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3-dimensional Delaunay tessellation



T = delaunay3(x,y,z) returns an array T, each row of which contains the indices of the points in (x,y,z) that make up a tetrahedron in the tessellation of (x,y,z). T is a numtes-by-4 array where numtes is the number of facets in the tessellation. x, y, and z are vectors of equal length. If the original data points are collinear or x, y, and z define an insufficient number of points, the triangles cannot be computed and delaunay3 returns an empty matrix.

delaunay3 uses Qhull.

T = delaunay3(x,y,z,options) specifies a cell array of strings options to be used in Qhull via delaunay3. The default options are {'Qt','Qbb','Qc'}.

If options is [], the default options are used. If options is {''}, no options are used, not even the default. For more information on Qhull and its options, see


Use tetramesh to plot delaunay3 output. tetramesh displays the tetrahedrons defined in T as mesh. tetramesh uses the default tranparency parameter value 'FaceAlpha' = 0.9.


Example 1. This example generates a 3-dimensional Delaunay tessellation, then uses tetramesh to plot the tetrahedrons that form the corresponding simplex. camorbit rotates the camera position to provide a meaningful view of the figure.

Example 2. The following example illustrates the options input for delaunay3.

The command

returns the following error message.

The error message indicates that you should add 'Qz' to the default Qhull options.


delaunay3 is based on Qhull [2]. For information about Qhull, see For copyright information, see

See Also

delaunay, delaunayn


[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. Available in HTML format at pubs/citations/journals/toms/1996-22-4/p469-barber/.

[2]  National Science and Technology Research Center for Computation and Visualization of Geometric Structures (The Geometry Center), University of Minnesota. 1993.

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