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<h3 class="section" id="Convex-Hull-1"><span>30.3 Convex Hull<a class="copiable-link" href="#Convex-Hull-1"> ¶</a></span></h3>
<p>The convex hull of a set of points is the minimum convex envelope
containing all of the points. Octave has the functions <code class="code">convhull</code>
and <code class="code">convhulln</code> to calculate the convex hull of 2-dimensional and
N-dimensional sets of points.
</p>
<a class="anchor" id="XREFconvhull"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-convhull"><span><code class="def-type"><var class="var">H</var> =</code> <strong class="def-name">convhull</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>)</code><a class="copiable-link" href="#index-convhull"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhull-1"><span><code class="def-type"><var class="var">H</var> =</code> <strong class="def-name">convhull</strong> <code class="def-code-arguments">(<var class="var">x</var>, <var class="var">y</var>, <var class="var">z</var>)</code><a class="copiable-link" href="#index-convhull-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhull-2"><span><code class="def-type"><var class="var">H</var> =</code> <strong class="def-name">convhull</strong> <code class="def-code-arguments">(<var class="var">x</var>)</code><a class="copiable-link" href="#index-convhull-2"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhull-3"><span><code class="def-type"><var class="var">H</var> =</code> <strong class="def-name">convhull</strong> <code class="def-code-arguments">(…, <var class="var">options</var>)</code><a class="copiable-link" href="#index-convhull-3"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhull-4"><span><code class="def-type">[<var class="var">H</var>, <var class="var">V</var>] =</code> <strong class="def-name">convhull</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-convhull-4"> ¶</a></span></dt>
<dd><p>Compute the convex hull of a 2-D or 3-D set of points.
</p>
<p>The hull <var class="var">H</var> is a linear index vector into the original set of points
that specifies which points form the enclosing hull. For 2-D inputs only,
the output is ordered in a counterclockwise manner around the hull.
</p>
<p>The input <var class="var">x</var> 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.
</p>
<p>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
<a class="url" href="http://www.qhull.org/html/qh-quick.htm#options">http://www.qhull.org/html/qh-quick.htm#options</a>.
The default option is <code class="code">{"Qt"}</code>.
</p>
<p>If <var class="var">options</var> is not present or <code class="code">[]</code> then the default arguments are
used. Otherwise, <var class="var">options</var> replaces the default argument list.
To append user options to the defaults it is necessary to repeat the
default arguments in <var class="var">options</var>. Use a null string to pass no arguments.
</p>
<p>If the second output <var class="var">V</var> is requested the volume of the enclosing
convex hull is calculated.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFconvhulln">convhulln</a>, <a class="ref" href="Delaunay-Triangulation.html#XREFdelaunay">delaunay</a>, <a class="ref" href="Voronoi-Diagrams.html#XREFvoronoi">voronoi</a>.
</p></dd></dl>
<a class="anchor" id="XREFconvhulln"></a><span style="display:block; margin-top:-4.5ex;"> </span>
<dl class="first-deftypefn">
<dt class="deftypefn" id="index-convhulln"><span><code class="def-type"><var class="var">h</var> =</code> <strong class="def-name">convhulln</strong> <code class="def-code-arguments">(<var class="var">pts</var>)</code><a class="copiable-link" href="#index-convhulln"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhulln-1"><span><code class="def-type"><var class="var">h</var> =</code> <strong class="def-name">convhulln</strong> <code class="def-code-arguments">(<var class="var">pts</var>, <var class="var">options</var>)</code><a class="copiable-link" href="#index-convhulln-1"> ¶</a></span></dt>
<dt class="deftypefnx def-cmd-deftypefn" id="index-convhulln-2"><span><code class="def-type">[<var class="var">h</var>, <var class="var">v</var>] =</code> <strong class="def-name">convhulln</strong> <code class="def-code-arguments">(…)</code><a class="copiable-link" href="#index-convhulln-2"> ¶</a></span></dt>
<dd><p>Compute the convex hull of the set of points <var class="var">pts</var>.
</p>
<p><var class="var">pts</var> is a matrix of size [n, dim] containing n points in a space of
dimension dim.
</p>
<p>The hull <var class="var">h</var> is an index vector into the set of points and specifies
which points form the enclosing hull.
</p>
<p>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
<a class="url" href="http://www.qhull.org/html/qh-quick.htm#options">http://www.qhull.org/html/qh-quick.htm#options</a>.
The default options depend on the dimension of the input:
</p>
<ul class="itemize mark-bullet">
<li>2D, 3D, 4D: <var class="var">options</var> = <code class="code">{"Qt"}</code>
</li><li>5D and higher: <var class="var">options</var> = <code class="code">{"Qt", "Qx"}</code>
</li></ul>
<p>If <var class="var">options</var> is not present or <code class="code">[]</code> then the default arguments are
used. Otherwise, <var class="var">options</var> replaces the default argument list.
To append user options to the defaults it is necessary to repeat the
default arguments in <var class="var">options</var>. Use a null string to pass no arguments.
</p>
<p>If the second output <var class="var">v</var> is requested the volume of the enclosing
convex hull is calculated.
</p>
<p><strong class="strong">See also:</strong> <a class="ref" href="#XREFconvhull">convhull</a>, <a class="ref" href="Delaunay-Triangulation.html#XREFdelaunayn">delaunayn</a>, <a class="ref" href="Voronoi-Diagrams.html#XREFvoronoin">voronoin</a>.
</p></dd></dl>
<p>An example of the use of <code class="code">convhull</code> is
</p>
<div class="example">
<div class="group"><pre class="example-preformatted">x = -3:0.05:3;
y = abs (sin (x));
k = convhull (x, y);
plot (x(k), y(k), "r-", x, y, "b+");
axis ([-3.05, 3.05, -0.05, 1.05]);
</pre></div></div>
<p>The output of the above can be seen in <a class="ref" href="#fig_003aconvhull">Figure 30.5</a>.
</p>
<div class="float" id="fig_003aconvhull">
<div class="center"><img class="image" src="convhull.png" alt="convhull">
</div><div class="caption"><p><strong class="strong">Figure 30.5: </strong>The convex hull of a simple set of points</p></div></div>
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