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voronoin

N-D Voronoi diagram

Syntax

[V,C] = voronoin(X)
[V,C] = voronoin(X,options)

Description

[V,C] = voronoin(X) returns Voronoi vertices V and the Voronoi cells C of the Voronoi diagram of X. V is a numv-by-n array of the numv Voronoi vertices in n-dimensional space, each row corresponds to a Voronoi vertex. C is a vector cell array where each element contains the indices into V of the vertices of the corresponding Voronoi cell. X is an m-by-n array, representing m n-dimensional points, where n > 1 and m >= n+1.

The first row of V is a point at infinity. If any index in a cell of the cell array is 1, then the corresponding Voronoi cell contains the first point in V, a point at infinity. This means the Voronoi cell is unbounded.

voronoin uses Qhull.

[V,C] = voronoin(X,options) specifies a cell array of Qhull options. The default options are:

  • {'Qbb'} for 2- and 3-dimensional input

  • {'Qbb','Qx'} for 4 and higher-dimensional input

If options is [], the default options are used. If code is {''}, no options are used, not even the default. For more information on Qhull and its options, see http://www.qhull.org.

Visualization

You can plot individual bounded cells of an n-dimensional Voronoi diagram. To do this, use convhulln to compute the vertices of the facets that make up the Voronoi cell. Then use patch and other plot functions to generate the figure.

Examples

collapse all

Compute Voronoi vertices and diagram cells.

Define a 2-D array of points and compute the vertices and diagram cells.

X = [0.5 0; 0 0.5; -0.5 -0.5; -0.2 -0.1; -0.1 0.1; 0.1 -0.1; 0.1 0.1]
X = 

    0.5000         0
         0    0.5000
   -0.5000   -0.5000
   -0.2000   -0.1000
   -0.1000    0.1000
    0.1000   -0.1000
    0.1000    0.1000

[V,C] = voronoin(X)
V = 

       Inf       Inf
    0.7000   -1.6500
   -0.0500   -0.0500
   -0.0500   -0.5250
   -1.4500    0.6500
   -1.7500    0.7500
         0    0.2875
    0.3833    0.3833
    0.2875         0
         0         0

C = 7x1 cell array
    {1x4 double}
    {1x5 double}
    {1x4 double}
    {1x4 double}
    {1x4 double}
    {1x5 double}
    {1x4 double}

Use a for loop to display the contents of the cell array C.

for i = 1:length(C)
    disp(C{i});
end
     9     2     1     8

     8     1     6     5     7

     6     1     2     4

     6     4     3     5

    10     3     5     7

    10     3     4     2     9

    10     7     8     9

Compute the Voronoi vertices and diagram cells of a 2-D set of points by specifying the convex hull parameters. The first row of C contains a point at infinity.

X = [-1 -1; 1 -1; 1 1; -1 1];
[V,C] = voronoin(X,{'Qbb','Qz'})
V = 

   Inf   Inf
     0     0

C = 4x1 cell array
    {1x2 double}
    {1x2 double}
    {1x2 double}
    {1x2 double}

Algorithms

voronoin is based on Qhull [1]. For information about Qhull, see http://www.qhull.org/.

References

[1] Barber, C. B., D.P. Dobkin, and H.T. Huhdanpaa, “The Quickhull Algorithm for Convex Hulls,” ACM Transactions on Mathematical Software, Vol. 22, No. 4, Dec. 1996, p. 469-483.

Introduced before R2006a

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