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<!DOCTYPE html> <html lang="en"> <head> <title>makeB'Section -- makeB'Section creates a hash table that represents a hyperplane.</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="software for numerical algebraic geometry" href="index.html">Bertini</a> :: <a title="makeB'Section creates a hash table that represents a hyperplane." href="_make__B_sq__Section.html">makeB'Section</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="_make__B_sq__Slice.html">next</a> | <a href="_make__B_sq__Input__File.html">previous</a> | <a href="_make__B_sq__Slice.html">forward</a> | <a href="_make__B_sq__Input__File.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>makeB'Section -- makeB'Section creates a hash table that represents a hyperplane.</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">makeB'Section(l)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">l</span>, <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, A list of variables.</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">B'Homogenization</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value 1</span></span></li> <li><span><span class="tt">B'NumberCoefficients</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span></span></li> <li><span><span class="tt">ContainsPoint</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span></span></li> <li><span><span class="tt">NameB'Section</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value null</span></span></li> <li><span><span class="tt">RandomCoefficientGenerator</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value FunctionClosure[../Bertini.m2:2314:37-2314:66]</span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>makeB'Section allows for easy creation of equations that define hyperplanes. The default creates a hash table with two keys: B'NumberCoefficients and B'SectionString. The first key is a list of numbers in CC that are coefficients, and the second key is a string representing the linear polynomial. The option RandomCoefficientGenerator can be set to a function to generate random numbers for the coefficients.</p> </div> <div> <p>To get affine linear equations include 1 in the input list.</p> </div> <div> <p>To have an affine linear equation that contains a particular point we set the ContainsPoint option to a list of coordinates or a point. To get an homogeneous equation that contains a projective point we have to set the ContainsPoint option as well as the B'Homogenization option.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : s=makeB'Section({x,y,z}) o1 = B'Section{...2...} o1 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : class s o2 = B'Section o2 : Type</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : randomRealCoefficientGenerator=()->random(RR) o3 = randomRealCoefficientGenerator o3 : FunctionClosure</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : sReal=makeB'Section({x,y,z},RandomCoefficientGenerator=>randomRealCoefficientGenerator) o4 = B'Section{...2...} o4 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : sReal#B'NumberCoefficients o5 = {.0741835, .808694, .362835} o5 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : randomRationalCoefficientGenerator=()->random(QQ) o6 = randomRationalCoefficientGenerator o6 : FunctionClosure</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : sRational=makeB'Section({x,y,z},RandomCoefficientGenerator=>randomRationalCoefficientGenerator) o7 = B'Section{...2...} o7 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : sRational#B'NumberCoefficients 7 1 7 o8 = {--, -, --} 10 2 10 o8 : List</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i9 : affineSection=makeB'Section({x,y,z,1}) o9 = B'Section{...2...} o9 : B'Section</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i10 : X={x,y,z} o10 = {x, y, z} o10 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : P={1,2,3} o11 = {1, 2, 3} o11 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : affineContainingPoint=makeB'Section({x,y,z},ContainsPoint=>P) o12 = B'Section{...3...} o12 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : r= affineContainingPoint#B'SectionString o13 = (1.18921+.849539*ii)*(x-(1)*(1))+(-.542371+.307137*ii)*(y-(1)*(2))+( 1.36945+.015633*ii)*(z-(1)*(3))</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i14 : print r (1.18921+.849539*ii)*(x-(1)*(1))+(-.542371+.307137*ii)*(y-(1)*(2))+(1.36945+.015633*ii)*(z-(1)*(3))</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i15 : rHomogeSection= makeB'Section({x,y,z},ContainsPoint=>P,B'Homogenization=>"x+y+z") o15 = B'Section{...3...} o15 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i16 : peek rHomogeSection o16 = B'Section{B'Homogenization => x+y+z B'NumberCoefficients => {.534614-.175945*ii, .426704 B'SectionString => (.534614-.175945*ii)*(x-(x+y+z)*( ----------------------------------------------------------------------- -.97539*ii, -.478803+.0416008*ii} 1))+(.426704-.97539*ii)*(y-(x+y+z)*(2))+(-.478803+.0416008*ii)*(z-(x+y+ ----------------------------------------------------------------------- } z)*(3))</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i17 : print rHomogeSection#B'SectionString (.534614-.175945*ii)*(x-(x+y+z)*(1))+(.426704-.97539*ii)*(y-(x+y+z)*(2))+(-.478803+.0416008*ii)*(z-(x+y+z)*(3))</code></pre> </td> </tr> </table> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i18 : f="y^3-x*y+1" o18 = y^3-x*y+1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i19 : s1=makeB'Section({x,y,1}) o19 = B'Section{...2...} o19 : B'Section</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i20 : makeB'InputFile(storeBM2Files, AffVariableGroup=>{x,y}, B'Polynomials=>{f,s1});</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i21 : runBertini(storeBM2Files)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i22 : #importSolutionsFile(storeBM2Files)==3 o22 = true</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">makeB'Section</span>:</h2> <ul> <li><kbd>makeB'Section(List)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="makeB'Section creates a hash table that represents a hyperplane." href="_make__B_sq__Section.html">makeB'Section</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Bertini.m2:3611:0</span>.</p> </div> </div> </div> </body> </html>