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<h2>poly.c, poly2.c -- polyhedron operations</h2>
<blockquote>
<p>Qhull uses dimension-free terminology. Qhull builds a
polyhedron in dimension <em>d. </em>A <em>polyhedron</em> is a
simplicial complex of faces with geometric information for the
top and bottom-level faces. A (<em>d-1</em>)-face is a <em>facet</em>,
a (<em>d-2</em>)-face is a <em>ridge</em>, and a <em>0</em>-face
is a <em>vertex</em>. For example in 3-d, a facet is a polygon
and a ridge is an edge. A facet is built from a ridge (the <em>base</em>)
and a vertex (the <em>apex</em>). See
<a href="../html/index.htm#structure">Qhull's data structures</a>.</p>
<p>Qhull's primary data structure is a polyhedron. A
polyhedron is a list of facets. Each facet has a set of
neighboring facets and a set of vertices. Each facet has a
hyperplane. For example, a tetrahedron has four facets.
If its vertices are <em>a, b, c, d</em>, and its facets
are <em>1, 2, 3, 4,</em> the tetrahedron is </p>
<blockquote>
<ul>
<li>facet 1 <ul>
<li>vertices: b c d </li>
<li>neighbors: 2 3 4 </li>
</ul>
</li>
<li>facet 2 <ul>
<li>vertices: a c d </li>
<li>neighbors: 1 3 4 </li>
</ul>
</li>
<li>facet 3 <ul>
<li>vertices: a b d </li>
<li>neighbors: 1 2 4 </li>
</ul>
</li>
<li>facet 4 <ul>
<li>vertices: a b c </li>
<li>neighbors: 1 2 3 </li>
</ul>
</li>
</ul>
</blockquote>
<p>A facet may be simplicial or non-simplicial. In 3-d, a
<i>simplicial facet</i> has three vertices and three
neighbors. A <i>nonsimplicial facet</i> has more than
three vertices and more than three neighbors. A
nonsimplicial facet has a set of ridges and a centrum. </p>
<p>
A simplicial facet has an orientation. An <i>orientation</i>
is either <i>top</i> or <i>bottom</i>.
The flag, <tt>facet->toporient,</tt>
defines the orientation of the facet's vertices. For example in 3-d,
'top' is left-handed orientation (i.e., the vertex order follows the direction
of the left-hand fingers when the thumb is pointing away from the center).
Except for axis-parallel facets in 5-d and higher, topological orientation
determines the geometric orientation of the facet's hyperplane.
<p>A nonsimplicial facet is due to merging two or more
facets. The facet's ridge set determine a simplicial
decomposition of the facet. Each ridge is a 1-face (i.e.,
it has two vertices and two neighboring facets). The
orientation of a ridge is determined by the order of the
neighboring facets. The flag, <tt>facet->toporient,</tt>is
ignored. </p>
<p>A nonsimplicial facet has a centrum for testing
convexity. A <i>centrum</i> is a point on the facet's
hyperplane that is near the center of the facet. Except
for large facets, it is the arithmetic average of the
facet's vertices. </p>
<p>A nonsimplicial facet is an approximation that is
defined by offsets from the facet's hyperplane. When
Qhull finishes, the <i>outer plane</i> is above all
points while the <i>inner plane</i> is below the facet's
vertices. This guarantees that any exact convex hull
passes between the inner and outer planes. The outer
plane is defined by <tt>facet->maxoutside</tt> while
the inner plane is computed from the facet's vertices.</p>
<p>Qhull 3.1 includes triangulation of non-simplicial facets
('<A href="../html/qh-optq.htm#Qt">Qt</A>').
These facets,
called <i>tricoplanar</i>, share the same normal. centrum, and Voronoi center.
One facet (keepcentrum) owns these data structures.
While tricoplanar facets are more accurate than the simplicial facets from
joggled input, they
may have zero area or flipped orientation.
</blockquote>
<p><b>Copyright © 1995-2003 The Geometry Center, Minneapolis MN</b></p>
<hr>
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