Geometry in Nature

Voronoi diagram

overview

In mathematics, a Voronoi diagram is a special kind of decomposition of a metric space[1] determined by distances to a specified discrete set of objects in the space, e.g., by a discrete set of points. It is named after Georgy Voronoi, also called a Voronoi tessellation, a Voronoi decomposition, or a Dirichlet tessellation (after Lejeune Dirichlet),

In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell, also called a Dirichlet cell, V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites.


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















Voronoi diagram

Definition:

Let S be a set of points in Euclidean space [2] with no limit points not in S. For almost any point x in the Euclidean space, there is one point of S closest to x. The word "almost" is used to indicate exceptions where 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 [3] 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.



















[3] Tessellation : A tessellated plane seen in street brickwork

Properties :

* The dual graph [4] for a Voronoi diagram corresponds to the Delaunay triangulation [5] for the same set of points S.
* The closest pair of points [6] corresponds to two adjacent cells in the Voronoi diagram.
* Two points are adjacent on the convex hull [7] if and only if their Voronoi cells share an infinitely long side.














[4]:Dual graph: G′ is the dual graph of G





















[5]: Delaunay triangulation : A Delaunay triangulation in the plane with circumcircles shown


















[6]: Closest pair of points shown in red














[7]: Convex hull: elastic band analogy


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 [8].















[8]: Fundamental parallelogram defined by a pair of vectors, generates the torus.


















[8]: Torus






















[8]: Real projective plane




















[8]:Klein bottle




















[8]:Sphere

Examples in geometry :

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' centers gives the arrangement of rhombus-capped hexagonal prisms seen in honeycomb
* A face-centred cubic [9] lattice gives a tessellation of space with rhombic dodecahedra[10]
* A body-centred cubic[9] lattice gives a tessellation of space with truncated octahedra[11]

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.

















[9]: Simple cubic



















[9]: Body-centered cubic



















[9]: Face-centered cubic

















[10 ]: Part of a Rhombic dodecahedral honeycomb
























[10 ]: Rhombic dodecahedron






















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. [12])


[12 ]: some polyhedra :




















[12 ]: Dodecahedron (Regular polyhedron)





















[12 ]:Small stellated dodecahedron (Regular star)




















[12]:Icosidodecahedron (Uniform)




















[12]Great cubicuboctahedron (Uniform star)


Generalization :

Voronoi cells can be defined for metrics other than Euclidean (such as the Mahalanobis or Manhattan) distances. 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 [13]. 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 [14].















[13 ]: A ellipse (red), its evolute (blue), and its medial axis (green). The symmetry set, a super-set of the medial axis is the green and yellow curves. One bi-tangent circle is shown.





















[14 ]: The straight skeleton, the shrinking process that creates it, and a three-dimensional roofline created from it.

source: www.wikipedia.org
www.emergentarchitecture.com