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| 6 | <title>merge.c -- facet merge operations</title>
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| 8 |
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| 9 | <body>
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| 10 | <!-- Navigation links -->
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| 11 | <p><a name="TOP"><b>Up:</b></a> <a
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| 12 | href="http://www.qhull.org">Home page</a> for Qhull<br>
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| 13 | <b>Up:</b> <a href="../../html/index.htm#TOC">Qhull manual</a>: Table of Contents <br>
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| 14 | <b>Up:</b> <a href="../../html/qh-quick.htm#programs">Programs</a>
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| 15 | • <a href="../../html/qh-quick.htm#options">Options</a>
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| 20 | • <a href="../../html/qh-optq.htm#qhull">Qhull</a>
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| 21 | • <a href="../../html/qh-optc.htm#prec">Precision</a>
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| 22 | • <a href="../../html/qh-optt.htm#trace">Trace</a><br>
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| 23 | <b>Up:</b> <a href="../../html/qh-code.htm#TOC">Qhull code: Table of Contents</a><br>
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| 24 | <b>To:</b> <a href="index.htm">Qhull functions</a>, macros, and data structures<br>
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| 25 | <b>To:</b> <a href="qh-geom.htm">Geom</a> • <a href="qh-globa.htm">Global</a>
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| 26 | • <a href="qh-io.htm">Io</a> • <a href="qh-mem.htm">Mem</a>
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| 27 | • <a href="qh-merge.htm#TOC">Merge</a> • <a href="qh-poly.htm">Poly</a>
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| 28 | • <a href="qh-qhull.htm">Qhull</a> • <a href="qh-set.htm">Set</a>
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| 29 | • <a href="qh-stat.htm">Stat</a> • <a href="qh-user.htm">User</a>
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| 30 | </p>
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| 31 | <hr>
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| 32 |
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| 33 | <h2>merge.c -- facet merge operations</h2>
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| 34 | <blockquote>
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| 35 | <p>Qhull handles precision problems by merged facets or joggled input.
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| 36 | Except for redundant vertices, it corrects a problem by
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| 37 | merging two facets. When done, all facets are clearly
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| 38 | convex. See <a href="../../html/qh-impre.htm">Imprecision in Qhull</a>
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| 39 | for further information. </p>
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| 40 | <p>Users may joggle the input ('<a href="../../html/qh-optq.htm#QJn">QJn</a>')
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| 41 | instead of merging facets. </p>
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| 42 | <p>Qhull detects and corrects the following problems: </p>
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| 43 | <ul>
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| 44 | <li><b>More than two facets meeting at a ridge. </b>When
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| 45 | Qhull creates facets, it creates an even number
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| 46 | of facets for each ridge. A convex hull always
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| 47 | has two facets for each ridge. More than two
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| 48 | facets may be created if non-adjacent facets
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| 49 | share a vertex. This is called a <em>duplicate
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| 50 | ridge</em>. In 2-d, a duplicate ridge would
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| 51 | create a loop of facets. </li>
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| 52 | </ul>
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| 53 | <ul>
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| 54 | <li><b>A facet contained in another facet. </b>Facet
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| 55 | merging may leave all vertices of one facet as a
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| 56 | subset of the vertices of another facet. This is
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| 57 | called a <em>redundant facet</em>. </li>
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| 58 | </ul>
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| 59 | <ul>
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| 60 | <li><b>A facet with fewer than three neighbors. </b>Facet
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| 61 | merging may leave a facet with one or two
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| 62 | neighbors. This is called a <em>degenerate facet</em>.
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| 63 | </li>
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| 64 | </ul>
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| 65 | <ul>
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| 66 | <li><b>A facet with flipped orientation. </b>A
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| 67 | facet's hyperplane may define a halfspace that
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| 68 | does not include the interior point.This is
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| 69 | called a <em>flipped facet</em>. </li>
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| 70 | </ul>
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| 71 | <ul>
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| 72 | <li><strong>A coplanar horizon facet.</strong> A
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| 73 | newly processed point may be coplanar with an
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| 74 | horizon facet. Qhull creates a new facet without
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| 75 | a hyperplane. It links new facets for the same
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| 76 | horizon facet together. This is called a <em>samecycle</em>.
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| 77 | The new facet or samecycle is merged into the
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| 78 | horizon facet. </li>
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| 79 | </ul>
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| 80 | <ul>
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| 81 | <li><b>Concave facets. </b>A facet's centrum may be
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| 82 | above a neighboring facet. If so, the facets meet
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| 83 | at a concave angle. </li>
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| 84 | </ul>
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| 85 | <ul>
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| 86 | <li><b>Coplanar facets. </b>A facet's centrum may be
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| 87 | coplanar with a neighboring facet (i.e., it is
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| 88 | neither clearly below nor clearly above the
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| 89 | facet's hyperplane). Qhull removes coplanar
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| 90 | facets in independent sets sorted by angle.</li>
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| 91 | </ul>
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| 92 | <ul>
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| 93 | <li><b>Redundant vertex. </b>A vertex may have fewer
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| 94 | than three neighboring facets. If so, it is
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| 95 | redundant and may be renamed to an adjacent
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| 96 | vertex without changing the topological
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| 97 | structure.This is called a <em>redundant vertex</em>.
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| 98 | </li>
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| 99 | </ul>
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| 100 | </blockquote>
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| 101 | <p><b>Copyright © 1995-2012 C.B. Barber</b></p>
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| 102 | <hr>
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| 103 | <p><a href="#TOP">»</a> <a href="qh-geom.htm#TOC">Geom</a>
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| 104 | <a name="TOC">•</a> <a href="qh-globa.htm#TOC">Global</a>
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| 105 | • <a href="qh-io.htm#TOC">Io</a> • <a href="qh-mem.htm#TOC">Mem</a>
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| 106 | • <b>Merge</b> • <a href="qh-poly.htm#TOC">Poly</a>
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| 107 | • <a href="qh-qhull.htm#TOC">Qhull</a> • <a href="qh-set.htm#TOC">Set</a>
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| 108 | • <a href="qh-stat.htm#TOC">Stat</a> • <a href="qh-user.htm#TOC">User</a>
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| 109 | </p>
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| 110 | <h3>Index to <a href="merge.c">merge.c</a> and
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| 111 | <a href="merge.h">merge.h</a></h3>
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| 112 | <ul>
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| 113 | <li><a href="#mtype">merge.h data types, macros, and
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| 114 | global sets</a> </li>
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| 115 | <li><a href="#mconst">merge.h constants</a> </li>
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| 116 | </ul>
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| 117 | <ul>
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| 118 | <li><a href="#mtop">top-level merge functions</a> </li>
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| 119 | <li><a href="#mset">functions for identifying merges</a></li>
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| 120 | <li><a href="#mbest">functions for determining the
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| 121 | best merge</a> </li>
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| 122 | <li><a href="#mmerge">functions for merging facets</a>
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| 123 | </li>
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| 124 | <li><a href="#mcycle">functions for merging a cycle
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| 125 | of facets</a> </li>
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| 126 | <li><a href="#mrename">functions for renaming a
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| 127 | vertex</a> </li>
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| 128 | <li><a href="#mvertex">functions for identifying
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| 129 | vertices for renaming</a> </li>
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| 130 | <li><a href="#mcheck">functions for check and trace</a> </li>
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| 131 | </ul>
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| 132 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mtype">merge.h data
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| 133 | types, macros, and global sets</a></h3>
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| 134 | <ul>
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| 135 | <li><a href="merge.h#mergeT">mergeT</a> structure to
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| 136 | identify a merge of two facets</li>
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| 137 | <li><a href="merge.h#FOREACHmerge_">FOREACHmerge_</a>
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| 138 | assign 'merge' to each merge in merges </li>
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| 139 | <li><a href="libqhull.h#qh-set">qh global sets</a>
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| 140 | qh.facet_mergeset contains non-convex merges
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| 141 | while qh.degen_mergeset contains degenerate and
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| 142 | redundant facets</li>
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| 143 | </ul>
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| 144 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mconst">merge.h
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| 145 | constants</a></h3>
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| 146 | <ul>
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| 147 | <li><a href="libqhull.h#qh-prec">qh precision constants</a>
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| 148 | precision constants for Qhull </li>
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| 149 | <li><a href="merge.h#MRG">MRG...</a> indicates the
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| 150 | type of a merge (mergeT->type)</li>
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| 151 | <li><a href="merge.h#qh_ANGLEredundant">qh_ANGLEredundant</a>
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| 152 | indicates redundant merge in mergeT->angle </li>
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| 153 | <li><a href="merge.h#qh_ANGLEdegen">qh_ANGLEdegen</a>
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| 154 | indicates degenerate facet in mergeT->angle </li>
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| 155 | <li><a href="merge.h#qh_ANGLEconcave">qh_ANGLEconcave</a>
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| 156 | offset to indicate concave facets in
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| 157 | mergeT->angle </li>
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| 158 | <li><a href="merge.h#qh_MERGEapex">qh_MERGEapex</a>
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| 159 | flag for qh_mergefacet() to indicate an apex
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| 160 | merge </li>
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| 161 | </ul>
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| 162 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mtop">top-level merge
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| 163 | functions</a></h3>
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| 164 | <ul>
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| 165 | <li><a href="merge.c#all_merges">qh_all_merges</a>
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| 166 | merge all non-convex facets </li>
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| 167 | <li><a href="merge.c#checkzero">qh_checkzero</a>
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| 168 | check that facets are clearly convex </li>
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| 169 | <li><a href="merge.c#flippedmerges">qh_flippedmerges</a>
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| 170 | merge flipped facets into best neighbor </li>
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| 171 | <li><a href="merge.c#forcedmerges">qh_forcedmerges</a>
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| 172 | merge all duplicate ridges </li>
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| 173 | <li><a href="merge.c#merge_degenredundant">qh_merge_degenredundant</a>
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| 174 | merge degenerate and redundant facets </li>
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| 175 | <li><a href="merge.c#merge_nonconvex">qh_merge_nonconvex</a>
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| 176 | merge a non-convex ridge </li>
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| 177 | <li><a href="merge.c#premerge">qh_premerge</a>
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| 178 | pre-merge non-convex facets </li>
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| 179 | <li><a href="merge.c#postmerge">qh_postmerge</a>
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| 180 | post-merge nonconvex facets as defined by
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| 181 | maxcentrum/maxangle </li>
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| 182 | </ul>
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| 183 |
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| 184 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mset">functions for
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| 185 | identifying merges</a></h3>
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| 186 | <ul>
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| 187 | <li><a href="merge.c#appendmergeset">qh_appendmergeset</a>
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| 188 | appends an entry to qh.facet_mergeset</li>
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| 189 | <li><a href="merge.c#compareangle">qh_compareangle</a>
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| 190 | used by qsort() to order merges </li>
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| 191 | <li><a href="merge.c#comparemerge">qh_comparemerge</a>
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| 192 | used by qsort() to order merges </li>
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| 193 | <li><a href="merge.c#degen_redundant_facet">qh_degen_redundant_facet</a>
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| 194 | check for a degenerate and redundant facet</li>
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| 195 | <li><a href="merge.c#degen_redundant_neighbors">qh_degen_redundant_neighbors</a>
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| 196 | append degenerate and redundant neighbors to
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| 197 | qh.degen_mergeset </li>
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| 198 | <li><a href="merge.c#getmergeset_initial">qh_getmergeset_initial</a>
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| 199 | build initial qh.facet_mergeset </li>
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| 200 | <li><a href="merge.c#getmergeset">qh_getmergeset</a>
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| 201 | update qh.facet_mergeset </li>
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| 202 | <li><a href="merge.c#mark_dupridges">qh_mark_dupridges</a>
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| 203 | add duplicated ridges to qh.facet_mergeset</li>
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| 204 | <li><a href="merge.c#maydropneighbor">qh_maydropneighbor</a>
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| 205 | drop neighbor relationship if no ridge between
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| 206 | facet and neighbor </li>
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| 207 | <li><a href="merge.c#test_appendmerge">qh_test_appendmerge</a>
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| 208 | test a pair of facets for convexity and append to
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| 209 | qh.facet_mergeset if non-convex </li>
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| 210 | <li><a href="merge.c#test_vneighbors">qh_test_vneighbors</a>
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| 211 | test vertex neighbors for convexity </li>
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| 212 | </ul>
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| 213 |
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| 214 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mbest">functions for
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| 215 | determining the best merge</a></h3>
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| 216 | <ul>
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| 217 | <li><a href="merge.c#findbest_test">qh_findbest_test</a>
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| 218 | test neighbor for best merge </li>
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| 219 | <li><a href="merge.c#findbestneighbor">qh_findbestneighbor</a>
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| 220 | finds best neighbor of a facet for merging (i.e.,
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| 221 | closest hyperplane) </li>
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| 222 | </ul>
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| 223 |
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| 224 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mmerge">functions for
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| 225 | merging facets</a></h3>
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| 226 | <ul>
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| 227 | <li><a href="merge.c#copynonconvex">qh_copynonconvex</a>
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| 228 | copy non-convex flag to another ridge for the
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| 229 | same neighbor </li>
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| 230 | <li><a href="merge.c#makeridges">qh_makeridges</a>
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| 231 | creates explicit ridges between simplicial facets
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| 232 | </li>
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| 233 | <li><a href="merge.c#mergefacet">qh_mergefacet</a>
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| 234 | merges one facet into another facet</li>
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| 235 | <li><a href="merge.c#mergeneighbors">qh_mergeneighbors</a>
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| 236 | merges the neighbors of two facets </li>
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| 237 | <li><a href="merge.c#mergeridges">qh_mergeridges</a>
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| 238 | merges the ridge sets of two facets </li>
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| 239 | <li><a href="merge.c#mergesimplex">qh_mergesimplex</a>
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| 240 | merge a simplicial facet into another simplicial
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| 241 | facet </li>
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| 242 | <li><a href="merge.c#mergevertex_del">qh_mergevertex_del</a>
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| 243 | delete a vertex due to merging one facet into
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| 244 | another facet </li>
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| 245 | <li><a href="merge.c#mergevertex_neighbors">qh_mergevertex_neighbors</a>
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| 246 | merge the vertex neighbors of two facets </li>
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| 247 | <li><a href="merge.c#mergevertices">qh_mergevertices</a>
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| 248 | merge the vertex sets of two facets </li>
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| 249 | <li><a href="merge.c#newvertices">qh_newvertices</a>
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| 250 | register all vertices as new vertices </li>
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| 251 | <li><a href="merge.c#updatetested">qh_updatetested</a>
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| 252 | clear tested flags and centrums involved in a
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| 253 | merge </li>
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| 254 | <li><a href="merge.c#willdelete">qh_willdelete</a>
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| 255 | moves facet to qh.visible_list; sets replacement
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| 256 | or NULL </li>
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| 257 | </ul>
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| 258 |
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| 259 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mcycle">functions for
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| 260 | merging a cycle of facets</a></h3>
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| 261 | <p>If a point is coplanar with an horizon facet, the
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| 262 | corresponding new facets are linked together (a <em>samecycle</em>)
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| 263 | for merging.</p>
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| 264 | <ul>
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| 265 | <li><a href="merge.c#basevertices">qh_basevertices</a>
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| 266 | return temporary set of base vertices for a
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| 267 | samecycle </li>
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| 268 | <li><a href="merge.c#mergecycle">qh_mergecycle</a>
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| 269 | merge a samecycle into a horizon facet </li>
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| 270 | <li><a href="merge.c#mergecycle_all">qh_mergecycle_all</a>
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| 271 | merge all samecycles into horizon facets</li>
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| 272 | <li><a href="merge.c#mergecycle_facets">qh_mergecycle_facets</a>
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| 273 | finish merge of samecycle </li>
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| 274 | <li><a href="merge.c#mergecycle_neighbors">qh_mergecycle_neighbors</a>
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| 275 | merge neighbor sets for samecycle </li>
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| 276 | <li><a href="merge.c#mergecycle_ridges">qh_mergecycle_ridges</a>
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| 277 | merge ridge sets for samecycle </li>
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| 278 | <li><a href="merge.c#mergecycle_vneighbors">qh_mergecycle_vneighbors</a>
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| 279 | merge vertex neighbor sets for samecycle </li>
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| 280 | </ul>
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| 281 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mrename">functions
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| 282 | for renaming a vertex</a></h3>
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| 283 | <ul>
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| 284 | <li><a href="merge.c#comparevisit">qh_comparevisit</a>
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| 285 | used by qsort() to order vertices by visitid</li>
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| 286 | <li><a href="merge.c#reducevertices">qh_reducevertices</a>
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| 287 | reduce vertex sets </li>
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| 288 | <li><a href="merge.c#redundant_vertex">qh_redundant_vertex</a>
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| 289 | returns true if detect and rename redundant
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| 290 | vertex </li>
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| 291 | <li><a href="merge.c#rename_sharedvertex">qh_rename_sharedvertex</a>
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| 292 | detect and rename a shared vertex </li>
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| 293 | <li><a href="merge.c#renameridgevertex">qh_renameridgevertex</a>
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| 294 | rename oldvertex to newvertex in a ridge </li>
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| 295 | <li><a href="merge.c#renamevertex">qh_renamevertex</a>
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| 296 | rename oldvertex to newvertex in ridges </li>
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| 297 | <li><a href="merge.c#remove_extravertices">qh_remove_extravertices</a>
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| 298 | remove extra vertices in non-simplicial facets </li>
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| 299 | </ul>
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| 300 |
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| 301 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mvertex">functions
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| 302 | for identifying vertices for renaming</a></h3>
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| 303 | <ul>
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| 304 | <li><a href="merge.c#find_newvertex">qh_find_newvertex</a>
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| 305 | locate new vertex for renaming old vertex </li>
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| 306 | <li><a href="merge.c#hashridge">qh_hashridge</a> add
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| 307 | ridge to hashtable </li>
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| 308 | <li><a href="merge.c#hashridge_find">qh_hashridge_find</a>
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| 309 | returns matching ridge in hashtable</li>
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| 310 | <li><a href="merge.c#neighbor_intersections">qh_neighbor_intersections</a>
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| 311 | return intersection of vertex sets for
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| 312 | neighboring facets </li>
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| 313 | <li><a href="merge.c#vertexridges">qh_vertexridges</a>
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| 314 | return temporary set of ridges adjacent to a
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| 315 | vertex </li>
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| 316 | <li><a href="merge.c#vertexridges_facet">qh_vertexridges_facet</a>
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| 317 | add adjacent ridges for a vertex in facet </li>
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| 318 | </ul>
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| 319 |
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| 320 | <h3><a href="qh-merge.htm#TOC">»</a><a name="mcheck">functions for check and
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| 321 | trace</a></h3>
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| 322 | <ul>
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| 323 | <li><a href="merge.c#checkconnect">qh_checkconnect</a>
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| 324 | check that new facets are connected </li>
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| 325 | <li><a href="merge.c#tracemerge">qh_tracemerge</a>
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| 326 | print trace message after merge </li>
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| 327 | <li><a href="merge.c#tracemerging">qh_tracemerging</a>
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| 328 | print trace message during post-merging </li>
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| 329 | </ul>
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| 330 |
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| 331 | <p><!-- Navigation links --> </p>
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| 332 | <hr>
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| 333 | <p><b>Up:</b>
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| 334 | <a href="http://www.qhull.org">Home page for
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| 335 | Qhull</a> <br>
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| 336 | <b>Up:</b> <a href="../../html/index.htm#TOC">Qhull manual: Table of Contents</a> <br>
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| 337 | <b>Up:</b> <a href="../../html/qh-quick.htm#programs">Programs</a>
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| 338 | • <a href="../../html/qh-quick.htm#options">Options</a>
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| 339 | • <a href="../../html/qh-opto.htm#output">Output</a>
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| 340 | • <a href="../../html/qh-optf.htm#format">Formats</a>
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| 341 | • <a href="../../html/qh-optg.htm#geomview">Geomview</a>
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| 342 | • <a href="../../html/qh-optp.htm#print">Print</a>
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| 343 | • <a href="../../html/qh-optq.htm#qhull">Qhull</a>
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| 344 | • <a href="../../html/qh-optc.htm#prec">Precision</a>
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| 345 | • <a href="../../html/qh-optt.htm#trace">Trace</a><br>
|
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| 346 | <b>Up:</b> <a href="../../html/qh-code.htm#TOC">Qhull code: Table of Contents</a> <br>
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| 347 | <b>To:</b> <a href="index.htm">Qhull functions</a>, macros, and data structures<br>
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| 348 | <b>To:</b> <a href="qh-geom.htm">Geom</a> •
|
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| 349 | <a href="qh-globa.htm">Global</a> • <a href="qh-io.htm">Io</a>
|
|---|
| 350 | • <a href="qh-mem.htm">Mem</a> • <a href="qh-merge.htm">Merge</a>
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| 351 | • <a href="qh-poly.htm">Poly</a> • <a href="qh-qhull.htm#TOC">Qhull</a>
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| 352 | • <a href="qh-set.htm">Set</a> • <a href="qh-stat.htm">Stat</a>
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| 353 | • <a href="qh-user.htm">User</a><br>
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| 354 | </p>
|
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| 355 | <p><!-- GC common information --> </p>
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| 356 | <hr>
|
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| 357 | <p><a href="http://www.geom.uiuc.edu/"><img
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| 358 | src="../../html/qh--geom.gif" align="middle" width="40" height="40"></a><i>The
|
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| 359 | Geometry Center Home Page </i></p>
|
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| 360 | <p>Comments to: <a href=mailto:qhull@qhull.org>qhull@qhull.org</a>
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| 361 | </a><br>
|
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| 362 | Created: May 2, 1997 --- <!-- hhmts start --> Last modified: see top <!-- hhmts end --> </p>
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| 363 | </body>
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| 364 | </html>
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