Voronoi diagram

20 points and their Voronoi cells (larger version below).

In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. That set of points (called seeds, sites, or generators) is specified beforehand, and for each seed there is a corresponding region consisting of all points closer to that seed than to any other. These regions are called Voronoi cells. The Voronoi diagram of a set of points is dual to its Delaunay triangulation.

It is named after Georgy Voronoi, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi diagrams have practical and theoretical applications to a large number of fields, mainly in science and technology but also including visual art.[1][2] 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, …, pn} in the Euclidean plane. In this case each site pk is simply a point, and its corresponding Voronoi cell Rk consists of every point in the Euclidean plane whose distance to pk is less than or equal to its distance to any other pk. Each such cell is obtained from the intersection of half-spaces, and hence it is a convex polygon. 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 (nodes) are the points equidistant to three (or more) sites.

Formal definition

Let be a metric space with distance function . Let be a set of indices and let be a tuple (ordered collection) of nonempty subsets (the sites) in the space . The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from . In other words, if denotes the distance between the point and the subset , then

The Voronoi diagram is simply the tuple of cells . 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 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, 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 is associated with a generator point . Let be the set of all points in the Euclidean space. Let be a point that generates its Voronoi region , that generates , and that generates , 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 Euclidian plane".


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 of a given shop 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: or the Manhattan distance:. The corresponding Voronoi diagrams look different for different distance metrics.

Voronoi diagrams of 20 points under two different metrics


History and research

Informal use of Voronoi diagrams can be traced back to Descartes in 1644. Peter Gustav Lejeune Dirichlet used 2-dimensional and 3-dimensional Voronoi diagrams in his study of quadratic forms in 1850. British physician John Snow used a Voronoi 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.

Voronoi diagrams are named after Russian and Ukrainian mathematician Georgy Fedosievych Voronyi (or Voronoy) who defined and studied the general n-dimensional case in 1908. Voronoi diagrams that are used in geophysics and meteorology to analyse spatially distributed data (such as rainfall measurements) are called Thiessen polygons after American meteorologist Alfred H. Thiessen. In condensed matter physics, such tessellations are also known as Wigner–Seitz unit cells. Voronoi tessellations of the reciprocal lattice of momenta are called Brillouin zones. For general lattices in Lie groups, the cells are simply called fundamental domains. In the case of general metric spaces, the cells are often called metric fundamental polygons. 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).


This is a slice of the Voronoi diagram of a random set of points in a 3D box. In general a cross section of a 3D Voronoi tessellation is not a 2D Voronoi tessellation itself. (The cells are all convex polyhedra.)

Voronoi tessellations of regular lattices of points in two or three dimensions give rise to many familiar tessellations.

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 nth-order Voronoi diagram from set S, start with the (n  1)th-order diagram and replace each cell generated by X = {x1, x2, ..., xn−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, ..., pn} 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, ..., hk} 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  S with hipj, 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 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.[11]

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. 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.

Approximate Voronoi diagram of a set of points. Notice the blended colors in the fuzzy boundary of the Voronoi cells.

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, 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 distance from the circle's center.[12]

The Voronoi diagram of n points in d-dimensional space requires storage space. Therefore, Voronoi diagrams are often not feasible for d > 2. An alternative is to use approximate Voronoi diagrams, where the Voronoi cells have a fuzzy boundary, which can be approximated.[13] Another alternative is when any site is a fuzzy circle and as a result the cells become fuzzy too.[14]

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.[15] 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.[16]


Natural sciences






Direct algorithms:

Starting with a Delaunay triangulation (obtain the dual):

See also


  1. Franz Aurenhammer (1991). Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, 23(3):345–405, 1991
  2. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd edition. John Wiley, 2000, 671 pages ISBN 0-471-98635-6
  3. Principles of Geographical Information Systems, By Peter A. Burrough, Rachael McDonnell, Rachael A. McDonnell, Christopher D. Lloyd
  4. Geographic Information Systems and Science, By Paul Longley
  5. Spatial Modeling Principles in Earth Sciences, Zekai Sen
  6. Q.T.Tran, D.Tainar and M.Safar (2009) "Transactions on Large-Scale Data- and Knowledge-Centered Systems", pag357. ISBN 9783642037214.
  7. Daniel Reem, An algorithm for computing Voronoi diagrams of general generators in general normed spaces, In Proceedings of the sixth International Symposium on Voronoi Diagrams in science and engineering (ISVD 2009), 2009, pp. 144–152
  8. Daniel Reem, 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
  9. 1 2 Mark de Berg; Marc van Kreveld; Mark Overmars; Otfried Schwarzkopf (2008). Computational Geometry (Third ed.). Springer-Verlag. ISBN 978-3-540-77974-2. 7.4 Farthest-Point Voronoi Diagrams. Includes a description of the algorithm.
  10. Skyum, Sven (18 February 1991). "A simple algorithm for computing the smallest enclosing circle". Information Processing Letters. 37 (3): 121–125. doi:10.1016/0020-0190(91)90030-L., contains a simple algorithm to compute the farthest-point Voronoi diagram.
  11. 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).
  12. Edelsbrunner, Herbert (1987), "13.6 Power Diagrams", Algorithms in Combinatorial Geometry, EATCS Monographs on Theoretical Computer Science, 10, Springer-Verlag, pp. 327–328.
  13. S. Arya, T. Malamatos, and D. M. Mount, Space-Efficient Approximate Voronoi Diagrams, Proc. 34th ACM Symp. on Theory of Computing (STOC 2002), pp. 721–730.
  14. Jooyandeh, Mohammadreza; Mohades, Ali; Mirzakhah, Maryam (2009). "Uncertain Voronoi Diagram" (PDF). Information Processing Letters. Elsevier. 109 (13): 709–712. doi:10.1016/j.ipl.2009.03.007.
  15. Geraerts, Roland (2010), Planning Short Paths with Clearance using Explicit Corridors (PDF), International Conference on Robotics and Automation, IEEE, pp. 1997–2004.
  16. van Toll, Wouter G.; Cook IV, Atlas F.; Geraerts, Roland (2011), Navigation Meshes for Realistic Multi-Layered Environments (PDF), International Conference on Intelligent Robots and Systems, IEEE/RSJ, pp. 3526–3532.
  17. Martin Bock (2009). "Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics".
  18. Hui Li (2012). "Spatial Modeling of Bone Microarchitecture".
  19. 1 2 Sanchez-Gutierrez, D.; Tozluoglu, M.; Barry, J. D.; Pascual, A.; Mao, Y.; Escudero, L. M. (2016-01-04). "Fundamental physical cellular constraints drive self-organization of tissues". The EMBO Journal. 35 (1): 77–88. doi:10.15252/embj.201592374. PMC 4718000Freely accessible. PMID 26598531.
  20. Springel, Volker (2010). "E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh". MNRAS. 401 (2): 791–851. doi:10.1111/j.1365-2966.2009.15715.x.
  22. Tom M. Mitchell (1997). Machine Learning (International Edition 1997 ed.). McGraw-Hill. p. 233. ISBN 0-07-042807-7.
  23. "User Interface Algorithms".


  • G. Lejeune Dirichlet (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. 133 (133): 97–178. doi:10.1515/crll.1908.133.97. 
  • Atsuyuki Okabe, Barry Boots, Kokichi Sugihara & Sung Nok Chiu (2000). Spatial Tessellations – Concepts and Applications of Voronoi Diagrams. 2nd edition. John Wiley, 2000, 671 pages, ISBN 0-471-98635-6
  • Franz Aurenhammer, Rolf Klein & Der-Tsai Lee (2013) "Voronoi Diagrams and Delaunay Triangulations". World Scientific, 2013, 337 pages, ISBN 978-9814447638
  • Bowyer, Adrian (1981). "Computing Dirichlet tessellations". Comput. J. 24 (2): 162–166. 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. 
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