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<h1><a
href="http://www.geom.uiuc.edu/graphics/pix/Special_Topics/Computational_Geometry/delaunay.html"><img
src="qh--dt.gif" alt="[delaunay]" align="middle" width="100"
height="100"></a> Qhull output options</h1>

<p>This section lists the output options for Qhull. These options
are indicated by lower case characters. See <a
href="qh-optf.htm#format">Formats</a>, <a
href="qh-optp.htm#print">Print</a>, and <a
href="qh-optg.htm#geomview">Geomview</a> for other output
options. </p>

<p><b>Copyright &copy; 1995-2015 C.B. Barber</b></p>

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<p><a href="index.htm#TOC">&#187;</a> <a href="qh-quick.htm#programs">Programs</a>
<a name="output">&#149;</a> <a href="qh-quick.htm#options">Options</a>
&#149; <a href="qh-opto.htm#output">Output</a>
&#149; <a href="qh-optf.htm#format">Formats</a>
&#149; <a href="qh-optg.htm#geomview">Geomview</a>
&#149; <a href="qh-optp.htm#print">Print</a>
&#149; <a href="qh-optq.htm#qhull">Qhull</a>
&#149; <a href="qh-optc.htm#prec">Precision</a>
&#149; <a href="qh-optt.htm#trace">Trace</a>
&#149; <a href="../src/libqhull_r/index.htm">Functions</a></p>

<h2>Output options</h2>

<p>Qhull prints its output to standard out. All output is printed
text. The default output is a summary (option '<a href="#s">s</a>').
Other outputs may be specified as follows. </p>

<dl compact>
    <dt><a href="#f">f</a></dt>
    <dd>print all fields of all facets</dd>
    <dt><a href="#n">n</a></dt>
    <dd>print hyperplane normals with offsets</dd>
    <dt><a href="#m">m</a></dt>
    <dd>print Mathematica output (2-d and 3-d)</dd>
    <dt><a href="#o">o</a></dt>
    <dd>print OFF file format (dim, points and facets)</dd>
    <dt><a href="#s">s</a></dt>
    <dd>print summary to stderr</dd>
    <dt><a href="#p">p</a></dt>
    <dd>print vertex and point coordinates</dd>
    <dt><a href="#i">i</a></dt>
    <dd>print vertices incident to each facet </dd>
    <dt>&nbsp;</dt>
    <dt>&nbsp;</dt>
    <dd><b>Related options</b></dd>
    <dt><a href="qh-optf.htm#format">F</a></dt>
    <dd>additional input/output formats</dd>
    <dt><a href="qh-optg.htm#geomview">G</a></dt>
    <dd>Geomview output</dd>
    <dt><a href="qh-optp.htm#print">P</a></dt>
    <dd>Print options</dd>
    <dt><a href="qh-optf.htm#Ft">Ft</a></dt>
    <dd>print triangulation with added points</dd>
    <dt>&nbsp;</dt>
</dl>

<hr>

<h3><a href="#output">&#187;</a><a name="f">f - print all fields of
all facets </a></h3>

<p>Print <a href=../src/libqhull.h#facetT>all fields</a> of all facets.
The facet is the primary <a href=index.htm#structure>data structure</a> for
Qhull.

<p>Option 'f' is for
debugging. Most of the fields are available via the '<a
href="qh-optf.htm#format">F</a>' options. If you need specialized
information from Qhull, you can use the <a
href="qh-code.htm#library">Qhull library</a> or <a
href="qh-code.htm#cpp">C++ interface</a>.</p>

<p>Use the '<a href="qh-optf.htm#FF">FF</a>' option to print the
facets but not the ridges. </p>

<h3><a href="#output">&#187;</a><a name="i">i - print vertices
incident to each facet </a></h3>

<p>The first line is the number of facets. The remaining lines
list the vertices for each facet, one facet per line. The indices
are 0-relative indices of the corresponding input points. The
facets are oriented. Option '<a href="qh-optf.htm#Fv">Fv</a>'
displays an unoriented list of vertices with a vertex count per
line. Options '<a href="qh-opto.htm#o">o</a>' and '<a
href="qh-optf.htm#Ft">Ft</a>' displays coordinates for each
vertex prior to the vertices for each facet. </p>

<p>Simplicial facets (e.g., triangles in 3-d) consist of <i>d</i>
vertices. Non-simplicial facets in 3-d consist of 4 or more
vertices. For example, a facet of a cube consists of 4 vertices.
Use option '<a href="qh-optq.htm#Qt">Qt</a>' to triangulate non-simplicial facets.</p>

<p>For 4-d and higher convex hulls and 3-d and higher Delaunay
triangulations, <i>d</i> vertices are listed for all facets. A
non-simplicial facet is triangulated with its centrum and each
ridge. The index of the centrum is higher than any input point.
Use option '<a href="qh-optf.htm#Fv">Fv</a>' to list the vertices
of non-simplicial facets as is. Use option '<a
href="qh-optf.htm#Ft">Ft</a>' to print the coordinates of the
centrums as well as those of the input points. </p>

<h3><a href="#output">&#187;</a><a name="m">m - print Mathematica
output </a></h3>

<p>Qhull writes a Mathematica file for 2-d and 3-d convex hulls,
2-d and 3-d halfspace intersections,
and 2-d Delaunay triangulations. Qhull produces a list of
objects that you can assign to a variable in Mathematica, for
example: &quot;<tt>list= &lt;&lt; &lt;outputfilename&gt; </tt>&quot;.
If the object is 2-d, it can be visualized by &quot;<tt>Show[Graphics[list]]
</tt>&quot;. For 3-d objects the command is &quot;<tt>Show[Graphics3D[list]]
</tt>&quot;. Now the object can be manipulated by commands of the
form <tt>&quot;Show[%, &lt;parametername&gt; -&gt;
&lt;newvalue&gt;]</tt>&quot;. </p>

<p>For Delaunay triangulation orthogonal projection is better.
This can be specified, for example, by &quot;<tt>BoxRatios:
Show[%, BoxRatios -&gt; {1, 1, 1e-8}]</tt>&quot;. To see the
meaningful side of the 3-d object used to visualize 2-d Delaunay,
you need to change the viewpoint: &quot;<tt>Show[%, ViewPoint
-&gt; {0, 0, -1}]</tt>&quot;. By specifying different viewpoints
you can slowly rotate objects. </p>

<p>For halfspace intersections, Qhull produces the dual
convex hull.

<p>See <a href="qh-faq.htm#math">Is Qhull available for Mathematica?</a>
for URLs.

<h3><a href="#output">&#187;</a><a name="n">n - print hyperplane
normals with offsets </a></h3>

<p>The first line is the dimension plus one. The second line is
the number of facets. The remaining lines are the normals for
each facet, one normal per line. The facet's offset follows its
normal coefficients.</p>

<p>The normals point outward, i.e., the convex hull satisfies <i>Ax
&lt;= -b </i>where <i>A</i> is the matrix of coefficients and <i>b</i>
is the vector of offsets.</p>

<p>A point is <i>inside</i> or <i>below</i> a hyperplane if its distance
to the hyperplane is negative.  A point is <i>outside</i> or <i>above</i> a hyperplane
if its distance to the hyperplane is positive.  Otherwise a point is <i>on</i> or
<i>coplanar to</i> the hyperplane.

<p>If cdd output is specified ('<a href="qh-optf.htm#FD">FD</a>'),
Qhull prints the command line, the keyword &quot;begin&quot;, the
number of facets, the dimension (plus one), the keyword
&quot;real&quot;, and the normals for each facet. The facet's
negative offset precedes its normal coefficients (i.e., if the
origin is an interior point, the offset is positive). Qhull ends
the output with the keyword &quot;end&quot;. </p>

<h3><a href="#output">&#187;</a><a name="o">o - print OFF file format
</a></h3>

<p>The output is: </p>

<ul>
    <li>The first line is the dimension </li>
    <li>The second line is the number of points, the number of
        facets, and the number of ridges. </li>
    <li>All of the input points follow, one per line. </li>
    <li>Then Qhull prints the vertices for each facet. Each facet
        is on a separate line. The first number is the number of
        vertices. The remainder is the indices of the
        corresponding points. The vertices are oriented in 2-d,
        3-d, and in simplicial facets. </li>
</ul>

<p>Option '<a href="qh-optf.htm#Ft">Ft</a>' prints the same
information with added points for non-simplicial facets.</p>

<p>Option '<a href="qh-opto.htm#i">i</a>' displays vertices
without the point coordinates. Option '<a href="qh-opto.htm#p">p</a>'
displays the point coordinates without vertex and facet information.</p>

<p>In 3-d, Geomview can load the file directly if you delete the
first line (e.g., by piping through '<tt>tail +2</tt>').</p>

<p>For Voronoi diagrams (<a href=qvoronoi.htm>qvoronoi</a>), option
'o' prints Voronoi vertices and Voronoi regions instead of input
points and facets. The first vertex is the infinity vertex
[-10.101, -10.101, ...]. Then, option 'o' lists the vertices in
the Voronoi region for each input site. The regions appear in
site ID order. In 2-d, the vertices of a Voronoi region are
sorted by adjacency (non-oriented). In 3-d and higher, the
Voronoi vertices are sorted by index. See the '<a
href="qh-optf.htm#FN">FN</a>' option for listing Voronoi regions
without listing Voronoi vertices.</p>

<p>If you are using the Qhull library, options 'v o' have the
side effect of reordering the neighbors for a vertex.</p>

<h3><a href="#output">&#187;</a><a name="p">p - print vertex and
point coordinates </a></h3>

<p>The first line is the dimension. The second line is the number
of vertices. The remaining lines are the vertices, one vertex per
line. A vertex consists of its point coordinates</p>

<p>With the '<a href="qh-optg.htm#Gc">Gc</a>' and '<a
href="qh-optg.htm#Gi">Gi</a>' options, option 'p' also prints
coplanar and interior points respectively.</p>

<p>For <a href=qvoronoi.htm>qvoronoi</a>, it prints the
coordinates of each Voronoi vertex.</p>

<p>For <a href=qdelaun.htm>qdelaunay</a>, it prints the
input sites as lifted to a paraboloid.  For <a href=qhalf.htm>qhalf</a>
it prints the dual points.  For both, option 'p' is the same as the first
section of option '<a href="qh-opto.htm#o">o</a>'.</p>

<p>Use '<a href="qh-optf.htm#Fx">Fx</a>' to list the point ids of
the extreme points (i.e., vertices). </p>

<p>If a subset of the facets is selected ('<a
href="qh-optp.htm#Pdk">Pdk</a>', '<a href="qh-optp.htm#PDk">PDk</a>',
'<a href="qh-optp.htm#Pg">Pg</a>' options), option 'p' only
prints vertices and points associated with those facets.</p>

<p>If cdd-output format is selected ('<a href="qh-optf.htm#FD">FD</a>'),
the first line is &quot;begin&quot;. The second line is the
number of vertices, the dimension plus one, and &quot;real&quot;.
The vertices follow with a leading &quot;1&quot;. Output ends
with &quot;end&quot;. </p>

<h3><a href="#output">&#187;</a><a name="s">s - print summary to
stderr </a></h3>

<p>The default output of Qhull is a summary to stderr. Options '<a
href="qh-optf.htm#FS">FS</a>' and '<a href="qh-optf.htm#Fs">Fs</a>'
produce the same information for programs.  <b>Note</b>: Windows 95 and 98
treats stderr the same as stdout.  Use option '<a href="qh-optt.htm#TO">TO file</a>' to separate
stderr and stdout.</p>

<p>The summary lists the number of input points, the dimension,
the number of vertices in the convex hull, and the number of
facets in the convex hull. It lists the number of selected
(&quot;good&quot;) facets for options '<a href="qh-optp.htm#Pg">Pg</a>',
'<a href="qh-optp.htm#Pdk">Pdk</a>', <a href=qdelaun.htm>qdelaunay</a>,
or <a href=qvoronoi.htm>qvoronoi</a> (Delaunay triangulations only
use the lower half of a convex hull). It lists the number of
coplanar points. For Delaunay triangulations without '<a
href="qh-optq.htm#Qc">Qc</a>', it lists the total number of
coplanar points. It lists the number of simplicial facets in
the output.</p>

<p>The terminology depends on the output structure. </p>

<p>The summary lists these statistics:</p>

<ul>
    <li>number of points processed by Qhull </li>
    <li>number of hyperplanes created</li>
    <li>number of distance tests (not counting statistics,
        summary, and checking) </li>
    <li>number of merged facets (if any)</li>
    <li>number of distance tests for merging (if any)</li>
    <li>CPU seconds to compute the hull</li>
    <li>the maximum joggle for '<a href="qh-optq.htm#QJn">QJ</a>'<br>
        or, the probability of precision errors for '<a
        href="qh-optq.htm#QJn">QJ</a> <a href="qh-optt.htm#TRn">TRn</a>'
    </li>
    <li>total area and volume (if computed, see '<a
        href="qh-optf.htm#FS">FS</a>' '<a href="qh-optf.htm#FA">FA</a>'
        '<a href="qh-optf.htm#Fa">Fa</a>' '<a
        href="qh-optp.htm#PAn">PAn</a>')</li>
    <li>max. distance of a point above a facet (if non-zero)</li>
    <li>max. distance of a vertex below a facet (if non-zero)</li>
</ul>

<p>The statistics include intermediate hulls. For example 'rbox d
D4 | qhull' reports merged facets even though the final hull is
simplicial. </p>

<p>Qhull starts counting CPU seconds after it has read and
projected the input points. It stops counting before producing
output. In the code, CPU seconds measures the execution time of
function qhull() in <tt>libqhull.c</tt>. If the number of CPU
seconds is clearly wrong, check qh_SECticks in <tt>user.h</tt>. </p>

<p>The last two figures measure the maximum distance from a point
or vertex to a facet. They are not printed if less than roundoff
or if not merging. They account for roundoff error in computing
the distance (c.f., option '<a href="qh-optc.htm#Rn">Rn</a>').
Use '<a href="qh-optf.htm#Fs">Fs</a>' to report the maximum outer
and inner plane. </p>

<p>A number may appear in parentheses after the maximum distance
(e.g., 2.1x). It is the ratio between the maximum distance and
the worst-case distance due to merging two simplicial facets. It
should be small for 2-d, 3-d, and 4-d, and for higher dimensions
with '<a href="qh-optq.htm#Qx">Qx</a>'. It is not printed if less
than 0.05. </p>
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