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In mathematics, a Voronoi diagram is a
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
of a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
into regions close to each of a given set of objects. 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 In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
, 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 In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in
science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...
and
technology Technology is the application of knowledge to reach practical goals in a specifiable and Reproducibility, reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in me ...
, but also in
visual art The visual arts are art forms such as painting, drawing, printmaking, sculpture, ceramics, photography, video, filmmaking, design, crafts and architecture. Many artistic disciplines such as performing arts, conceptual art, and textile arts ...
.


The simplest case

In the simplest case, shown in the first picture, we are given a finite set of points in the Euclidean plane. In this case each site ''p''''k'' is simply a point, and its corresponding Voronoi cell ''R''''k'' consists of every point in the Euclidean plane whose distance to ''p''''k'' is less than or equal to its distance to any other ''p''''k''. Each such cell is obtained from the intersection of half-spaces, and hence it is a (convex) polyhedron. The line segments of the Voronoi diagram are all the points in the plane that are equidistant to the two nearest sites. The Voronoi vertices (
node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines ...
s) are the points equidistant to three (or more) sites.


Formal definition

Let X be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
with distance function d. Let K be a set of indices and let (P_k)_ be a
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
(ordered collection) of nonempty subsets (the sites) in the space X. The Voronoi cell, or Voronoi region, R_k, associated with the site P_k is the set of all points in X whose distance to P_k is not greater than their distance to the other sites P_j, where j is any index different from k. In other words, if d(x,\, A) = \inf\ denotes the distance between the point x and the subset A, then R_k = \ The Voronoi diagram is simply the
tuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of cells (R_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
geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informatio ...
and crystallography), but again, in many cases only finitely many sites are considered. In the particular case where the space is a finite-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, 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. In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon R_k is associated with a generator point P_k. Let X be the set of all points in the Euclidean space. Let P_1 be a point that generates its Voronoi region R_1, P_2 that generates R_2, and P_3 that generates R_3, and so on. Then, as expressed by Tran ''et al'', "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

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 of a given shop 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
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
: :\ell_2 = d\left left(a_1, a_2\right), \left(b_1, b_2\right)\right= \sqrt or the
Manhattan distance A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
: :d\left left(a_1, a_2\right), \left(b_1, b_2\right)\right= \left, a_1 - b_1\ + \left, a_2 - b_2\. The corresponding Voronoi diagrams look different for different distance metrics.


Properties

* The
dual graph In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-lo ...
for a Voronoi diagram (in the case of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
with point sites) corresponds to the
Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
for the same set of points. * The
closest pair of points The closest pair of points problem or closest pair problem is a problem of computational geometry: given n points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean ...
corresponds to two adjacent cells in the Voronoi diagram. * Assume 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. * If the space is a normed space and the distance to each site is attained (e.g., when a site is a
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
or a closed ball), then each Voronoi cell can be represented as a union of line segments emanating from the sites.. 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
uniformly convex space In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a ...
, 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.. 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

Informal use of Voronoi diagrams can be traced back to Descartes in 1644.
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
used two-dimensional and three-dimensional Voronoi diagrams in his study of quadratic forms in 1850. British 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 Soho is an area of the City of Westminster, part of the West End of London. Originally a fashionable district for the aristocracy, it has been one of the main entertainment districts in the capital since the 19th century. The area was develop ...
than to any other water pump. Voronoi diagrams are named after Georgy Feodosievych Voronoy who defined and studied the general ''n''-dimensional case in 1908. Voronoi diagrams that are used in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
and
meteorology Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...
to analyse spatially distributed data (such as rainfall measurements) are called Thiessen polygons after American meteorologist Alfred H. Thiessen. 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

Voronoi tessellations of regular
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
s 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
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
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 simple cubic lattice gives the
cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a r ...
. * A
hexagonal close-packed In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...
lattice gives a tessellation of space with trapezo-rhombic dodecahedra. * A
face-centred cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...
lattice gives a tessellation of space with rhombic dodecahedra. * A
body-centred cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...
lattice gives a tessellation of space with
truncated octahedra In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 ...
. * Parallel planes with regular triangular lattices aligned with each other's centers give the
hexagonal prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...
. * Certain body-centered tetragonal lattices give a tessellation of space with 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

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'' − 1)th-order diagram and replace each cell generated by ''X'' =  with a Voronoi diagram generated on the set ''S'' − ''X''.


Farthest-point Voronoi diagram

For a set of ''n'' points the (''n'' − 1)th-order Voronoi diagram is called a farthest-point Voronoi diagram. For a given set of points ''S'' =  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'' =  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'' ∈ ''S'' with ''h''''i'' ≠ ''p''''j'', where d(''p'', ''q'') is the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
between two points ''p'' and ''q''. 7.4 Farthest-Point Voronoi Diagrams. Includes a description of the algorithm. The boundaries of the cells in the farthest-point Voronoi diagram have the structure of a topological tree, with infinite
rays Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gra ...
as its leaves. Every finite tree is isomorphic to the tree formed in this way from a farthest-point Voronoi diagram.


Generalizations and variations

As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as the Mahalanobis distance or
Manhattan distance A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
. 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. 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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
, 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 Power distance is a dimension theorized and proven by Geert Hofstede, who outlined multiple cultural dimensions throughout his work. This term refers to inequality and unequal distributions of power between parties; whether it is within the work ...
; 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 In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two Point (geometry), points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theo ...
from the circle's center. The Voronoi diagram of n points in d-dimensional space can have O(n^) 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. Voronoi diagrams are also related to other geometric structures such as the
medial axis The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recogn ...
(which has found applications in image segmentation,
optical character recognition Optical character recognition or optical character reader (OCR) is the electronic or mechanical conversion of images of typed, handwritten or printed text into machine-encoded text, whether from a scanned document, a photo of a document, a sc ...
, and other computational applications),
straight skeleton In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a pol ...
, and zone diagrams.


Applications


Meteorology/Hydrology

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) touching station point is known as influence area of the station. The average precipitation is calculated by the formula \bar=\frac


Humanities

*In classical archaeology, specifically
art history Art history is the study of aesthetic objects and visual expression in historical and stylistic context. Traditionally, the discipline of art history emphasized painting, drawing, sculpture, architecture, ceramics and decorative arts; yet today ...
, 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 The Sabouroff head is a Late Archaic Greece, Archaic Greek marble sculpture. It is dated to circa 550 BC, 550–525 BC. This head of a Kouros was named after Peter Alexandrovich Saburov, a collector of ancient Greek sculpture and antiquities. It i ...
, which made use of a high-resolution
polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles ( triangle mesh), quadrilaterals (quads), or other simple convex p ...
. *In
dialectometry Dialectometry is the quantitative and computational branch of dialectology, the study of dialect. This sub-field of linguistics studies language variation using the methods of statistics; it arose in the 1970s and 80s as a result of seminal work ...
, Voronoi cells are used to indicate a supposed linguistic continuity between survey points.


Natural sciences

*In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, Voronoi diagrams are used to model a number of different biological structures, including cells and bone microarchitecture. Indeed, Voronoi tessellations work as a geometrical tool to understand the physical constraints that drive the organization of biological tissues. *In
hydrology Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...
, 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
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
, 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 computational chemistry, ligand-binding sites are transformed into Voronoi diagrams for
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
applications (e.g., to classify binding pockets in proteins). 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. *In 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. *In
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
, 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. *In
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...
, Voronoi diagrams are used to calculate profiles of an object with
Shadowgraph Shadowgraph is an optical method that reveals non-uniformities in transparent media like air, water, or glass. It is related to, but simpler than, the schlieren and schlieren photography methods that perform a similar function. Shadowgraph is a ...
and proton radiography in
High energy density physics High-energy-density physics (HEDP) is a new subfield of physics intersecting condensed matter physics, nuclear physics, astrophysics and plasma physics. It has been defined as the physics of matter and radiation at energy densities in excess of a ...
.


Health

*In
medical diagnosis Medical diagnosis (abbreviated Dx, Dx, or Ds) is the process of determining which disease or condition explains a person's symptoms and signs. It is most often referred to as diagnosis with the medical context being implicit. The information r ...
, models of muscle tissue, based on Voronoi diagrams, can be used to detect neuromuscular diseases. *In
epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population. It is a cornerstone of public health, and shapes policy decisions and evide ...
, 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.


Engineering

*In polymer physics, Voronoi diagrams can be used to represent free volumes of polymers. *In 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. In solid-state physics, the Wigner-Seitz cell is the Voronoi tessellation of a solid, and the
Brillouin zone In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
is the Voronoi tessellation of reciprocal (
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
) space of crystals which have the symmetry of a space group. *In
aviation Aviation includes the activities surrounding mechanical flight and the aircraft industry. ''Aircraft'' includes fixed-wing and rotary-wing types, morphable wings, wing-less lifting bodies, as well as lighter-than-air craft such as hot a ...
, Voronoi diagrams are superimposed on oceanic plotting charts to identify the nearest airfield for in-flight diversion (see
ETOPS ETOPS () is an acronym for ''Extended-range Twin-engine Operations Performance Standards''—a special part of flight rules for one-engine-inoperative flight conditions. The International Civil Aviation Organization (ICAO) coined the acronym for ...
), as an aircraft progresses through its flight plan. *In
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
, Voronoi patterns were the basis for the winning entry for the redevelopment of
The Arts Centre Gold Coast Home of the Arts (HOTA), opened as the Keith Hunt Community Entertainment and Arts Centre in 1986 and subsequently renamed The Arts Centre Gold Coast (TAC) and Gold Coast Arts Centre, is a cultural precinct situated in Surfers Paradise, City of ...
. *In
urban planning Urban planning, also known as town planning, city planning, regional planning, or rural planning, is a technical and political process that is focused on the development and design of land use and the built environment, including air, water, ...
, Voronoi diagrams can be used to evaluate the Freight Loading Zone system. *In
mining Mining is the extraction of valuable minerals or other geological materials from the Earth, usually from an ore body, lode, vein, seam, reef, or placer deposit. The exploitation of these deposits for raw material is based on the economic ...
, 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
surface metrology Surface metrology is the measurement of small-scale features on surfaces, and is a branch of metrology. Surface primary form, surface Fractal dimension, fractality, and surface finish (including surface roughness) are the parameters most commonly a ...
, Voronoi tessellation can be used for surface roughness modeling. *In
robotics Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrate ...
, some of the control strategies and path planning algorithms of multi-robot systems are based on the Voronoi partitioning of the environment.


Geometry

*A
point location The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided d ...
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 In computing, a database is an organized collection of data stored and accessed electronically. Small databases can be stored on a file system, while large databases are hosted on computer clusters or cloud storage. The design of databases s ...
. A large application is
vector quantization Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the distribution of prototype vectors. It was originally used for data compression. It works by di ...
, commonly used in
data compression In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressio ...
. *In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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. *Voronoi diagrams together with farthest-point Voronoi diagrams are used for efficient algorithms to compute the roundness of a set of points. The Voronoi approach is also put to use in the evaluation of circularity/ roundness while assessing the dataset from a
coordinate-measuring machine A coordinate measuring machine (CMM) is a device that measures the geometry of physical objects by sensing discrete points on the surface of the object with a probe. Various types of probes are used in CMMs, the most common being mechanical and l ...
.


Informatics

*In networking, Voronoi diagrams can be used in derivations of the capacity of a
wireless network A wireless network is a computer network that uses wireless data connections between network nodes. Wireless networking is a method by which homes, telecommunications networks and business installations avoid the costly process of introducing ...
. *In
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, Voronoi diagrams are used to calculate 3D shattering / fracturing geometry patterns. It is also used to procedurally generate organic or lava-looking textures. * In autonomous
robot navigation Robot localization denotes the robot's ability to establish its own position and orientation within the frame of reference. Path planning is effectively an extension of localisation, in that it requires the determination of the robot's current pos ...
, 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
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
, Voronoi diagrams are used to do 1-NN classifications. *In 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. *In
user interface In the industrial design field of human–computer interaction, a user interface (UI) is the space where interactions between humans and machines occur. The goal of this interaction is to allow effective operation and control of the machine f ...
development, Voronoi patterns can be used to compute the best hover state for a given point.


Civics and planning

* In
Melbourne Melbourne ( ; Boonwurrung/Woiwurrung: ''Narrm'' or ''Naarm'') is the capital and most populous city of the Australian state of Victoria, and the second-most populous city in both Australia and Oceania. Its name generally refers to a met ...
, government school students are always eligible to attend the nearest primary school or high school to where they live, as measured by a straight-line distance. The map of school zones is therefore a Voronoi diagram.


Bakery

* Ukrainian Pastry chef Dinara Kasko uses the mathematical principles of the Voronoi diagram to create silicone molds made with a 3D printer to shape her original cakes.


Algorithms

Several efficient algorithms are known for constructing Voronoi diagrams, either directly (as the diagram itself) or indirectly by starting with a
Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
and then obtaining its dual. Direct algorithms include
Fortune's algorithm Fortune's algorithm is a sweep line algorithm for generating a Voronoi diagram from a set of points in a plane using O(''n'' log ''n'') time and O(''n'') space. Section 7.2: Computing the Voronoi Diagram: pp.151–160. It was origina ...
, an O(''n'' log(''n'')) algorithm for generating a Voronoi diagram from a set of points in a plane.
Bowyer–Watson algorithm In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is ...
, an O(''n'' log(''n'')) to 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 Jump Flooding Algorithm can generate approximate Voronoi diagrams in constant time and is suited for use on commodity graphics hardware.
Lloyd's algorithm In electrical engineering and computer science, Lloyd's algorithm, also known as Voronoi iteration or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of Euclidean spaces and partitions of t ...
and its generalization via the
Linde–Buzo–Gray algorithm The Linde–Buzo–Gray algorithm (introduced by Yoseph Linde, Andrés Buzo and Robert M. Gray in 1980) is a vector quantization algorithm to derive a good codebook. It is similar to the k-means method in data clustering. The algorithm At each ...
(aka k-means clustering), use the construction of Voronoi diagrams as a subroutine. These 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.


See also

*
Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...
* Map segmentation * Natural element method *
Natural neighbor interpolation image:Natural-neighbors-coefficients-example.png, 200px, Natural neighbor interpolation with Sibson weights. The area of the green circles are the interpolating weights, ''w'i''. The purple-shaded region is the new Voronoi cell, after inserting ...
*
Nearest-neighbor interpolation Nearest-neighbor interpolation (also known as proximal interpolation or, in some contexts, point sampling) is a simple method of multivariate interpolation in one or more dimensions. Interpolation is the problem of approximating the value of ...
* Power diagram * Voronoi pole


Notes


References

* * * ''Includes a description of Fortune's algorithm.'' * * * * * * * *


External links

*
Voronoi Diagrams
in CGAL, the Computational Geometry Algorithms Library {{DEFAULTSORT:Voronoi Diagram Discrete geometry Computational geometry Diagrams Ukrainian inventions Russian inventions