<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>latticePoints -- computes the lattice points of a polytope</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_lin__Space.html">next</a> | <a href="_is__Smooth.html">previous</a> | <a href="_lin__Space.html">forward</a> | <a href="_is__Smooth.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>latticePoints -- computes the lattice points of a polytope</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt> L = latticePoints P</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>P</tt>, <span>a <a href="___Polyhedron.html">convex polyhedron</a></span>, which must be compact</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, containing the lattice points as matrices over <a href="../../Macaulay2Doc/html/___Z__Z.html" title="the class of all integers">ZZ</a> with only
one column</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><p/>
<tt>latticePoints</tt> can only be applied to polytopes, i.e. compact polyhedra. It
embeds the polytope on height 1 in a space of dimension plus 1 and takes the Cone over
this polytope. Then it projects the elements of height 1 of the Hilbert basis back again.<table class="examples"><tr><td><pre>i1 : P = crossPolytope 3
o1 = {ambient dimension => 3 }
dimension of lineality space => 0
dimension of polyhedron => 3
number of facets => 8
number of rays => 0
number of vertices => 6
o1 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i2 : latticePoints P
o2 = {| -1 |, | 0 |, | 0 |, 0, | 0 |, | 0 |, | 1 |}
| 0 | | -1 | | 0 | | 0 | | 1 | | 0 |
| 0 | | 0 | | -1 | | 1 | | 0 | | 0 |
o2 : List</pre>
</td></tr>
<tr><td><pre>i3 : Q = cyclicPolytope(2,4)
o3 = {ambient dimension => 2 }
dimension of lineality space => 0
dimension of polyhedron => 2
number of facets => 4
number of rays => 0
number of vertices => 4
o3 : Polyhedron</pre>
</td></tr>
<tr><td><pre>i4 : latticePoints Q
o4 = {0, | 1 |, | 1 |, | 1 |, | 2 |, | 2 |, | 2 |, | 3 |}
| 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 9 |
o4 : List</pre>
</td></tr>
</table>
</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>latticePoints</tt> :</h2>
<ul><li>latticePoints(Polyhedron)</li>
</ul>
</div>
</div>
</body>
</html>
