delaunay3 (MATLAB Functions)

3-dimensional Delaunay tessellation

Syntax

T = delaunay3(x,y,z)
T = delaunay3(x,y,z,options)

Description

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 http://www.qhull.org.

Visualization

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.

Examples

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.

d = [-1 1];
[x,y,z] = meshgrid(d,d,d);  % A cube
x = [x(:);0];
y = [y(:);0];
z = [z(:);0];
% [x,y,z] are corners of a cube plus the center.
Tes = delaunay3(x,y,z)

Tes =

     9   1   5   6
     3   9   1   5
     2   9   1   6
     2   3   9   4
     2   3   9   1
     7   9   5   6
     7   3   9   5
     8   7   9   6
     8   2   9   6
     8   2   9   4
     8   3   9   4
     8   7   3   9

X = [x(:) y(:) z(:)];
tetramesh(Tes,X);camorbit(20,0)

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

X = [-0.5 -0.5 -0.5 -0.5 0.5 0.5 0.5 0.5];
Y = [-0.5 -0.5 0.5 0.5 -0.5 -0.5 0.5 0.5];
Z = [-0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5];

The command

```
T = delaunay3(X);
```

returns the following error message.

??? qhull input error: can not scale last coordinate. Input is 
cocircular
   or cospherical. Use option 'Qz' to add a point at infinity.

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

T = delaunay3( X, Y, Z, {'Qt', 'Qbb', 'Qc', 'Qz'} )

T =

     4     3     5     1
     4     2     5     1
     4     7     3     5
     4     7     8     5
     4     6     2     5
     4     6     8     5

Algorithm

delaunay3 is based on Qhull [2]. For information about Qhull, see http://www.qhull.org/. For copyright information, see http://www.qhull.org/COPYING.txt.

See Also

delaunay, delaunayn

Reference

[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 http://www.acm.org/ 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.

delaunay delaunayn