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<h3 class="section">29.1 Delaunay Triangulation</h3>
<p>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.
<p>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.
<!-- ./geometry/delaunay.m -->
<p><a name="doc_002ddelaunay"></a>
<div class="defun">
— Function File: <var>tri</var> = <b>delaunay</b> (<var>x, y</var>)<var><a name="index-delaunay-2084"></a></var><br>
— Function File: <var>tri</var> = <b>delaunay</b> (<var>x, y, opt</var>)<var><a name="index-delaunay-2085"></a></var><br>
<blockquote><p>The return matrix of size [n, 3] contains a set triangles which are
described by the indices to the data point x and y vector.
The triangulation satisfies the Delaunay circum-circle criterion.
No other data point is in the circum-circle of the defining triangle.
<p>A third optional argument, which must be a string, contains extra options
passed to the underlying qhull command. See the documentation for the
Qhull library for details.
<pre class="example"> x = rand (1, 10);
y = rand (size (x));
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*");
</pre>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dvoronoi.html#doc_002dvoronoi">voronoi</a>, <a href="doc_002ddelaunay3.html#doc_002ddelaunay3">delaunay3</a>, <a href="doc_002ddelaunayn.html#doc_002ddelaunayn">delaunayn</a>.
</p></blockquote></div>
<p>The 3- and N-dimensional extension of the Delaunay triangulation are
given by <code>delaunay3</code> and <code>delaunayn</code> respectively.
<code>delaunay3</code> returns a set of tetrahedra that satisfy the
Delaunay circum-circle criteria. Similarly, <code>delaunayn</code> returns the
N-dimensional simplex satisfying the Delaunay circum-circle criteria.
The N-dimensional extension of a triangulation is called a tessellation.
<!-- ./geometry/delaunay3.m -->
<p><a name="doc_002ddelaunay3"></a>
<div class="defun">
— Function File: <var>T</var> = <b>delaunay3</b> (<var>x, y, z</var>)<var><a name="index-delaunay3-2086"></a></var><br>
— Function File: <var>T</var> = <b>delaunay3</b> (<var>x, y, z, opt</var>)<var><a name="index-delaunay3-2087"></a></var><br>
<blockquote><p>A matrix of size [n, 4] is returned. Each row contains a
set of tetrahedron which are
described by the indices to the data point vectors (x,y,z).
<p>A fourth optional argument, which must be a string or cell array of strings,
contains extra options passed to the underlying qhull command. See the
documentation for the Qhull library for details.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002ddelaunay.html#doc_002ddelaunay">delaunay</a>, <a href="doc_002ddelaunayn.html#doc_002ddelaunayn">delaunayn</a>.
</p></blockquote></div>
<!-- ./geometry/delaunayn.m -->
<p><a name="doc_002ddelaunayn"></a>
<div class="defun">
— Function File: <var>T</var> = <b>delaunayn</b> (<var>P</var>)<var><a name="index-delaunayn-2088"></a></var><br>
— Function File: <var>T</var> = <b>delaunayn</b> (<var>P, opt</var>)<var><a name="index-delaunayn-2089"></a></var><br>
<blockquote><p>Form the Delaunay triangulation for a set of points.
The Delaunay triangulation is a tessellation of the convex hull of the
points such that no n-sphere defined by the n-triangles contains
any other points from the set.
The input matrix <var>P</var> of size <code>[n, dim]</code> contains <var>n</var>
points in a space of dimension dim. The return matrix <var>T</var> has the
size <code>[m, dim+1]</code>. It contains for each row a set of indices to
the points, which describes a simplex of dimension dim. For example,
a 2d simplex is a triangle and 3d simplex is a tetrahedron.
<p>Extra options for the underlying Qhull command can be specified by the
second argument. This argument is a cell array of strings. The default
options depend on the dimension of the input:
<ul>
<li>2D and 3D: <var>opt</var> = <code>{"Qt", "Qbb", "Qc"}</code>
<li>4D and higher: <var>opt</var> = <code>{"Qt", "Qbb", "Qc", "Qz"}</code>
</ul>
<p>If <var>opt</var> is [], then the default arguments are used. If <var>opt</var>
is <code>{"<!-- /@w -->"}</code>, then none of the default arguments are used by Qhull.
See the Qhull documentation for the available options.
<p>All options can also be specified as single string, for example
<code>"Qt Qbb Qc Qz"</code>.
</blockquote></div>
<p>An example of a Delaunay triangulation of a set of points is
<pre class="example"> rand ("state", 2);
x = rand (10, 1);
y = rand (10, 1);
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*");
</pre>
<ul class="menu">
<li><a accesskey="1" href="Plotting-the-Triangulation.html#Plotting-the-Triangulation">Plotting the Triangulation</a>
<li><a accesskey="2" href="Identifying-points-in-Triangulation.html#Identifying-points-in-Triangulation">Identifying points in Triangulation</a>
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