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The Delaunay triangulation is constructed from a set of circum-circles. These circum-circles are chosen so that there are at least three of the points in the set to triangulation on the circumference of the circum-circle. None of the points in the set of points falls within any of the circum-circles.

In general there are only three points on the circumference of any circum-circle. However, in some cases, and in particular for the case of a regular grid, 4 or more points can be on a single circum-circle. In this case the Delaunay triangulation is not unique.

- :
`tri`=**delaunay***(*¶`x`,`y`) - :
`tetr`=**delaunay***(*¶`x`,`y`,`z`) - :
`tri`=**delaunay***(*¶`x`) - :
`tri`=**delaunay***(…,*¶`options`) Compute the Delaunay triangulation for a 2-D or 3-D set of points.

For 2-D sets, the return value

`tri`is a set of triangles which satisfies the Delaunay circum-circle criterion, i.e., no data point from [`x`,`y`] is within the circum-circle of the defining triangle. The set of triangles`tri`is a matrix of size [n, 3]. Each row defines a triangle and the three columns are the three vertices of the triangle. The value of

is an index into`tri`(i,j)`x`and`y`for the location of the j-th vertex of the i-th triangle.For 3-D sets, the return value

`tetr`is a set of tetrahedrons which satisfies the Delaunay circum-circle criterion, i.e., no data point from [`x`,`y`,`z`] is within the circum-circle of the defining tetrahedron. The set of tetrahedrons is a matrix of size [n, 4]. Each row defines a tetrahedron and the four columns are the four vertices of the tetrahedron. The value of

is an index into`tetr`(i,j)`x`,`y`,`z`for the location of the j-th vertex of the i-th tetrahedron.The input

`x`may also be a matrix with two or three columns where the first column contains x-data, the second y-data, and the optional third column contains z-data.An optional final argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The default options are

`{"Qt", "Qbb", "Qc"}`

. If Qhull fails for 2-D input the triangulation is attempted again with the options`{"Qt", "Qbb", "Qc", "Qz"}`

which may result in reduced accuracy.If

`options`is not present or`[]`

then the default arguments are used. Otherwise,`options`replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in`options`. Use a null string to pass no arguments.x = rand (1, 10); y = rand (1, 10); tri = delaunay (x, y); triplot (tri, x, y); hold on; plot (x, y, "r*"); axis ([0,1,0,1]);

**See also:**delaunayn, convhull, voronoi, triplot, trimesh, tetramesh, trisurf.

For 3-D inputs `delaunay`

returns a set of tetrahedra that satisfy the
Delaunay circum-circle criteria. Similarly, `delaunayn`

returns the
N-dimensional simplex satisfying the Delaunay circum-circle criteria.
The N-dimensional extension of a triangulation is called a tessellation.

- :
`T`=**delaunayn***(*¶`pts`) - :
`T`=**delaunayn***(*¶`pts`,`options`) Compute the Delaunay triangulation for an N-dimensional set of points.

The Delaunay triangulation is a tessellation of the convex hull of a set of points such that no N-sphere defined by the N-triangles contains any other points from the set.

The input matrix

`pts`of size [n, dim] contains n points in a space of dimension dim. The return matrix`T`has size [m, dim+1]. Each row of`T`contains a set of indices back into the original set of points`pts`which describes a simplex of dimension dim. For example, a 2-D simplex is a triangle and 3-D simplex is a tetrahedron.An optional second argument, which must be a string or cell array of strings, contains options passed to the underlying qhull command. See the documentation for the Qhull library for details http://www.qhull.org/html/qh-quick.htm#options. The default options depend on the dimension of the input:

- 2-D and 3-D:
`options`=`{"Qt", "Qbb", "Qc"}`

- 4-D and higher:
`options`=`{"Qt", "Qbb", "Qc", "Qx"}`

If Qhull fails for 2-D input the triangulation is attempted again with the options

`{"Qt", "Qbb", "Qc", "Qz"}`

which may result in reduced accuracy.If

`options`is not present or`[]`

then the default arguments are used. Otherwise,`options`replaces the default argument list. To append user options to the defaults it is necessary to repeat the default arguments in`options`. Use a null string to pass no arguments.**See also:**delaunay, convhulln, voronoin, trimesh, tetramesh.- 2-D and 3-D:

An example of a Delaunay triangulation of a set of points is

rand ("state", 1); x = rand (1, 10); y = rand (1, 10); T = delaunay (x, y); X = [ x(T(:,1)); x(T(:,2)); x(T(:,3)); x(T(:,1)) ]; Y = [ y(T(:,1)); y(T(:,2)); y(T(:,3)); y(T(:,1)) ]; axis ([0, 1, 0, 1]); plot (X, Y, "b", x, y, "r*");

The result of which can be seen in Figure 30.1.

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