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<title>poly.c, poly2.c -- polyhedron operations</title>
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<p><a name="TOP"><b>Up:</b></a> <a
href="http://www.geom.umn.edu/locate/qhull">Home page</a> for Qhull<br>
<b>Up:</b> <a href="../html/index.htm#TOC">Qhull manual</a>: Table of Contents <br>
<b>Up:</b> <a href="../html/qh-in.htm#TOC">Qhull internals: Table of Contents</a><br>
<b>Up:</b> <a href="../html/qh-quick.htm#programs">Programs</a>
&#149;</a> <a href="../html/qh-quick.htm#options">Options</a>
&#149; <a href="../html/qh-opto.htm#output">Output</a>
&#149; <a href="../html/qh-optf.htm#format">Formats</a>
&#149; <a href="../html/qh-optg.htm#geomview">Geomview</a>
&#149; <a href="../html/qh-optp.htm#print">Print</a>
&#149; <a href="../html/qh-optq.htm#qhull">Qhull</a>
&#149; <a href="../html/qh-optc.htm#prec">Precision</a>
&#149; <a href="../html/qh-optt.htm#trace">Trace</a><br>
<b>To:</b> <a href="index.htm">Qhull functions</a>, macros, and data structures<br>
<b>To:</b> <a href="qh-geom.htm">Geom</a> &#149; <a href="qh-globa.htm">Global</a>
&#149; <a href="qh-io.htm">Io</a> &#149; <a href="qh-mem.htm">Mem</a>
&#149; <a href="qh-merge.htm">Merge</a> &#149; <a href="qh-poly.htm#TOC">Poly</a>
&#149; <a href="qh-qhull.htm">Qhull</a> &#149; <a href="qh-set.htm">Set</a>
&#149; <a href="qh-stat.htm">Stat</a> &#149; <a href="qh-user.htm">User</a>
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<hr>
<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="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
neighoring 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-&gt;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.
</b>
<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-&gt;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-&gt;maxoutside</tt> while
the inner plane is computed from the facet's vertices.</p>
</blockquote>
<p><b>Copyright ｩ 1995-2001 The Geometry Center, Minneapolis MN</b></p>
<hr>
<p><a href="#TOP">ｻ</a> <a href="qh-geom.htm#TOC">Geom</a>
<a name="TOC">&#149;</a> <a href="qh-globa.htm#TOC">Global</a>
&#149; <a href="qh-io.htm#TOC">Io</a> &#149; <a href="qh-mem.htm#TOC">Mem</a>
&#149; <a href="qh-merge.htm#TOC">Merge</a> &#149; <b>Poly</b>
&#149; <a href="qh-qhull.htm#TOC">Qhull</a> &#149; <a href="qh-set.htm#TOC">Set</a>
&#149; <a href="qh-stat.htm#TOC">Stat</a> &#149; <a href="qh-user.htm#TOC">User</a>
</p>
<h3>Index to <a href="poly.c#TOP">poly.c</a>,
<a href="poly2.c#TOP">poly2.c</a>, <a href="poly.h#TOP">poly.h</a>,
and <a href="qhull.h#TOP">qhull.h</a></h3>
<ul>
<li><a href="#ptype">Data types and global
lists for polyhedrons</a> </li>
<li><a href="#pconst">poly.h constants</a> </li>
<li><a href="#pgall">Global FORALL macros</a> </li>
<li><a href="#pall">FORALL macros</a> </li>
<li><a href="#peach">FOREACH macros</a> </li>
<li><a href="#pieach">Indexed FOREACH macros</a> </li>
<li><a href="#pmacro">Other macros for polyhedrons</a><p>&nbsp;</li>
<li><a href="#plist">Facetlist functions</a> </li>
<li><a href="#pfacet">Facet functions</a> </li>
<li><a href="#pvertex">Vertex, ridge, and point
functions</a> </li>
<li><a href="#phash">Hashtable functions</a> </li>
<li><a href="#pnew">Allocation and deallocation
functions</a> </li>
<li><a href="#pcheck">Check functions</a> </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="ptype">Data
types and global lists for polyhedrons</a></h3>
<ul>
<li><a href="qhull.h#facetT">facetT</a> defines a
facet </li>
<li><a href="qhull.h#ridgeT">ridgeT</a> defines a
ridge </li>
<li><a href="qhull.h#vertexT">vertexT</a> defines a
vertex </li>
<li><a href="qhull.h#qh-lists">qh facet and vertex
lists</a> lists of facets and vertices </li>
<li><a href="qhull.h#qh-set">qh global sets</a>
global sets for merging, hashing, input, etc. </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pconst">poly.h constants</a></h3>
<ul>
<li><a href="poly.h#ALGORITHMfault">ALGORITHMfault</a>
flag to not report errors in qh_checkconvex() </li>
<li><a href="poly.h#DATAfault">DATAfault</a> flag to
report errors in qh_checkconvex() </li>
<li><a href="poly.h#DUPLICATEridge">DUPLICATEridge</a>
special value for facet-&gt;neighbor to indicate
a duplicate ridge </li>
<li><a href="poly.h#MERGEridge">MERGEridge</a>
special value for facet-&gt;neighbor to indicate
a merged ridge </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pgall">Global FORALL
macros</a></h3>
<ul>
<li><a href="qhull.h#FORALLfacets">FORALLfacets</a>
assign 'facet' to each facet in qh.facet_list </li>
<li><a href="poly.h#FORALLnew_facets">FORALLnew_facets</a>
assign 'facet' to each facet in qh.newfacet_list </li>
<li><a href="poly.h#FORALLvisible_facets">FORALLvisible_facets</a>
assign 'visible' to each visible facet in
qh.visible_list </li>
<li><a href="qhull.h#FORALLpoints">FORALLpoints</a>
assign 'point' to each point in qh.first_point,
qh.num_points </li>
<li><a href="qhull.h#FORALLvertices">FORALLvertices</a>
assign 'vertex' to each vertex in qh.vertex_list </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pall">FORALL macros</a></h3>
<ul>
<li><a href="poly.h#FORALLfacet_">FORALLfacet_</a>
assign 'facet' to each facet in facetlist </li>
<li><a href="qhull.h#FORALLpoint_">FORALLpoint_</a>
assign 'point' to each point in points array</li>
<li><a href="poly.h#FORALLsame_">FORALLsame_</a>
assign 'same' to each facet in samecycle</li>
<li><a href="poly.h#FORALLsame_cycle_">FORALLsame_cycle_</a>
assign 'same' to each facet in samecycle</li>
<li><a href="poly.h#FORALLvertex_">FORALLvertex_</a>
assign 'vertex' to each vertex in vertexlist </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="peach">FOREACH macros</a></h3>
<ul>
<li><a href="qhull.h#FOREACHfacet_">FOREACHfacet_</a>
assign 'facet' to each facet in facets </li>
<li><a href="qhull.h#FOREACHneighbor_">FOREACHneighbor_</a>
assign 'neighbor' to each facet in
facet-&gt;neighbors or vertex-&gt;neighbors</li>
<li><a href="poly.h#FOREACHnewfacet_">FOREACHnewfacet_</a>
assign 'newfacet' to each facet in facet set </li>
<li><a href="qhull.h#FOREACHpoint_">FOREACHpoint_</a>
assign 'point' to each point in points set </li>
<li><a href="qhull.h#FOREACHridge_">FOREACHridge_</a>
assign 'ridge' to each ridge in ridge set </li>
<li><a href="qhull.h#FOREACHvertex_">FOREACHvertex_</a>
assign 'vertex' to each vertex in vertex set </li>
<li><a href="poly.h#FOREACHvertexA_">FOREACHvertexA_</a>
assign 'vertexA' to each vertex in vertex set</li>
<li><a href="poly.h#FOREACHvisible_">FOREACHvisible_</a>
assign 'visible' to each facet in facet set </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pieach">Indexed
FOREACH macros</a></h3>
<ul>
<li><a href="qhull.h#FOREACHfacet_i_">FOREACHfacet_i_</a>
assign 'facet' and 'facet_i' to each facet in
facet set </li>
<li><a href="qhull.h#FOREACHneighbor_i_">FOREACHneighbor_i_</a>
assign 'neighbor' and 'neighbor_i' to each facet
in facet-&gt;neighbors or vertex-&gt;neighbors</li>
<li><a href="qhull.h#FOREACHpoint_i_">FOREACHpoint_i_</a>
assign 'point' and 'point_i' to each point in
points set </li>
<li><a href="qhull.h#FOREACHridge_i_">FOREACHridge_i_</a>
assign 'ridge' and 'ridge_i' to each ridge in
ridges set </li>
<li><a href="qhull.h#FOREACHvertex_i_">FOREACHvertex_i_</a>
assign 'vertex' and 'vertex_i' to each vertex in
vertices set </li>
<li><a href="poly.h#FOREACHsetelementreverse12_">FOREACHvertexreverse12_</a>
assign 'vertex' to each vertex in vertex set;
reverse the order of first two vertices </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pmacro">Other macros for polyhedrons</a></h3>
<ul>
<li><a href="qhull.h#getid_">getid_</a> return ID for
a facet, ridge, or vertex </li>
<li><a href="qhull.h#otherfacet_">otherfacet_</a>
return neighboring facet for a ridge in a facet </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="plist">Facetlist
functions</a></h3>
<ul>
<li><a href="poly.c#appendfacet">qh_appendfacet</a>
appends facet to end of qh.facet_list</li>
<li><a href="poly.c#attachnewfacets">qh_attachnewfacets</a>
attach new facets in qh.newfacet_list to the
horizon </li>
<li><a href="poly2.c#findgood">qh_findgood</a>
identify good facets for qh.PRINTgood </li>
<li><a href="poly2.c#findgood_all">qh_findgood_all</a>
identify more good facets for qh.PRINTgood </li>
<li><a href="poly2.c#furthestnext">qh_furthestnext</a>
move facet with furthest of furthest points to
facet_next </li>
<li><a href="poly2.c#initialhull">qh_initialhull</a>
construct the initial hull as a simplex of
vertices </li>
<li><a href="poly2.c#nearcoplanar">qh_nearcoplanar</a>
remove near-inside points from coplanar sets</li>
<li><a href="poly2.c#nonupper">qh_nonupper</a> return
first non-flipped facet </li>
<li><a href="poly2.c#prependfacet">qh_prependfacet</a>
prepends facet to start of facetlist </li>
<li><a href="poly.c#removefacet">qh_removefacet</a>
unlinks facet from qh.facet_list</li>
<li><a href="poly2.c#resetlists">qh_resetlists</a>
reset qh.newvertex_list, qh.newfacet_list, and
qh.visible_list </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pfacet">Facet
functions</a></h3>
<ul>
<li><a href="poly2.c#createsimplex">qh_createsimplex</a>
create a simplex of facets from a set of vertices
</li>
<li><a href="poly2.c#furthestout">qh_furthestout</a>
make furthest outside point the last point of a
facet's outside set </li>
<li><a href="poly.c#makenew_nonsimplicial">qh_makenew_nonsimplicial</a>
make new facets from ridges of visible facets </li>
<li><a href="poly.c#makenew_simplicial">qh_makenew_simplicial</a>
make new facets for horizon neighbors </li>
<li><a href="poly.c#makenewfacet">qh_makenewfacet</a>
create a facet from vertices and apex </li>
<li><a href="poly2.c#makenewfacets">qh_makenewfacets</a>
make new facets from vertex, horizon facets, and
visible facets </li>
<li><a href="poly.c#makenewplanes">qh_makenewplanes</a>
make new hyperplanes for facets </li>
<li><a href="poly2.c#outcoplanar">qh_outcoplanar</a>
move points from outside set to coplanar set </li>
<li><a href="poly2.c#setvoronoi_all">qh_setvoronoi_all</a>
compute Voronoi centers for all facets </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pvertex">Vertex,
ridge, and point functions</a></h3>
<ul>
<li><a href="poly.c#appendvertex">qh_appendvertex</a>
append vertex to end of qh.vertex_list, </li>
<li><a href="poly2.c#facet3vertex">qh_facet3vertex</a>
return an oriented vertex set for a 3-d facet </li>
<li><a href="poly.c#facetintersect">qh_facetintersect</a>
return intersection of simplicial facets </li>
<li><a href="poly2.c#initialvertices">qh_initialvertices</a>
return non-singular set of initial vertices </li>
<li><a href="poly2.c#isvertex">qh_isvertex</a> true
if point is in a vertex set </li>
<li><a href="poly2.c#nearvertex">qh_nearvertex</a>
return nearest vertex to point </li>
<li><a href="poly2.c#nextridge3d">qh_nextridge3d</a>
iterate over each ridge and vertex for a 3-d
facet </li>
<li><a href="poly2.c#point">qh_point</a> return point
for a point ID </li>
<li><a href="poly2.c#pointfacet">qh_pointfacet</a>
return temporary set of facets indexed by point
ID </li>
<li><a href="poly.c#pointid">qh_pointid</a> return ID
for a point</li>
<li><a href="poly2.c#pointvertex">qh_pointvertex</a>
return temporary set of vertices indexed by point
ID </li>
<li><a href="poly.c#removevertex">qh_removevertex</a>
unlink vertex from qh.vertex_list, </li>
<li><a href="poly.c#updatevertices">qh_updatevertices</a>
update vertex neighbors and delete interior
vertices </li>
<li><a href="poly2.c#vertexintersect">qh_vertexintersect</a>
intersect two vertex sets </li>
<li><a href="poly2.c#vertexintersect_new">qh_vertexintersect_new</a>
return intersection of two vertex sets </li>
<li><a href="poly2.c#vertexneighhbors">qh_vertexneighhbors</a>
for each vertex in hull, determine facet
neighbors </li>
<li><a href="poly2.c#vertexsubset">qh_vertexsubset</a>
returns True if vertexsetA is a subset of
vertexsetB </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="phash">Hashtable functions</a></h3>
<ul>
<li><a href="poly2.c#addhash">qh_addhash</a> add hash
element to linear hash table</li>
<li><a href="poly.c#gethash">qh_gethash</a> return
hash value for a set</li>
<li><a href="poly2.c#matchduplicates">qh_matchduplicates</a>
match duplicate ridges in hash table </li>
<li><a href="poly.c#matchneighbor">qh_matchneighbor</a>
try to match subridge of new facet with a
neighbor </li>
<li><a href="poly.c#matchnewfacets">qh_matchnewfacets</a>
match new facets with their new facet neighbors </li>
<li><a href="poly.c#matchvertices">qh_matchvertices</a>
tests whether a facet and hash entry match at a
ridge </li>
<li><a href="poly2.c#newhashtable">qh_newhashtable</a>
allocate a new qh.hash_table </li>
<li><a href="poly2.c#printhashtable">qh_printhashtable</a>
print hash table </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pnew">Allocation and
deallocation functions</a></h3>
<ul>
<li><a href="poly2.c#clearcenters">qh_clearcenters</a>
clear old data from facet-&gt;center </li>
<li><a href="poly.c#deletevisible">qh_deletevisible</a>
delete visible facets and vertices </li>
<li><a href="poly.c#delfacet">qh_delfacet</a> free up
the memory occupied by a facet </li>
<li><a href="poly2.c#delridge">qh_delridge</a> delete
ridge</li>
<li><a href="poly2.c#delvertex">qh_delvertex</a>
delete vertex </li>
<li><a href="poly.c#newfacet">qh_newfacet</a> create
and allocate space for a facet </li>
<li><a href="poly.c#newridge">qh_newridge</a> create
and allocate space for a ridge </li>
<li><a href="poly2.c#newvertex">qh_newvertex</a>
create and allocate space for a vertex </li>
</ul>
<h3><a href="qh-poly.htm#TOC">ｻ</a><a name="pcheck">Check
functions</a></h3>
<ul>
<li><a href="poly2.c#check_bestdist">qh_check_bestdist</a>
check that points are not outside of facets </li>
<li><a href="poly2.c#check_maxout">qh_check_maxout</a>
updates qh.max_outside and checks all points
against bestfacet </li>
<li><a href="poly2.c#check_output">qh_check_output</a>
check topological and geometric output</li>
<li><a href="poly2.c#check_point">qh_check_point</a>
check that point is not outside of facet </li>
<li><a href="poly2.c#check_points">qh_check_points</a>
check that all points are inside all facets </li>
<li><a href="poly2.c#checkconvex">qh_checkconvex</a>
check that each ridge in facetlist is convex </li>
<li><a href="poly2.c#checkfacet">qh_checkfacet</a>
check for consistency errors in facet </li>
<li><a href="poly.c#checkflipped">qh_checkflipped</a>
check facet orientation to the interior point </li>
<li><a href="poly2.c#checkflipped_all">qh_checkflipped_all</a>
check facet orientation for a facet list </li>
<li><a href="poly2.c#checkpolygon">qh_checkpolygon</a>
check topological structure </li>
<li><a href="poly2.c#checkvertex">qh_checkvertex</a>
check vertex for consistency </li>
<li><a href="poly2.c#infiniteloop">qh_infiniteloop</a>
report error for a loop of facets </li>
<li><a href="poly2.c#printlists">qh_printlists</a>
print out facet list for debugging </li>
</ul>
<p><!-- Navigation links --> </p>
<hr>
<p><b>Up:</b>
<a href="http://www.geom.umn.edu/locate/qhull">Home page for
Qhull</a> <br>
<b>Up:</b> <a href="../html/index.htm#TOC">Qhull manual: Table of Contents</a> <br>
<b>Up:</b> <a href="../html/qh-in.htm#TOC">Qhull internals: Table of Contents</a> <br>
<b>Up:</b> <a href="../html/qh-quick.htm#programs">Programs</a>
&#149;</a> <a href="../html/qh-quick.htm#options">Options</a>
&#149; <a href="../html/qh-opto.htm#output">Output</a>
&#149; <a href="../html/qh-optf.htm#format">Formats</a>
&#149; <a href="../html/qh-optg.htm#geomview">Geomview</a>
&#149; <a href="../html/qh-optp.htm#print">Print</a>
&#149; <a href="../html/qh-optq.htm#qhull">Qhull</a>
&#149; <a href="../html/qh-optc.htm#prec">Precision</a>
&#149; <a href="../html/qh-optt.htm#trace">Trace</a><br>
<b>To:</b> <a href="index.htm">Qhull functions</a>, macros, and data structures<br>
<b>To:</b> <a href="qh-geom.htm">Geom</a> &#149;
<a href="qh-globa.htm">Global</a> &#149; <a href="qh-io.htm">Io</a>
&#149; <a href="qh-mem.htm">Mem</a> &#149; <a href="qh-merge.htm">Merge</a>
&#149; <a href="qh-poly.htm">Poly</a> &#149; <a href="qh-qhull.htm#TOC">Qhull</a>
&#149; <a href="qh-set.htm">Set</a> &#149; <a href="qh-stat.htm">Stat</a>
&#149; <a href="qh-user.htm">User</a><br>
</p>
<p><!-- GC common information --> </p>
<hr>
<p><a href="http://www.geom.umn.edu/"><img
src="../html/qh--geom.gif" align="middle" width="40" height="40"></a><i>The
Geometry Center Home Page </i></p>
<p>Comments to:
<a href="http://www.geom.umn.edu/software/qhull/qhull-mail.html">qhull@geom.umn.edu
</a><br>
Created: May 2, 1997 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
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