One Hat Cyber Team
Your IP :
216.73.216.135
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Points
/
html
/
View File Name :
_points.html
<!DOCTYPE html> <html lang="en"> <head> <title>points -- make the ideal of a set of points</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="A package for making and studying points in affine and projective spaces" href="index.html">Points</a> :: <a title="make the ideal of a set of points" href="_points.html">points</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_projective__Fat__Points.html">next</a> | <a href="_omega__Points.html">previous</a> | <a href="_projective__Fat__Points.html">forward</a> | <a href="_omega__Points.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>points -- make the ideal of a set of points</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">i = points pointsMat</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">pointsMat</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, matrix whose columns are the homogeneous coordinates of the points</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">i</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = ZZ/101[vars(0..4)] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : pointsMat = randomPointsMat(R,11) o2 = | 1 0 0 0 0 1 24 19 -29 21 -18 | | 0 1 0 0 0 1 -36 -10 -24 34 -13 | | 0 0 1 0 0 1 -30 -29 -38 19 -43 | | 0 0 0 1 0 1 -29 -8 -16 -47 -15 | | 0 0 0 0 1 1 19 -22 39 -39 -28 | 5 11 o2 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : points pointsMat o3 = ideal (a*d - 2b*d + 26c*d - 15a*e + 37b*e + 8c*e + 46d*e, b*c - 36b*d + ------------------------------------------------------------------------ 23c*d + 48a*e + 48b*e - 50c*e - 34d*e, a*c + 50b*d + 41c*d + 28a*e + ------------------------------------------------------------------------ 50b*e - 42c*e - 27d*e, a*b - 10b*d - 44c*d + 37a*e + 20b*e - 8c*e + ------------------------------------------------------------------------ 2 2 2 2 2 2 4d*e, b*d*e + 20c*d*e + 23d e - 26a*e - b*e + 12c*e - 29d*e , c e + ------------------------------------------------------------------------ 2 2 2 2 2 2 2 19c*d*e + 14d e + 17a*e - 29b*e + 7c*e - 29d*e , b e - 2c*d*e - 3d e ------------------------------------------------------------------------ 2 2 2 2 2 2 2 - 11a*e + 25b*e + 33c*e - 43d*e , a e - c*d*e - 30d e - 27a*e + ------------------------------------------------------------------------ 2 2 2 11b*e + 5c*e + 41d*e ) o3 : Ideal of R</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="matrix of homogeneous coordinates of random points in projective space" href="_random__Points__Mat.html">randomPointsMat</a> -- matrix of homogeneous coordinates of random points in projective space</span></li> </ul> </div> <div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="make the ideal of a set of points" href="_points.html">points</a> is <span>a <a title="the class of all function closures" href="../../Macaulay2Doc/html/___Function__Closure.html">function closure</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Points.m2:897:0</span>.</p> </div> </div> </div> </body> </html>