Diagram of Voronoï

In Mathematical, a diagram of Voronoï (also called decomposition of Voronoï or partition of Voronoï of the name of the Russian Mathématicien Georgi Fedoseevich Voronoï (1868 - 1908) is a particular decomposition of a metric Espace determined by the Distance S with a discrete Ensemble of objects of space, in general a discrete set of points.

Definition

One places in a Euclidean Espace E . that is to say S a finished whole of N points of E ; the elements of S are called centers, sites or germs.

One calls area of Voronoï or cell of Voronoï associated with an element p with S the unit with the points which are closer to p than of any other point of S .

$Vor_s (p) = \ {X \ in E \/\ \ forall Q \ in S \ D (X, p) \ D (X, Q) \}$

For two points has and B of S , the unit $\ pi (has, b)$ equidistant points of has and B is a Hyperplan Affine (a subspace closely connected of Co-dimension 1). This hyperplane is the border between the whole of the points closer to has than B , and the whole of the points closer to B than of has .

$\ pi (p, Q) = \ {X \ in E \/\ D (X, p) = D (X, Q) \}$

One notes H (has, b) the half spaces delimited by this hyperplane containing has , it then contains all the points closer to has than B . The area of Voronoï associated with has is then the intersection of the H (has, b) where B traverses S \ {has} .

$H (p, Q) = \ {X \ in E \/\ D (X, p) \ D (X, Q) \}$

$Vor_s (p) = \ bigcap_ {Q \ in S \ backslash \ {p \}} H (p, Q)$

The areas of Voronoï are Polytope S convex S as an intersection of half spaces. The whole of such polygons partitionne E, and is the partition of Voronoï corresponding to the unit S . In dimension 2 it is easy to trace these partitions, one calls them in this case sometimes diagrams of Voronoi. One bases oneself on the fact that the border between the cells of Voronoi of two distinct germs is located inevitably on the mediator which separates these two germs. Indeed, the points of this mediator are equidistant of the two germs thus one cannot affirm that it are located in one or the other cell of Voronoi. For a whole of germs, the diagram of Voronoi is thus built by determining mediating each couple of germs. A point of a mediator belongs then to a border of Voronoi if it is equidistant of at least two germs and that there does not exist weaker distance between this point and another germ of the unit.

The diagram of Voronoï is the dual of the Triangulation of Delaunay, one can define the triangulation of Delaunay starting from the diagram of Voronoï, two points p and Q create an edge in the graph of Delaunay if and only if the areas of Voronoï associated to p and Q are adjacent.

$DEL (S) = \ {(p, Q) \ in S^2 \/\ Vor_s (p) \ course Vor_s (Q) \ \ empty \}$

History

The abstract use of the diagrams of Voronoï goes up with Descartes in 1644. Dirichlet used diagrams of Voronoï in dimension 2 or 3 in its study of the quadratic forms in 1850.

The British physicist John Snow used a diagram of Voronoï in 1854 to show that the majority of the people died in the epidemic of Choléra of Soho lived more close to the infected pump of Broad Street than to any other pump.

The diagrams of Voronoï bear the name of the Russian mathematician Georgy Fedoseevich Voronoï (or Voronoy) who defined and studied the general case in dimension N in 1908. The diagrams of Voronoi which are used in Géophysique and in Météorologie to analyze data of spatial distributions (as measurements of falls of rain) are called polygons of Thiessen name of the American meteorologist Alfred H. Thiessen.

Example

The following example takes again the same points as the example of the Triangulation of Delaunay:

Algorithms

The algorithm of Steven Fortune (1987, Laboratories Beautiful AT&T), shown like optimal, makes it possible to calculate the diagram of Voronoï of a whole of N points in time

O (N \ log (N))

Applications

The diagrams of Voronoï are used, or reinvented under many names, in various fields. They often intervene when one seeks with partitionner space in Spheres of influence:

Rebuilding of optimal geographical data, for a flight simulator for example.
Effect Mosaic in a Software of Final improvement of image.
Construction of a dual geodetic dome
Partition of the space structures of the cancerous populations of star S.
Diagnosis of cell S .
Modeling of microstructures such as some Steel S.
Simulation of the circulation of the fluids in the porous environments.
Calculations of trajectory in mobile robotics.
…

References

Steven Fortune, " In Sweepline Algorithm for Voronoi Diagrams" , Algorithmica, volume 2,1987, pp. 153-174

External bonds

Mr. Faidney, C. Poultney & D. Shasha, ''' Play with the diagrams of Voronoï ''', article accompanied by an applet of the play on Interstices

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