Voronoi diagram

This is the Voronoi diagram of a random set of points in the plane (all points lie within the image).

In mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation (after Lejeune Dirichlet), is special kind of decomposition of a metric space determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points.

Definition

For any (topologically) discrete set S of points in Euclidean space and for almost any point x, there is one point of S closest to x. The word "almost" is occasioned by the fact that a point x may be equally close to two or more points of S.

If S contains only two points, a and b, then the set of all points equidistant from a and b is a hyperplane — an affine subspace of codimension 1. That hyperplane is the boundary between the set of all points closer to a than to b, and the set of all points closer to b than to a. It is the perpendicular bisector of the line segment from a and b.

In general, the set of all points closer to a point c of S than to any other point of S is the interior of a (in some cases unbounded) convex polytope called the Dirichlet domain or Voronoi cell for c. The set of such polytopes tessellates the whole space, and is the Voronoi tessellation corresponding to the set S. If the dimension of the space is only 2, then it is easy to draw pictures of Voronoi tessellations, and in that case they are sometimes called Voronoi diagrams.

Properties

• The dual graph for a Voronoi diagram corresponds to the Delaunay triangulation for the same set of points S.

• The closest pair of points corresponds to two adjacent cells in the Voronoi diagram.

• There are 4 cocircular points if and only if the Voronoi diagram has a vertex of [[Degree (graph theory)|degree] 4.

• The Voronoi diagram of a set of points in d-dimensional spaces corresponds to the convex hull of the projection of the points onto a (d+1)-dimensional paraboloid.

History

Informal use of Voronoi diagrams can be traced back to Descartes in 1644. 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 Soho cholera epidemic lived closer to the infected Broad Street pump than to any other water pump.

Voronoi diagrams are named after Russian mathematician Georgy Fedoseevich Voronoi (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.

Examples

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 tesselation is not a 2D Voronoi tesselation 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.

• 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.
• A pair of planes with triangular lattices aligned with each others' centres gives the arrangement of rhombus-capped hexagonal prisms seen in honeycomb
• 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

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.

Centroidal Voronoi tessellations

A Voronoi tessellation is called centroidal when the generating point of each cell is also its mean (center of mass).

Generalizations

Voronoi cells can be defined for metrics other than Euclidean. However in these cases the Voronoi tessellation is not guaranteed to exist (or to be a "true" tessellation), since the equidistant locus for two points may fail to be subspace of codimension 1, even in the 2-dimensional case.

Voronoi cells can also be defined by measuring distances to objects that are not points. The Voronoi diagram with these cells is also called the medial axis. Even when the objects are line segments, the Voronoi cells are not bounded by straight lines. The medial axis is used in image segmentation, optical character recognition and other computational applications. In materials science, polycrystalline microstructures in metallic alloys are commonly represented using Voronoi tessellations. A simplified version of the Voronoi diagram of line segments is the straight skeleton.

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

The Voronoi diagram of n points in d-dimensional space uses $O(n^{\lceil d/2\rceil })$ 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.

See also

• Delaunay triangulation
• Computational geometry
• Medial axis

References

• Gustav 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.
• Georgy Voronoi (1907). Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Journal für die Reine und Angewandte Mathematik, 133:97-178, 1907
• 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 (1991). Voronoi Diagrams - A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, 23(3):345-405, 1991.

External links

• Applet for calculation and visualization of convex hull, Delaunay triangulations and Voronoi diagrams in space
• Demo for various metrics
• Mathworld on Voronoi diagrams
• Parameterized and programmed architectural object using the Voronoi Diagram
• Qhull for computing the Voronoi diagram in 2-d, 3-d, etc.
• Voronoi Diagrams: Applications from Archaeology to Zoology
• Voronoi Diagrams in CGAL, the Computational Geometry Algorithms Library
• Voronoi Web Site : using Voronoi diagrams for spatial analysis