<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Projective cone</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Projective_cone"> <link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Projective_cone rootpage-Projective_cone skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Projective cone</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1251242444">
/* start https://en.wikipedia.org/ */
.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}
/* end https://en.wikipedia.org/ */
</style>
<p>A <b>projective cone</b> (or just <b>cone</b>) in <a href="Projective_geometry" title="Projective geometry">projective geometry</a> is the union of all lines that intersect a projective subspace <i>R</i> (the apex of the cone) and an arbitrary subset <i>A</i> (the basis) of some other subspace <i>S</i>, disjoint from <i>R</i>.
</p><p>In the special case that <i>R</i> is a single point, <i>S</i> is a plane, and <i>A</i> is a <a href="Conic_section" title="Conic section">conic section</a> on <i>S</i>, the projective cone is a <a href="Conical_surface" title="Conical surface">conical surface</a>; hence the name.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Let <i>X</i> be a projective space over some field <i>K</i>, and <i>R</i>, <i>S</i> be disjoint subspaces of <i>X</i>. Let <i>A</i> be an arbitrary subset of <i>S</i>. Then we define <i>RA</i>, the cone with top <i>R</i> and basis <i>A</i>, as follows :
</p>
<ul><li>When <i>A</i> is empty, <i>RA</i> = <i>A</i>.</li>
<li>When <i>A</i> is not empty, <i>RA</i> consists of all those <a href="Point_(geometry)" title="Point (geometry)">points</a> on a line connecting a point on <i>R</i> and a point on <i>A</i>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>
<ul><li>As <i>R</i> and <i>S</i> are disjoint, one may deduce from <a href="Linear_algebra" title="Linear algebra">linear algebra</a> and the definition of a projective space that every point on <i>RA</i> not in <i>R</i> or <i>A</i> is on exactly one line connecting a point in <i>R</i> and a point in <i>A</i>.</li>
<li>(<i>RA</i>) <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cap }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∩<!-- ∩ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cap }</annotation>
</semantics>
</math></span><img src="./9d4e886e6f5a28a33e073fb108440c152ecfe2d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \cap }" loading="lazy"></span> <i>S</i> = <i>A</i></li>
<li>When <i>K</i> is the <a href="Finite_field" title="Finite field">finite field</a> of order <i>q</i>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |RA|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>R</mi>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |RA|}</annotation>
</semantics>
</math></span><img src="./4bdc2cb1cf26498454e33991477fa55c8ce25878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.801ex; height:2.843ex;" alt="{\displaystyle |RA|}" loading="lazy"></span> = <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q^{r+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle q^{r+1}}</annotation>
</semantics>
</math></span><img src="./cc04878099f5ebf691ca0d1e5729d4493f908483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.154ex; height:3.009ex;" alt="{\displaystyle q^{r+1}}" loading="lazy"></span><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |A|}</annotation>
</semantics>
</math></span><img src="./648fce92f29d925f04d39244ccfe435320dfc6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle |A|}" loading="lazy"></span> + <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {q^{r+1}-1}{q-1}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>r</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>q</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {q^{r+1}-1}{q-1}}}</annotation>
</semantics>
</math></span><img src="./0ea3f75690cd1063a2b1a8d67065cd18fd8a5f84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.993ex; height:6.176ex;" alt="{\displaystyle {\frac {q^{r+1}-1}{q-1}}}" loading="lazy"></span>, where <i>r</i> = dim(<i>R</i>).</li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Cone_(geometry)" class="mw-redirect" title="Cone (geometry)">Cone (geometry)</a></li>
<li><a href="Cone_(algebraic_geometry)" title="Cone (algebraic geometry)">Cone (algebraic geometry)</a></li>
<li><a href="Cone_(topology)" title="Cone (topology)">Cone (topology)</a></li>
<li><a href="Cone_(linear_algebra)" class="mw-redirect" title="Cone (linear algebra)">Cone (linear algebra)</a></li>
<li><a href="Conic_section" title="Conic section">Conic section</a></li>
<li><a href="Ruled_surface" title="Ruled surface">Ruled surface</a></li>
<li><a href="Hyperboloid" title="Hyperboloid">Hyperboloid</a></li></ul>
<p><br>
</p>
<style data-mw-deduplicate="TemplateStyles:r1271159938">
/* start https://en.wikipedia.org/ */
.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox-body{font-style:italic}.mw-parser-output .asbox-note{font-size:smaller}.mw-parser-output .asbox .navbar{position:absolute;top:-0.75em;right:1em;display:none}.mw-parser-output :not(p):not(.asbox)+style+.asbox,.mw-parser-output :not(p):not(.asbox)+link+.asbox{margin-top:3em}
/* end https://en.wikipedia.org/ */
</style></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2022-10-24" href="https://en.wikipedia.org/wiki/?title=Projective_cone&oldid=1117995323">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>