of+Voronoi+Diagrams&rft.edition=2nd&rft.pub=John+Wiley&rft.date=2000&rft.isbn=978-0-471-98635-5&rft.aulast=Okabe&rft.aufirst=Atsuyuki&rft.au=Boots%2C+Barry&rft.au=Sugihara%2C+Kokichi&rft.au=Chiu%2C+Sung+Nok&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization . Exercise 2.9: Cambridge University Press. p.\u00a060. (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Convex+Optimization&rft.place=Exercise+2.9&rft.pages=60&rft.pub=Cambridge+University+Press&rft.date=2004&rft.aulast=Boyd&rft.aufirst=Stephen&rft.au=Vandenberghe%2C+Lieven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) {{(/wiki/Template:Cite_book) (Template:Cite book) cite book }} : CS1 maint: location ((/wiki/Category:CS1_maint:_location) (Category:CS1 maint: location) link ) ^ (mw-data:TemplateStyles:r1238218222) Tran, Q. T.; Tainar, D.; Safar, M. (2009). Transactions on Large-Scale Data- and Knowledge-Centered Systems . Springer. p.\u00a0357. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/9783642037214) (Special:BookSources/9783642037214) 9783642037214 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transactions+on+Large-Scale+Data-+and+Knowledge-Centered+Systems&rft.pages=357&rft.pub=Springer&rft.date=2009&rft.isbn=9783642037214&rft.aulast=Tran&rft.aufirst=Q.+T.&rft.au=Tainar%2C+D.&rft.au=Safar%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ Reem 2009 . ^ Reem 2011 . ^ (mw-data:TemplateStyles:r1238218222) Senechal, Marjorie (1993-05-21). (https://www.science.org/doi/10.1126/science.260.5111.1170) \"Mathematical Structures: Spatial Tessellations . Concepts and Applications of Voronoi Diagrams. Atsuyuki Okabe, Barry Boots, and Kokichi Sugihara. Wiley, New York, 1992. xii, 532 pp., illus. $89.95. Wiley Series in Probability and Mathematical Statistics\" . Science . 260 (5111): 1170\u2013 1173. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1126%2Fscience.260.5111.1170) 10.1126/science.260.5111.1170 . (/wiki/ISSN_(identifier)) (ISSN (identifier)) ISSN (https://search.worldcat.org/issn/0036-8075) 0036-8075 . (/wiki/PMID_(identifier)) (PMID (identifier)) PMID (https://pubmed.ncbi.nlm.nih.gov/17806355) 17806355 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Mathematical+Structures%3A+Spatial+Tessellations+.+Concepts+and+Applications+of+Voronoi+Diagrams.+Atsuyuki+Okabe%2C+Barry+Boots%2C+and+Kokichi+Sugihara.+Wiley%2C+New+York%2C+1992.+xii%2C+532+pp.%2C+illus.+%2489.95.+Wiley+Series+in+Probability+and+Mathematical+Statistics.&rft.volume=260&rft.issue=5111&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1170-%3C%2Fspan%3E1173&rft.date=1993-05-21&rft.issn=0036-8075&rft_id=info%3Apmid%2F17806355&rft_id=info%3Adoi%2F10.1126%2Fscience.260.5111.1170&rft.aulast=Senechal&rft.aufirst=Marjorie&rft_id=https%3A%2F%2Fwww.science.org%2Fdoi%2F10.1126%2Fscience.260.5111.1170&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ Vorono\u00ef 1908a and Vorono\u00ef 1908b . ^ a b (mw-data:TemplateStyles:r1238218222) (/wiki/Mark_de_Berg) (Mark de Berg) de Berg, Mark ; (/wiki/Marc_van_Kreveld) (Marc van Kreveld) van Kreveld, Marc ; (/wiki/Mark_Overmars) (Mark Overmars) Overmars, Mark ; (/wiki/Otfried_Schwarzkopf) (Otfried Schwarzkopf) Schwarzkopf, Otfried (2008). Computational Geometry (Third\u00a0ed.). (/wiki/Springer-Verlag) (Springer-Verlag) Springer-Verlag . (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-3-540-77974-2) (Special:BookSources/978-3-540-77974-2) 978-3-540-77974-2 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Geometry&rft.edition=Third&rft.pub=Springer-Verlag&rft.date=2008&rft.isbn=978-3-540-77974-2&rft.aulast=de+Berg&rft.aufirst=Mark&rft.au=van+Kreveld%2C+Marc&rft.au=Overmars%2C+Mark&rft.au=Schwarzkopf%2C+Otfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) 7.4 Farthest-Point Voronoi Diagrams. Includes a description of the algorithm. ^ (mw-data:TemplateStyles:r1238218222) Skyum, Sven (18 February 1991). \"A simple algorithm for computing the smallest enclosing circle\". Information Processing Letters . 37 (3): 121\u2013 125. 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(ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Mark+DiMarco%3A+User+Interface+Algorithms+%5BJSConf2014%5D&rft.date=2014-06-11&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D90NsjKvz9Ns&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) Rong, Guodong; Tan, Tiow Seng (2006). (https://www.comp.nus.edu.sg/~tants/jfa/i3d06.pdf) \"Jump flooding in GPU with applications to Voronoi diagram and distance transform\" (PDF) . In Olano, Marc; S\u00e9quin, Carlo H. (eds.). Proceedings of the 2006 Symposium on Interactive 3D Graphics, SI3D 2006, March 14-17, 2006, Redwood City, California, USA . ACM. pp. 109\u2013 116. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1145%2F1111411.1111431) 10.1145/1111411.1111431 . (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/1-59593-295-X) (Special:BookSources/1-59593-295-X) 1-59593-295-X . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Jump+flooding+in+GPU+with+applications+to+Voronoi+diagram+and+distance+transform&rft.btitle=Proceedings+of+the+2006+Symposium+on+Interactive+3D+Graphics%2C+SI3D+2006%2C+March+14-17%2C+2006%2C+Redwood+City%2C+California%2C+USA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E109-%3C%2Fspan%3E116&rft.pub=ACM&rft.date=2006&rft_id=info%3Adoi%2F10.1145%2F1111411.1111431&rft.isbn=1-59593-295-X&rft.aulast=Rong&rft.aufirst=Guodong&rft.au=Tan%2C+Tiow+Seng&rft_id=https%3A%2F%2Fwww.comp.nus.edu.sg%2F~tants%2Fjfa%2Fi3d06.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) ^ (mw-data:TemplateStyles:r1238218222) (https://www.shadertoy.com/view/4syGWK) \"Shadertoy\" . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Shadertoy&rft_id=https%3A%2F%2Fwww.shadertoy.com%2Fview%2F4syGWK&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) References (/w/index.php?title=Voronoi_diagram&action=edit§ion=22) (Edit section: References) edit (mw-data:TemplateStyles:r1238218222) (/wiki/Franz_Aurenhammer) (Franz Aurenhammer) Aurenhammer, Franz ; Klein, Rolf; (/wiki/Der-Tsai_Lee) (Der-Tsai Lee) Lee, Der-Tsai (2013). Voronoi Diagrams and Delaunay Triangulations . World Scientific. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-9814447638) (Special:BookSources/978-9814447638) 978-9814447638 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Voronoi+Diagrams+and+Delaunay+Triangulations&rft.pub=World+Scientific&rft.date=2013&rft.isbn=978-9814447638&rft.aulast=Aurenhammer&rft.aufirst=Franz&rft.au=Klein%2C+Rolf&rft.au=Lee%2C+Der-Tsai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) (/wiki/Adrian_Bowyer) (Adrian Bowyer) Bowyer, Adrian (1981). (https://doi.org/10.1093%2Fcomjnl%2F24.2.162) \"Computing Dirichlet tessellations\" . (/wiki/The_Computer_Journal) (The Computer Journal) Comput. J. 24 (2): 162\u2013 166. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(Freely accessible) (https://doi.org/10.1093%2Fcomjnl%2F24.2.162) 10.1093/comjnl/24.2.162 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comput.+J.&rft.atitle=Computing+Dirichlet+tessellations&rft.volume=24&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E162-%3C%2Fspan%3E166&rft.date=1981&rft_id=info%3Adoi%2F10.1093%2Fcomjnl%2F24.2.162&rft.aulast=Bowyer&rft.aufirst=Adrian&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fcomjnl%252F24.2.162&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) de Berg, Mark; van Kreveld, Marc; (/wiki/Mark_Overmars) (Mark Overmars) Overmars, Mark ; Schwarzkopf, Otfried (2000). (https://archive.org/details/computationalgeo00berg) \"7. Voronoi Diagrams\" . Computational Geometry (2nd revised\u00a0ed.). Springer. pp. 47\u2013 163. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-3-540-65620-3) (Special:BookSources/978-3-540-65620-3) 978-3-540-65620-3 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7.+Voronoi+Diagrams&rft.btitle=Computational+Geometry&rft.pages=%3Cspan+class%3D%22nowrap%22%3E47-%3C%2Fspan%3E163&rft.edition=2nd+revised&rft.pub=Springer&rft.date=2000&rft.isbn=978-3-540-65620-3&rft.aulast=de+Berg&rft.aufirst=Mark&rft.au=van+Kreveld%2C+Marc&rft.au=Overmars%2C+Mark&rft.au=Schwarzkopf%2C+Otfried&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcomputationalgeo00berg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) Includes a description of Fortune's algorithm. (mw-data:TemplateStyles:r1238218222) Klein, Rolf (1988). \"Abstract voronoi diagrams and their applications: Extended abstract\". Computational Geometry and its Applications . (/wiki/Lecture_Notes_in_Computer_Science) (Lecture Notes in Computer Science) Lecture Notes in Computer Science . Vol.\u00a0333. Springer. pp. 148\u2013 157. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1007%2F3-540-50335-8_31) 10.1007/3-540-50335-8_31 . (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-3-540-52055-9) (Special:BookSources/978-3-540-52055-9) 978-3-540-52055-9 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Abstract+voronoi+diagrams+and+their+applications%3A+Extended+abstract&rft.btitle=Computational+Geometry+and+its+Applications&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E148-%3C%2Fspan%3E157&rft.pub=Springer&rft.date=1988&rft_id=info%3Adoi%2F10.1007%2F3-540-50335-8_31&rft.isbn=978-3-540-52055-9&rft.aulast=Klein&rft.aufirst=Rolf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) (/wiki/Peter_Gustav_Lejeune_Dirichlet) (Peter Gustav Lejeune Dirichlet) Lejeune Dirichlet, G. (1850). \"\u00dcber die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen\". Journal f\u00fcr die Reine und Angewandte Mathematik . 1850 (40): 209\u2013 227. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1515%2Fcrll.1850.40.209) 10.1515/crll.1850.40.209 . (/wiki/S2CID_(identifier)) (S2CID (identifier)) S2CID (https://api.semanticscholar.org/CorpusID:199546675) 199546675 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=%C3%9Cber+die+Reduktion+der+positiven+quadratischen+Formen+mit+drei+unbestimmten+ganzen+Zahlen&rft.volume=1850&rft.issue=40&rft.pages=%3Cspan+class%3D%22nowrap%22%3E209-%3C%2Fspan%3E227&rft.date=1850&rft_id=info%3Adoi%2F10.1515%2Fcrll.1850.40.209&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199546675%23id-name%3DS2CID&rft.aulast=Lejeune+Dirichlet&rft.aufirst=G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Okabe, Atsuyuki; Boots, Barry; (/wiki/Kokichi_Sugihara) (Kokichi Sugihara) Sugihara, Kokichi ; Chiu, Sung Nok (2000). Spatial Tessellations \u2014 Concepts and Applications of Voronoi Diagrams (2nd\u00a0ed.). Wiley. (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/0-471-98635-6) (Special:BookSources/0-471-98635-6) 0-471-98635-6 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spatial+Tessellations+%E2%80%94+Concepts+and+Applications+of+Voronoi+Diagrams&rft.edition=2nd&rft.pub=Wiley&rft.date=2000&rft.isbn=0-471-98635-6&rft.aulast=Okabe&rft.aufirst=Atsuyuki&rft.au=Boots%2C+Barry&rft.au=Sugihara%2C+Kokichi&rft.au=Chiu%2C+Sung+Nok&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Reem, Daniel (2009). \"An algorithm for computing Voronoi diagrams of general generators in general normed spaces\". Proceedings of the Sixth International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2009) . pp. 144\u2013 152. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1109%2FISVD.2009.23) 10.1109/ISVD.2009.23 . (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/978-1-4244-4769-5) (Special:BookSources/978-1-4244-4769-5) 978-1-4244-4769-5 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Sixth+International+Symposium+on+Voronoi+Diagrams+in+Science+and+Engineering+%28ISVD+2009%29&rft.atitle=An+algorithm+for+computing+Voronoi+diagrams+of+general+generators+in+general+normed+spaces&rft.pages=%3Cspan+class%3D%22nowrap%22%3E144-%3C%2Fspan%3E152&rft.date=2009&rft_id=info%3Adoi%2F10.1109%2FISVD.2009.23&rft.isbn=978-1-4244-4769-5&rft.aulast=Reem&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Reem, Daniel (2011). \"The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites\". Proceedings of the twenty-seventh annual symposium on Computational geometry . pp. 254\u2013 263. (/wiki/ArXiv_(identifier)) (ArXiv (identifier)) arXiv :(Freely accessible) (https://arxiv.org/abs/1103.4125) 1103.4125 . (/wiki/Bibcode_(identifier)) (Bibcode (identifier)) Bibcode :(https://ui.adsabs.harvard.edu/abs/2011arXiv1103.4125R) 2011arXiv1103.4125R . (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1145%2F1998196.1998234) 10.1145/1998196.1998234 . (/wiki/ISBN_(identifier)) (ISBN (identifier)) ISBN (/wiki/Special:BookSources/9781450306829) (Special:BookSources/9781450306829) 9781450306829 . (/wiki/S2CID_(identifier)) (S2CID (identifier)) S2CID (https://api.semanticscholar.org/CorpusID:14639512) 14639512 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+Geometric+Stability+of+Voronoi+Diagrams+with+Respect+to+Small+Changes+of+the+Sites&rft.btitle=Proceedings+of+the+twenty-seventh+annual+symposium+on+Computational+geometry&rft.pages=%3Cspan+class%3D%22nowrap%22%3E254-%3C%2Fspan%3E263&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14639512%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2011arXiv1103.4125R&rft_id=info%3Aarxiv%2F1103.4125&rft_id=info%3Adoi%2F10.1145%2F1998196.1998234&rft.isbn=9781450306829&rft.aulast=Reem&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) (/wiki/Alfred_H._Thiessen) (Alfred H. Thiessen) Thiessen, Alfred H. (July 1911). (https://doi.org/10.1175%2F1520-0493%281911%2939%3C1082b%3Apafla%3E2.0.co%3B2) \"Precipitation averages for large areas\" . Monthly Weather Review . 39 (7). American Meteorological Society: 1082\u2013 1089. 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(ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monthly+Weather+Review&rft.atitle=Precipitation+averages+for+large+areas&rft.volume=39&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1082-%3C%2Fspan%3E1089&rft.date=1911-07&rft_id=info%3Adoi%2F10.1175%2F1520-0493%281911%2939%3C1082b%3Apafla%3E2.0.co%3B2&rft_id=info%3Abibcode%2F1911MWRv...39R1082T&rft.aulast=Thiessen&rft.aufirst=Alfred+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1175%252F1520-0493%25281911%252939%253C1082b%253Apafla%253E2.0.co%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Vorono\u00ef, Georges (1908a). (https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0133/LOG_0010.pdf) \"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Premier m\u00e9moire. Sur quelques propri\u00e9t\u00e9s des formes quadratiques positives parfaites\" (PDF) . Journal f\u00fcr die Reine und Angewandte Mathematik . 1908 (133): 97\u2013 178. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1515%2Fcrll.1908.133.97) 10.1515/crll.1908.133.97 . (/wiki/S2CID_(identifier)) (S2CID (identifier)) S2CID (https://api.semanticscholar.org/CorpusID:116775758) 116775758 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Nouvelles+applications+des+param%C3%A8tres+continus+%C3%A0+la+th%C3%A9orie+des+formes+quadratiques.+Premier+m%C3%A9moire.+Sur+quelques+propri%C3%A9t%C3%A9s+des+formes+quadratiques+positives+parfaites.&rft.volume=1908&rft.issue=133&rft.pages=%3Cspan+class%3D%22nowrap%22%3E97-%3C%2Fspan%3E178&rft.date=1908&rft_id=info%3Adoi%2F10.1515%2Fcrll.1908.133.97&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A116775758%23id-name%3DS2CID&rft.aulast=Vorono%C3%AF&rft.aufirst=Georges&rft_id=https%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdownload%2Fpdf%2FPPN243919689_0133%2FLOG_0010.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Vorono\u00ef, Georges (1908b). (https://gdz.sub.uni-goettingen.de/download/pdf/PPN243919689_0134/LOG_0014.pdf) \"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Deuxi\u00e8me m\u00e9moire. Recherches sur les parall\u00e9llo\u00e8dres primitifs\" (PDF) . Journal f\u00fcr die Reine und Angewandte Mathematik . 1908 (134): 198\u2013 287. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(https://doi.org/10.1515%2Fcrll.1908.134.198) 10.1515/crll.1908.134.198 . (/wiki/S2CID_(identifier)) (S2CID (identifier)) S2CID (https://api.semanticscholar.org/CorpusID:118441072) 118441072 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Nouvelles+applications+des+param%C3%A8tres+continus+%C3%A0+la+th%C3%A9orie+des+formes+quadratiques.+Deuxi%C3%A8me+m%C3%A9moire.+Recherches+sur+les+parall%C3%A9llo%C3%A8dres+primitifs.&rft.volume=1908&rft.issue=134&rft.pages=%3Cspan+class%3D%22nowrap%22%3E198-%3C%2Fspan%3E287&rft.date=1908&rft_id=info%3Adoi%2F10.1515%2Fcrll.1908.134.198&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118441072%23id-name%3DS2CID&rft.aulast=Vorono%C3%AF&rft.aufirst=Georges&rft_id=https%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fdownload%2Fpdf%2FPPN243919689_0134%2FLOG_0014.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) (mw-data:TemplateStyles:r1238218222) Watson, David F. (1981). (https://doi.org/10.1093%2Fcomjnl%2F24.2.167) \"Computing the n -dimensional Delaunay tessellation with application to Voronoi polytopes\" . (/wiki/The_Computer_Journal) (The Computer Journal) Comput. J. 24 (2): 167\u2013 172. (/wiki/Doi_(identifier)) (Doi (identifier)) doi :(Freely accessible) (https://doi.org/10.1093%2Fcomjnl%2F24.2.167) 10.1093/comjnl/24.2.167 . (ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comput.+J.&rft.atitle=Computing+the+n-dimensional+Delaunay+tessellation+with+application+to+Voronoi+polytopes&rft.volume=24&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E167-%3C%2Fspan%3E172&rft.date=1981&rft_id=info%3Adoi%2F10.1093%2Fcomjnl%2F24.2.167&rft.aulast=Watson&rft.aufirst=David+F.&rft_id=https%3A%2F%2Fdoi.org%2F10.1093%252Fcomjnl%252F24.2.167&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVoronoi+diagram) External links (/w/index.php?title=Voronoi_diagram&action=edit§ion=23) (Edit section: External links) edit (/wiki/File:Commons-logo.svg) () Wikimedia Commons has media related to (https://commons.wikimedia.org/wiki/Category:Voronoi_diagrams) (commons:Category:Voronoi diagrams) Voronoi diagrams . (mw-data:TemplateStyles:r1238218222) (/wiki/Eric_W._Weisstein) (Eric W. Weisstein) Weisstein, Eric W. (https://mathworld.wolfram.com/VoronoiDiagram.html) \"Voronoi diagram\" . (/wiki/MathWorld) (MathWorld) MathWorld . 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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.\n20 points and their Voronoi cells (larger version below)\nThe 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.[1][2][3] Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art.[4][5]\n\n\n\n\nIn the simplest case, shown in the first picture, we are given a finite set of points \u00a0 in the Euclidean plane. In this case, each point \u00a0 has a corresponding cell \u00a0 consisting of the points in the Euclidean plane for which \u00a0 is the nearest site: the distance to \u00a0 is less than or equal to the minimum distance to any other site \u00a0. For one other site \u00a0, the points that are closer to \u00a0 than to \u00a0, or equally distant, form a closed half-space, whose boundary is the perpendicular bisector of line segment \u00a0. Cell \u00a0 is the intersection of all of these \u00a0 half-spaces, and hence it is a convex polygon.[6] When two cells in the Voronoi diagram share a boundary, it is a line segment, ray, or line, consisting of all the points in the plane that are equidistant to their two nearest sites. The vertices of the diagram, where three or more of these boundaries meet, are the points that have three or more equally distant nearest sites.\n\n\nLet \u00a0 be a metric space with distance function \u00a0. Let \u00a0 be a set of indices and let \u00a0 be a tuple (indexed collection) of nonempty subsets (the sites) in the space \u00a0. The Voronoi cell, or Voronoi region, \u00a0, associated with the site \u00a0 is the set of all points in \u00a0 whose distance to \u00a0 is not greater than their distance to the other sites \u00a0, where \u00a0 is any index different from \u00a0. In other words, if \u00a0 denotes the distance between the point \u00a0 and the subset \u00a0, then\nThe Voronoi diagram is simply the tuple of cells \u00a0. 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.\nIn 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 polytopes and they can be represented in a combinatorial way using their vertices, sides, two-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however, the Voronoi cells may not be convex or even connected.\nIn the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon \u00a0 is associated with a generator point \u00a0.\nLet \u00a0 be the set of all points in the Euclidean space. Let \u00a0 be a point that generates its Voronoi region \u00a0, \u00a0 that generates \u00a0, and \u00a0 that generates \u00a0, and so on. Then, as expressed by Tran et al,[7] \"all locations in the Voronoi polygon are closer to the generator point of that polygon than any other generator point in the Voronoi diagram in Euclidean plane\".\n\n\nAs a simple illustration, consider a group of shops in a 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 \u00a0 of a given shop \u00a0 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 city).\nFor most cities, the distance between points can be measured using the familiar\nEuclidean distance:\n\n\u00a0\nor the Manhattan distance:\n\n\u00a0.\nThe corresponding Voronoi diagrams look different for different distance metrics.\n\n\n\nThe 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.\nThe closest pair of points corresponds to two adjacent cells in the Voronoi diagram.\nIf the setting is the Euclidean plane and a discrete set of points is given, then two points of the set are adjacent on the convex hull if and only if their Voronoi cells share an infinitely long side.\nIf 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.[8] As shown there, this property does not necessarily hold when the distance is not attained.\nUnder 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.[9] 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.\nHistory and research\n\n \nedit\n\n\n\n\n\nInformal use of Voronoi diagrams can be traced back to Descartes in 1644.[10] Peter Gustav Lejeune Dirichlet used two-dimensional and three-dimensional Voronoi diagrams in his study of quadratic forms in 1850.\nBritish physician John Snow used a Voronoi-like diagram in 1854 to illustrate how the majority of people who died in the Broad Street cholera outbreak lived closer to the infected Broad Street pump than to any other water pump.\nVoronoi diagrams are named after Georgy Feodosievych Voronoy who defined and studied the general n-dimensional case in 1908.[11] Voronoi diagrams that are used in geophysics and meteorology to analyse spatially distributed data are called Thiessen polygons after American meteorologist Alfred H. Thiessen, who used them to estimate rainfall from scattered measurements in 1911. Other equivalent names for this concept (or particular important cases of it): Voronoi polyhedra, Voronoi polygons, domain(s) of influence, Voronoi decomposition, Voronoi tessellation(s), Dirichlet tessellation(s).\n\n\n\u00a0This 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 a power diagram, a weighted form of a 2d Voronoi diagram, rather than being an unweighted Voronoi diagram.\nVoronoi tessellations of regular lattices of points in two or three dimensions give rise to many familiar tessellations.\n\nA 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; 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).\nA simple cubic lattice gives the cubic honeycomb.\nA hexagonal close-packed lattice gives a tessellation of space with trapezo-rhombic dodecahedra.\nA face-centred cubic lattice gives a tessellation of space with rhombic dodecahedra.\nA body-centred cubic lattice gives a tessellation of space with truncated octahedra.\nParallel planes with regular triangular lattices aligned with each other's centers give the hexagonal prismatic honeycomb.\nCertain body-centered tetragonal lattices give a tessellation of space with rhombo-hexagonal dodecahedra.\nCertain body-centered tetragonal lattices give a tessellation of space with rhombo-hexagonal dodecahedra.\nFor the set of points (x,\u00a0y) 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.\n\nHigher-order Voronoi diagrams\n\n \nedit\n\n\n\n\n\nAlthough a normal Voronoi cell is defined as the set of points closest to a single point in S, an nth-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.\nHigher-order Voronoi diagrams can be generated recursively. To generate the nth-order Voronoi diagram from set\u00a0S, start with the (n\u00a0\u2212\u00a01)th-order diagram and replace each cell generated by X\u00a0=\u00a0{x1,\u00a0x2,\u00a0...,\u00a0xn\u22121} with a Voronoi diagram generated on the set\u00a0S\u00a0\u2212\u00a0X.\n\nFarthest-point Voronoi diagram\n\n \nedit\n\n\n\n\n\nFor a set of n points, the (n\u00a0\u2212\u00a01)th-order Voronoi diagram is called a farthest-point Voronoi diagram.\nFor a given set of points S\u00a0=\u00a0{p1,\u00a0p2,\u00a0...,\u00a0pn}, the farthest-point Voronoi diagram divides the plane into cells in which the same point of P is the farthest point. 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. Let H\u00a0=\u00a0{h1,\u00a0h2,\u00a0...,\u00a0hk} be the convex hull of P; then the farthest-point Voronoi diagram is a 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 hi if and only if d(q, hi) > d(q, pj) for each pj\u00a0\u2208\u00a0S with hi \u2260 pj, where d(p, q) is the Euclidean distance between two points p and\u00a0q.[12][13]\nThe boundaries of the cells in the farthest-point Voronoi diagram have the structure of a topological tree, with infinite rays as its leaves. Every finite tree is isomorphic to the tree formed in this way from a farthest-point Voronoi diagram.[14]\n\nGeneralizations and variations\n\n \nedit\n\n\n\n\n\nAs implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as the Mahalanobis distance or Manhattan distance. 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 two-dimensional case.\n\n\u00a0Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells.\nA 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, 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 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 squared Euclidean distance from the circle's center.[15]\nThe Voronoi diagram of \u00a0 points in \u00a0-dimensional space can have \u00a0 vertices, requiring the same bound for the amount of memory needed to store an explicit description of it. Therefore, Voronoi diagrams are often not feasible for moderate or high dimensions. A more space-efficient alternative is to use approximate Voronoi diagrams.[16]\nVoronoi 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.\n\n\nMeteorology/Hydrology\n\n \nedit\n\n\n\n\n\nIt is used in meteorology and engineering hydrology to find the weights for precipitation data of stations over an area (watershed). The points generating the polygons are the various station that record precipitation data. Perpendicular bisectors are drawn to the line joining any two stations. This results in the formation of polygons around the stations. The area \u00a0 touching station point is known as influence area of the station. The average precipitation is calculated by the formula \u00a0\nHumanities and social sciences\n\n \nedit\n\n\n\n\n\nIn classical archaeology, specifically art history, the symmetry of statue heads is analyzed to determine the type of statue a severed head may have belonged to. An example of this that made use of Voronoi cells was the identification of the Sabouroff head, which made use of a high-resolution polygon mesh.[17][18]\nIn dialectometry, Voronoi cells are used to indicate a supposed linguistic continuity between survey points.\nIn political science, Voronoi diagrams have been used to study multi-dimensional, multi-party competition.[19]\n\n\u00a0A Voronoi tessellation emerges by radial growth from seeds outward.\nIn biology, Voronoi diagrams are used to model a number of different biological structures, including cells[20] and bone microarchitecture.[21] Indeed, Voronoi tessellations work as a geometrical tool to understand the physical constraints that drive the organization of biological tissues.[22]\nIn hydrology, 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.\nIn 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.\nIn ethology, Voronoi diagrams are used to model domains of danger in the selfish herd theory.\nIn computational chemistry, ligand-binding sites are transformed into Voronoi diagrams for machine learning applications (e.g., to classify binding pockets in proteins).[23] In other applications, Voronoi cells defined by the positions of the nuclei in a molecule are used to compute atomic charges. This is done using the Voronoi deformation density method.\nIn astrophysics, Voronoi diagrams are used to generate adaptative smoothing zones on images, adding signal fluxes on each one. The main objective of these procedures is to maintain a relatively constant signal-to-noise ratio on all the images.\nIn computational fluid dynamics, the Voronoi tessellation of a set of points can be used to define the computational domains used in finite volume methods, e.g. as in the moving-mesh cosmology code AREPO.[24]\nIn computational physics, Voronoi diagrams are used to calculate profiles of an object with Shadowgraph and proton radiography in High energy density physics.[25]\n\nIn medical diagnosis, models of muscle tissue, based on Voronoi diagrams, can be used to detect neuromuscular diseases.[22]\nIn 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 to study the 1854 Broad Street cholera outbreak in Soho, England. He showed the correlation between residential areas on the map of Central London whose residents had been using a specific water pump, and the areas with the most deaths due to the outbreak.[26]\n\nIn polymer physics, Voronoi diagrams can be used to represent free volumes of polymers.\nIn materials science, polycrystalline microstructures in metallic alloys are commonly represented using Voronoi tessellations.\nIn island growth, the Voronoi diagram is used to estimate the growth rate of individual islands.[27][28][29][30][31]\nIn solid-state physics, the Wigner-Seitz cell is the Voronoi tessellation of a solid, and the Brillouin zone is the Voronoi tessellation of reciprocal (wavenumber) space of crystals which have the symmetry of a space group.\nIn aviation, Voronoi diagrams are superimposed on oceanic plotting charts to identify the nearest airfield for in-flight diversion (see ETOPS), as an aircraft progresses through its flight plan.\nIn architecture, Voronoi patterns were the basis for the winning entry for the redevelopment of The Arts Centre Gold Coast.[32]\nIn urban planning, Voronoi diagrams can be used to evaluate the Freight Loading Zone system.[33]\nIn 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.\nIn surface metrology, Voronoi tessellation can be used for surface roughness modeling.[34]\nIn robotics, some of the control strategies and path planning algorithms[35] of multi-robot systems are based on the Voronoi partitioning of the environment.[36][37]\n\nA point location data structure can be built on top of the Voronoi diagram in order to answer 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.\nIn geometry, Voronoi diagrams can be used to find the 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.\nVoronoi diagrams together with farthest-point Voronoi diagrams are used for efficient algorithms to compute the roundness of a set of points.[12] The Voronoi approach is also put to use in the evaluation of circularity/roundness while assessing the dataset from a coordinate-measuring machine.\nZeroes of iterated derivatives of a rational function on the complex plane accumulate on the edges of the Voronoi diagam of the set of the poles (P\u00f3lya's shires theorem[38]).\n\nIn networking, Voronoi diagrams can be used in derivations of the capacity of a wireless network.\nIn computer graphics, Voronoi diagrams are used to calculate 3D shattering / fracturing geometry patterns. It is also used to procedurally generate organic or lava-looking textures.\nIn 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).\nIn machine learning, Voronoi diagrams are used to do 1-NN classifications.[39]\nIn global scene reconstruction, including with random sensor sites and unsteady wake flow, geophysical data, and 3D turbulence data, Voronoi tesselations are used with deep learning.[40]\nIn user interface development, Voronoi patterns can be used to compute the best hover state for a given point.[41]\n\nSeveral efficient algorithms are known for constructing Voronoi diagrams, either directly (as the diagram itself) or indirectly by starting with a Delaunay triangulation and then obtaining its dual.\nDirect algorithms include Fortune's algorithm, an O(n log(n)) algorithm for generating a Voronoi diagram from a set of points in a plane.\nBowyer\u2013Watson algorithm, an O(n log(n)) to O(n2) algorithm for generating a Delaunay triangulation in any number of dimensions, can be used in an indirect algorithm for the Voronoi diagram. The Jump Flooding Algorithm can generate approximate Voronoi diagrams in constant time and is suited for use on commodity graphics hardware.[42][43]\nLloyd's algorithm and its generalization via the Linde\u2013Buzo\u2013Gray algorithm (aka k-means clustering) use the construction of Voronoi diagrams as a subroutine.\nThese methods alternate between steps in which one constructs the Voronoi diagram for a set of seed points, and steps in which the seed points are moved to new locations that are more central within their cells. These methods can be used in spaces of arbitrary dimension to iteratively converge towards a specialized form of the Voronoi diagram, called a Centroidal Voronoi tessellation, where the sites have been moved to points that are also the geometric centers of their cells.\n\n\nVoronoi meshes can also be generated in 3D.\n\n\n\t\t\n\t\t\t\n\t\t\tRandom points in 3D for forming a 3D Voronoi partition\n\t\t\n\t\t\n\t\t\t\n\t\t\t3D Voronoi mesh of 25 random points\n\t\t\n\t\t\n\t\t\t\n\t\t\t3D Voronoi mesh of 25 random points with 0.3 opacity and points\n\t\t\n\t\t\n\t\t\t\n\t\t\t3D Voronoi mesh of 25 random points convex polyhedra pieces\n\t\t\n\n\nDelaunay triangulation\nMap segmentation\nNatural element method\nNatural neighbor interpolation\nNearest-neighbor interpolation\nPower diagram\nVoronoi pole\n\n\n^ Burrough, Peter A.; McDonnell, Rachael; McDonnell, Rachael A.; Lloyd, Christopher D. (2015). \"8.11 Nearest neighbours: Thiessen (Dirichlet/Voroni) polygons\". 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Information Processing Letters. 37 (3): 121\u2013125. doi:10.1016/0020-0190(91)90030-L., contains a simple algorithm to compute the farthest-point Voronoi diagram.\n\n^ Biedl, Therese; Grimm, Carsten; Palios, Leonidas; Shewchuk, Jonathan; Verdonschot, Sander (2016). \"Realizing farthest-point Voronoi diagrams\". Proceedings of the 28th Canadian Conference on Computational Geometry (CCCG 2016).\n\n^ Edelsbrunner, Herbert (2012) [1987]. \"13.6 Power Diagrams\". Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science. Vol.\u00a010. Springer-Verlag. pp.\u00a0327\u2013328. ISBN\u00a09783642615689.\n\n^ Sunil Arya, Sunil; Malamatos, Theocharis; Mount, David M. (2002). \"Space-efficient approximate Voronoi diagrams\". Proceedings of the thiry-fourth annual ACM symposium on Theory of computing. pp.\u00a0721\u2013730. doi:10.1145/509907.510011. ISBN\u00a01581134959. 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ISSN\u00a01098-0121. S2CID\u00a013719907.\n\n^ L\u00f6bl, Matthias C.; Zhai, Liang; Jahn, Jan-Philipp; Ritzmann, Julian; Huo, Yongheng; Wieck, Andreas D.; Schmidt, Oliver G.; Ludwig, Arne; Rastelli, Armando; Warburton, Richard J. (2019-10-03). \"Correlations between optical properties and Voronoi-cell area of quantum dots\". Physical Review B. 100 (15): 155402. arXiv:1902.10145. Bibcode:2019PhRvB.100o5402L. doi:10.1103/physrevb.100.155402. ISSN\u00a02469-9950. S2CID\u00a0119443529.\n\n^ \"GOLD COAST CULTURAL PRECINCT\". ARM Architecture. Archived from the original on 2016-07-07. Retrieved 2014-04-28.\n\n^ Lopez, C.; Zhao, C.-L.; Magniol, S; Chiabaut, N; Leclercq, L (28 February 2019). \"Microscopic Simulation of Cruising for Parking of Trucks as a Measure to Manage Freight Loading Zone\". Sustainability. 11 (5), 1276 (5): 1276. Bibcode:2019Sust...11.1276L. doi:10.3390/su11051276.\n\n^ Singh, K.; Sadeghi, F.; Correns, M.; Blass, T. (December 2019). \"A microstructure based approach to model effects of surface roughness on tensile fatigue\". International Journal of Fatigue. 129: 105229. doi:10.1016/j.ijfatigue.2019.105229. S2CID\u00a0202213370.\n\n^ Niu, Hanlin; Savvaris, Al; Tsourdos, Antonios; Ji, Ze (2019). \"Voronoi-visibility roadmap-based path planning algorithm for unmanned surface vehicles\" (PDF). The Journal of Navigation. 72 (4): 850\u2013874. Bibcode:2019JNav...72..850N. doi:10.1017/S0373463318001005. S2CID\u00a067908628.\n\n^ Cortes, J.; Martinez, S.; Karatas, T.; Bullo, F. (April 2004). \"Coverage control for mobile sensing networks\". IEEE Transactions on Robotics and Automation. 20 (2): 243\u2013255. doi:10.1109/TRA.2004.824698. ISSN\u00a02374-958X. S2CID\u00a02022860.\n\n^ Teruel, Enrique; Aragues, Rosario; L\u00f3pez-Nicol\u00e1s, Gonzalo (April 2021). \"A Practical Method to Cover Evenly a Dynamic Region With a Swarm\". 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Proceedings of the 2006 Symposium on Interactive 3D Graphics, SI3D 2006, March 14-17, 2006, Redwood City, California, USA. ACM. pp.\u00a0109\u2013116. doi:10.1145/1111411.1111431. ISBN\u00a01-59593-295-X.\n\n^ \"Shadertoy\".\n\n\nAurenhammer, Franz; Klein, Rolf; Lee, Der-Tsai (2013). Voronoi Diagrams and Delaunay Triangulations. World Scientific. ISBN\u00a0978-9814447638.\nBowyer, Adrian (1981). \"Computing Dirichlet tessellations\". Comput. J. 24 (2): 162\u2013166. doi:10.1093/comjnl/24.2.162.\n Includes a description of Fortune's algorithm.\nKlein, Rolf (1988). \"Abstract voronoi diagrams and their applications: Extended abstract\". Computational Geometry and its Applications. Lecture Notes in Computer Science. Vol.\u00a0333. Springer. pp.\u00a0148\u2013157. doi:10.1007/3-540-50335-8_31. ISBN\u00a0978-3-540-52055-9.\nLejeune Dirichlet, G. (1850). \"\u00dcber die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen\". Journal f\u00fcr die Reine und Angewandte Mathematik. 1850 (40): 209\u2013227. doi:10.1515/crll.1850.40.209. S2CID\u00a0199546675.\nOkabe, Atsuyuki; Boots, Barry; Sugihara, Kokichi; Chiu, Sung Nok (2000). Spatial Tessellations \u2014 Concepts and Applications of Voronoi Diagrams (2nd\u00a0ed.). Wiley. ISBN\u00a00-471-98635-6.\nReem, Daniel (2009). \"An algorithm for computing Voronoi diagrams of general generators in general normed spaces\". Proceedings of the Sixth International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2009). pp.\u00a0144\u2013152. doi:10.1109/ISVD.2009.23. ISBN\u00a0978-1-4244-4769-5.\nReem, Daniel (2011). \"The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites\". Proceedings of the twenty-seventh annual symposium on Computational geometry. pp.\u00a0254\u2013263. arXiv:1103.4125. Bibcode:2011arXiv1103.4125R. doi:10.1145/1998196.1998234. ISBN\u00a09781450306829. S2CID\u00a014639512.\nThiessen, Alfred H. (July 1911). \"Precipitation averages for large areas\". Monthly Weather Review. 39 (7). American Meteorological Society: 1082\u20131089. Bibcode:1911MWRv...39R1082T. doi:10.1175/1520-0493(1911)39<1082b:pafla>2.0.co;2.\nVorono\u00ef, Georges (1908a). \"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Premier m\u00e9moire. Sur quelques propri\u00e9t\u00e9s des formes quadratiques positives parfaites\" (PDF). Journal f\u00fcr die Reine und Angewandte Mathematik. 1908 (133): 97\u2013178. doi:10.1515/crll.1908.133.97. S2CID\u00a0116775758.\nVorono\u00ef, Georges (1908b). \"Nouvelles applications des param\u00e8tres continus \u00e0 la th\u00e9orie des formes quadratiques. Deuxi\u00e8me m\u00e9moire. Recherches sur les parall\u00e9llo\u00e8dres primitifs\" (PDF). Journal f\u00fcr die Reine und Angewandte Mathematik. 1908 (134): 198\u2013287. doi:10.1515/crll.1908.134.198. S2CID\u00a0118441072.\nWatson, David F. (1981). \"Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes\". Comput. J. 24 (2): 167\u2013172. doi:10.1093/comjnl/24.2.167.\n\n\nWeisstein, Eric W. \"Voronoi diagram\". MathWorld.\nVoronoi Diagrams in CGAL, the Computational Geometry Algorithms Library\nDemo program for SFTessellation algorithm, which creates Voronoi diagram using a Steppe Fire Model" ], "output": "readability/", "pwd": "/data/archive/1743133040.525563", "schema": "ArchiveResult", "start_ts": "2025-03-28T03:37:49.741470+00:00", "status": "succeeded" } ], "screenshot": [], "singlefile": [], "title": [ { "cmd": [ "/usr/bin/curl", "--silent", "--location", "--compressed", "--proxy", "socks5://tor-socks-proxy:9150", "--max-time", "60", "--user-agent", "Mozilla/5.0 (Macintosh; Intel Mac OS X 10_15_7) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/128.0.0.0 Safari/537.36 ArchiveBox/{VERSION} (+https://github.com/ArchiveBox/ArchiveBox/)", "https://en.m.wikipedia.org/wiki/Voronoi_diagram" ], "cmd_version": "8.10.1", "end_ts": "2025-03-28T03:37:47.953335+00:00", "index_texts": null, "output": "Voronoi diagram - Wikipedia", "pwd": "/data/archive/1743133040.525563", "schema": "ArchiveResult", "start_ts": "2025-03-28T03:37:47.295232+00:00", "status": "succeeded" } ], "wget": [] }, "icons": null, "is_archived": true, "is_static": false, "latest": { "archive_org": "https://web.archive.org/web/20250328033842/https://en.m.wikipedia.org/wiki/Voronoi_diagram", "dom": "output.html", "favicon": "favicon.ico", "git": null, "media": "media/", "pdf": null, "screenshot": null, "singlefile": null, "title": "Voronoi diagram - Wikipedia", "warc": null, "wget": null }, "link_dir": "/data/archive/1743133040.525563", "newest_archive_date": "2025-03-28T03:38:28.144141+00:00", "num_failures": 0, "num_outputs": 9, "oldest_archive_date": "2025-03-28T03:37:26.831271+00:00", "path": "/wiki/Voronoi_diagram", "schema": "Link", "scheme": "https", "snapshot_abid": "snp_01JQDD6R063614EF0E01Z1ECVG", "snapshot_id": "cb8b9f52-48e2-4c49-be57-2cc63e173370", "sources": [ "/data/sources/1743133036-import.txt" ], "tags": null, "tags_str": "", "timestamp": "1743133040.525563", "title": "Voronoi diagram - Wikipedia", "url": "https://en.m.wikipedia.org/wiki/Voronoi_diagram" }