MATLAB Function Reference |

**Syntax**

**Description**

```
TES = delaunay3(x,y,z)
```

returns an array `TES`

, 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)`

. `TES`

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.

**Visualization**

Use `tetramesh`

to plot `delaunay3`

output. `tetramesh`

displays the tetrahedrons defined in `TES`

as mesh. `tetramesh`

uses the default tranparency parameter value `'FaceAlpha' = 0.9`

.

**Example**

This example generates a 3-D 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)

**Algorithm**

`delaunay3`

is based on Qhull [2]. It uses the Qhull joggle option (`'QJ'`

). For information about `qhull`

, see http://www.geom.umn.edu/software/qhull/. For copyright information, see http://www.geom.umn.edu/software/download/COPYING.html.

**See Also**

**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/ and in PostScript
format at ftp://geom.umn.edu/pub/software/qhull-96.ps.

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

delaunay | delaunayn |