Voronoi diagram: Difference between revisions

In

.

It is named after

They are also known as Thiessen polygons.[3]^[4][5]

The simplest case

In the simplest case, shown in the first picture, we are given a finite set of points {p₁, …, p_n} in the

) are the points equidistant to three (or more) sites.

Formal definition

Let ${\displaystyle \scriptstyle X}$ be a metric space with distance function ${\displaystyle \scriptstyle d}$. Let ${\displaystyle \scriptstyle K}$ be a set of indices and let ${\displaystyle \scriptstyle (P_{k})_{k\in K}}$ be a

(the sites) in the space ${\displaystyle \scriptstyle X}$. The Voronoi cell, or Voronoi region, ${\displaystyle \scriptstyle R_{k}}$, associated with the site ${\displaystyle \scriptstyle P_{k}}$ is the set of all points in ${\displaystyle \scriptstyle X}$ whose distance to ${\displaystyle \scriptstyle P_{k}}$ is not greater than their distance to the other sites ${\displaystyle \scriptstyle P_{j}}$, where ${\displaystyle \scriptstyle j}$ is any index different from ${\displaystyle \scriptstyle k}$. In other words, if ${\displaystyle \scriptstyle d(x,\,A)\;=\;\inf\{d(x,\,a)\mid a\,\in \,A\}}$ denotes the distance between the point ${\displaystyle \scriptstyle x}$ and the subset ${\displaystyle \scriptstyle A}$, then

${\displaystyle R_{k}=\{x\in X\mid d(x,P_{k})\leq d(x,P_{j})\;{\text{for all}}\;j\neq k\}}$

The Voronoi diagram is simply the tuple of cells ${\displaystyle \scriptstyle (R_{k})_{k\in K}}$. 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 geometry of numbers and crystallography), but again, in many cases only finitely many sites are considered.

In the particular case where the space is a

and they can be represented in a combinatorial way using their vertices, sides, 2-dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. However, in general the Voronoi cells may not be convex or even connected.

In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon ${\displaystyle \scriptstyle R_{k}}$ is associated with a generator point ${\displaystyle \scriptstyle P_{k}}$. Let ${\displaystyle \scriptstyle X}$ be the set of all points in the Euclidean space. Let ${\displaystyle \scriptstyle P_{1}}$ be a point that generates its Voronoi region ${\displaystyle \scriptstyle R_{1}}$, ${\displaystyle \scriptstyle P_{2}}$ that generates ${\displaystyle \scriptstyle R_{2}}$, and ${\displaystyle \scriptstyle P_{3}}$ that generates ${\displaystyle \scriptstyle R_{3}}$, and so on. Then, as expressed by Tran et al[6] "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 ${\displaystyle \scriptstyle R_{k}}$ of a given shop ${\displaystyle \scriptstyle P_{k}}$ 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: ${\displaystyle \ell _{2}=d\left[\left(a_{1},a_{2}\right),\left(b_{1},b_{2}\right)\right]={\sqrt {\left(a_{1}-b_{1}\right)^{2}+\left(a_{2}-b_{2}\right)^{2}}}}$ or the

:${\displaystyle d\left[\left(a_{1},a_{2}\right),\left(b_{1},b_{2}\right)\right]=\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\right|}$. The corresponding Voronoi diagrams look different for different distance metrics.

Properties

• The dual graph for a Voronoi diagram (in the case of a Euclidean space with point sites) corresponds to the Delaunay triangulation for the same set of points.
• The
  closest pair of points
  corresponds to two adjacent cells in the Voronoi diagram.
• Assume the setting is the Euclidean plane and a group of different points are given. Then two points are adjacent on the convex hull if and only if their Voronoi cells share an infinitely long side.
• If the space 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.^[7]
  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, 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.^[8] 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

used 2-dimensional and 3-dimensional Voronoi diagrams in his study of quadratic forms in 1850. British physician than to any other water pump.

Voronoi diagrams are named after Ukrainian mathematician living in the Russian Empire

. 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 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
  square lattice gives the regular tessellation of squares; note that the rectangles and the squares can also be generated by other lattices (for example the lattice defined by the vectors (1,0) and (1/2,1/2) gives squares). See here
  for a dynamic visual example.
• A
  simple cubic lattice gives the cubic honeycomb
  .
• A
  hexagonal close-packed lattice gives a tessellation of space with trapezo-rhombic dodecahedra
  .
• A
  face-centred cubic lattice gives a tessellation of space with rhombic dodecahedra
  .
• A
  body-centred cubic lattice gives a tessellation of space with truncated octahedra
  .
• Parallel planes with regular triangular lattices aligned with each other's centers give the
  hexagonal prismatic honeycomb
  .
• 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 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.

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 = {x₁, x₂, ..., x_n−1} 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 = {p₁, p₂, ..., 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 convex hull of P. Let H = {h₁, h₂, ..., 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 ∈ S with h_i ≠ p_j, where d(p, q) is the Euclidean distance between two points p and q.^[9][10]

The boundaries of the cells in the farthest-point Voronoi diagram have the structure of a

Generalizations and variations

As implied by the definition, Voronoi cells can be defined for metrics other than Euclidean, such as the

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 2-dimensional case.

A

The Voronoi diagram of n points in d-dimensional space requires ${\displaystyle \scriptstyle O\left(n^{\left\lceil {\frac {1}{2}}d\right\rceil }\right)}$ storage space.^[clarification needed] Therefore, Voronoi diagrams are often not feasible for d > 2.^{[clarification needed]} An alternative is to use approximate Voronoi diagrams, where the Voronoi cells have a fuzzy boundary, which can be approximated.^[13]

Voronoi diagrams are also related to other geometric structures such as the medial axis (which has found applications in image segmentation, optical character recognition, and other computational applications), straight skeleton, and zone diagrams. Besides points, such diagrams use lines and polygons as seeds. By augmenting the diagram with line segments that connect to nearest points on the seeds, a planar subdivision of the environment is obtained.^[14] This structure can be used as a navigation mesh for path-finding through large spaces. The navigation mesh has been generalized to support 3D multi-layered environments, such as an airport or a multi-storey building.^[15]

Applications

Natural sciences

• In
  bone microarchitecture.^[17] Indeed, Voronoi tessellations work as a geometrical tool to understand the physical constraints that drive the organization of biological tissues.^[18]
• In 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 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 computational chemistry, 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 for these procedures is to maintain a relatively constant signal-to-noise ratio on all the image.
• In
  finite volume methods, e.g. as in the moving-mesh cosmology code AREPO.^[19]
• In computational physics, Voronoi diagrams are used to calculate profiles of an object with Shadowgraph and proton radiography in High energy density physics.^[20]

Health

• In medical diagnosis, models of muscle tissue, based on Voronoi diagrams, can be used to detect neuromuscular diseases.^[18]

  

• In
  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 most deaths due to the outbreak.^[21]

Engineering

• In polymer physics, Voronoi diagrams can be used to represent free volumes of polymers.
• In
  Wigner-Seitz cell is the Voronoi tessellation of a solid, and the Brillouin zone
  is the Voronoi tessellation of reciprocal (wave number) space of crystals which have the symmetry of a space group.
• In 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.
• In
  The Arts Centre Gold Coast.^[22]
• In mining, Voronoi polygons are used to estimate the reserves of valuable materials, minerals, or other resources. Exploratory drillholes are used as the set of points in the Voronoi polygons.

Geometry

• A 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.
• In 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.
• Voronoi diagrams together with farthest-point Voronoi diagrams are used for efficient algorithms to compute the
  roundness while assessing the dataset from a coordinate-measuring machine
  .
• Modern computational geometry has provided efficient algorithms for constructing Voronoi diagrams, and has allowed them to be used in mesh generation, point location, cluster analysis, machining plans and many other computational tasks.^[23]

Informatics

• In networking, Voronoi diagrams can be used in derivations of the capacity of a wireless network.
• In 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.
• In autonomous robot navigation, Voronoi diagrams are used to find clear routes. If the points are obstacles, then the edges of the graph will be the routes furthest from obstacles (and theoretically any collisions).
• In
  1-NN classifications.^[24]
• In user interface development, Voronoi patterns can be used to compute the best hover state for a given point.^[25]

Civics and planning

• In Victoria, Australia, government schools typically admit eligible students to the nearest primary school or high school to where they live.^[26] Students and parents can see which school "catchment zone" they live in by using this Voronoi representation.

Algorithms

Direct algorithms:

• Fortune's algorithm, an O(n log(n)) algorithm for generating a Voronoi diagram from a set of points in a plane.
• Lloyd's algorithm and its generalization via the Linde–Buzo–Gray algorithm (aka k-means clustering), produces a Voronoi tessellation in a space of arbitrary dimensions.

Starting with a Delaunay triangulation (obtain the dual):

• Bowyer–Watson algorithm, an O(n log(n)) to O(n²) algorithm for generating a Delaunay triangulation in any number of dimensions, from which the Voronoi diagram can be obtained.

See also

• Centroidal Voronoi tessellation
• Computational geometry
• Delaunay triangulation
• Mathematical diagram
• Natural neighbor interpolation
• Nearest neighbor search
• Nearest-neighbor interpolation
• Voronoi pole
• Power diagram
• Map segmentation

Notes

References

• Dirichlet, G. Lejeune (1850). "Über die Reduktion der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen". Journal für die Reine und Angewandte Mathematik. 40: 209–227.
• Voronoi, Georgy (1908). "Nouvelles applications des paramètres continus à la théorie des formes quadratiques". Journal für die Reine und Angewandte Mathematik. 1908 (133): 97–178.
  doi:10.1515/crll.1908.133.97
  .
• Atsuyuki Okabe, Barry Boots,
  ISBN 0-471-98635-6
• ISBN 978-9814447638
• doi:10.1093/comjnl/24.2.162
  .
• 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. 144–152.
  doi:10.1109/ISVD.2009.23
  .

• Daniel Reem (2011). The geometric stability of Voronoi diagrams with respect to small changes of the sites. Full version: arXiv 1103.4125 (2011), Extended abstract: in Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG 2011), pp. 254–263.
• Watson, David F. (1981). "Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes".
  doi:10.1093/comjnl/24.2.167
  .
• de Berg, Mark; van Kreveld, Marc;
  ISBN 3-540-65620-0
  . Chapter 7: Voronoi Diagrams: pp. 147–163. Includes a description of Fortune's algorithm.
• Klein, Rolf (1989). Abstract voronoi diagrams and their applications.
  ISBN 3-540-52055-4
  .

External links

Wikimedia Commons has media related to Voronoi diagrams.

• Real time interactive Voronoi and Delaunay diagrams with source code
• Demo for various metrics
• Mathworld on Voronoi diagrams
• Voronoi Diagrams: Applications from Archaeology to Zoology
• Voronoi Diagrams in CGAL, the Computational Geometry Algorithms Library
• More discussions and picture gallery on centroidal Voronoi tessellations
• Voronoi Diagrams by
  Ed Pegg, Jr., Jeff Bryant, and Theodore Gray, Wolfram Demonstrations Project
  .
• A Voronoi diagram on a sphere
• Plot a Voronoi diagram with Mathematica
• Voronoi Tessellation – Interactive Voronoi tessellation with D3.js