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href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-方程式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#方程式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>方程式</span> </div> </a> <ul id="toc-方程式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-面積及周長" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#面積及周長"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>面積及周長</span> </div> </a> <ul id="toc-面積及周長-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-歷史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#歷史"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>歷史</span> </div> </a> <ul id="toc-歷史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-應用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#應用"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>應用</span> </div> </a> <ul id="toc-應用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-相關條目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#相關條目"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>相關條目</span> </div> </a> <ul id="toc-相關條目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考资料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参考资料</span> </div> </a> <ul id="toc-参考资料-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">三尖瓣线</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。17种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Corba_deltoide" title="Corba deltoide – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Corba deltoide" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Deltoid_curve" title="Deltoid curve – 英语" lang="en" hreflang="en" data-title="Deltoid curve" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Curva_deltoide" title="Curva deltoide – 西班牙语" lang="es" hreflang="es" data-title="Curva deltoide" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Deltoide_(kurba)" title="Deltoide (kurba) – 巴斯克语" lang="eu" hreflang="eu" data-title="Deltoide (kurba)" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%84%D8%AA%D8%A7%DA%AF%D9%88%D9%86" title="دلتاگون – 波斯语" lang="fa" hreflang="fa" data-title="دلتاگون" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Delto%C3%AFde_(courbe)" title="Deltoïde (courbe) – 法语" lang="fr" hreflang="fr" data-title="Deltoïde (courbe)" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%9C%D7%AA%D7%95%D7%90%D7%99%D7%93%D7%94" title="דלתואידה – 希伯来语" lang="he" hreflang="he" data-title="דלתואידה" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Steiner-ciklois" title="Steiner-ciklois – 匈牙利语" lang="hu" hreflang="hu" data-title="Steiner-ciklois" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Deltoide_(curva)" title="Deltoide (curva) – 意大利语" lang="it" hreflang="it" data-title="Deltoide (curva)" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%83%AB%E3%83%88%E3%82%A4%E3%83%89" title="デルトイド – 日语" lang="ja" hreflang="ja" data-title="デルトイド" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B8%ED%86%A0%EC%9D%B4%EB%93%9C_%EA%B3%A1%EC%84%A0" title="델토이드 곡선 – 韩语" lang="ko" hreflang="ko" data-title="델토이드 곡선" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Delto%C3%AFde" title="Deltoïde – 荷兰语" lang="nl" hreflang="nl" data-title="Deltoïde" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%BE%D0%B8%D0%B4%D0%B0" title="Дельтоида – 俄语" lang="ru" hreflang="ru" data-title="Дельтоида" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Deltoida" title="Deltoida – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Deltoida" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Deltoid_e%C4%9Frisi" title="Deltoid eğrisi – 土耳其语" lang="tr" hreflang="tr" data-title="Deltoid eğrisi" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D0%BB%D1%8C%D1%82%D0%BE%D1%97%D0%B4%D0%B0" title="Дельтоїда – 乌克兰语" lang="uk" hreflang="uk" data-title="Дельтоїда" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%C6%B0%E1%BB%9Dng_cong_delta" title="Đường cong delta – 越南语" lang="vi" hreflang="vi" data-title="Đường cong delta" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2348724#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div 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href="/zh-hans/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF" lang="zh-Hans-MY" 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data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Deltoid2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Deltoid2.gif/400px-Deltoid2.gif" decoding="async" width="400" height="397" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b5/Deltoid2.gif 1.5x" data-file-width="452" data-file-height="449" /></a><figcaption>紅色的即為三尖瓣线</figcaption></figure> <p><b>三尖瓣线</b>（tricuspoid）也稱為<b>施泰纳曲線</b>（Steiner curve），是有三個<a href="/wiki/%E5%B0%96%E9%BB%9E" title="尖點">尖點</a>的<a href="/wiki/%E5%9C%86%E5%86%85%E8%9E%BA%E7%BA%BF" title="圆内螺线">圆内螺线</a>，是一個圓繞著直徑為其三倍的圓內側無滑動滾動時，圓上一點產生的<a href="/wiki/%E4%B8%80%E8%88%AC%E6%97%8B%E8%BD%AE%E7%BA%BF" title="一般旋轮线">一般旋轮线</a> </p><p>三尖瓣线也可以指有三個頂點，之間用向內彎曲的曲線相連的封閉空間，因此三尖瓣线內的空間是非凸集合<sup id="cite_ref-#1_1-0" class="reference"><a href="#cite_note-#1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="方程式"><span id=".E6.96.B9.E7.A8.8B.E5.BC.8F"></span>方程式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=1" title="编辑章节：方程式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>三尖瓣线可以用以下的<a href="/wiki/%E5%8F%83%E6%95%B8%E6%96%B9%E7%A8%8B" title="參數方程">參數方程</a>表示： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=(b-a)\cos(t)+a\cos \left({\frac {b-a}{a}}t\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=(b-a)\cos(t)+a\cos \left({\frac {b-a}{a}}t\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1257dd597a6d604df76bcdd8879c96e78f40e295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.573ex; height:6.176ex;" alt="{\displaystyle x=(b-a)\cos(t)+a\cos \left({\frac {b-a}{a}}t\right)\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=(b-a)\sin(t)-a\sin \left({\frac {b-a}{a}}t\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=(b-a)\sin(t)-a\sin \left({\frac {b-a}{a}}t\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0ec38d899f02a1b1ff5372b9cc6d412ecadc872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.534ex; height:6.176ex;" alt="{\displaystyle y=(b-a)\sin(t)-a\sin \left({\frac {b-a}{a}}t\right)\,,}"></span></dd></dl> <p>其中<i>a</i>是小圓的半徑，b是大圓（也就是小圓在其內側無滑動滾動）的半徑（此處<i>b = 3a</i>）。 </p><p>在複變座標下可得 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=2ae^{it}+ae^{-2it}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=2ae^{it}+ae^{-2it}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aad5247abb75cf45347ce22662154988a5bb321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.703ex; height:2.843ex;" alt="{\displaystyle z=2ae^{it}+ae^{-2it}}"></span>.</dd></dl> <p>上述的<i>t</i>可以消去，得到以下的笛卡爾座標下的方程 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2}),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>27</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>8</mn> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2}),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60eac01dc11fa173958de2acf3fa6666822d65f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.715ex; height:3.176ex;" alt="{\displaystyle (x^{2}+y^{2})^{2}+18a^{2}(x^{2}+y^{2})-27a^{4}=8a(x^{3}-3xy^{2}),\,}"></span></dd></dl> <p>因此三尖瓣线是四階的<a href="/wiki/%E4%BB%A3%E6%95%B8%E6%9B%B2%E7%B7%9A" title="代數曲線">代數曲線</a>，在<a href="/wiki/%E6%9E%81%E5%9D%90%E6%A0%87" class="mw-redirect" title="极坐标">極坐標</a>下為 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{4}+18a^{2}r^{2}-27a^{4}=8ar^{3}\cos 3\theta \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>8</mn> <mi>a</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{4}+18a^{2}r^{2}-27a^{4}=8ar^{3}\cos 3\theta \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17a49e8af6707712d2207f82ee8a9546f9c0efc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:33.87ex; height:2.843ex;" alt="{\displaystyle r^{4}+18a^{2}r^{2}-27a^{4}=8ar^{3}\cos 3\theta \,.}"></span></dd></dl> <p>曲線有三個奇點，是對應<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0,\,\pm {\tfrac {2\pi }{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0,\,\pm {\tfrac {2\pi }{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb68e92be14bf4d115304f7ba6e71385ef7853b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.93ex; height:3.676ex;" alt="{\displaystyle t=0,\,\pm {\tfrac {2\pi }{3}}}"></span>的尖點。上述的參數式意味者曲線為有理曲線，也就表示其<span class="ilh-all" data-orig-title="幾何虧格" data-lang-code="en" data-lang-name="英语" data-foreign-title="geometric genus"><span class="ilh-page"><a href="/w/index.php?title=%E5%B9%BE%E4%BD%95%E8%99%A7%E6%A0%BC&action=edit&redlink=1" class="new" title="幾何虧格（页面不存在）">幾何虧格</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/geometric_genus" class="extiw" title="en:geometric genus"><span lang="en" dir="auto">geometric genus</span></a></span>）</span></span>為零。 </p><p>三尖瓣线的<span class="ilh-all" data-orig-title="對偶曲線" data-lang-code="en" data-lang-name="英语" data-foreign-title="dual curve"><span class="ilh-page"><a href="/w/index.php?title=%E5%B0%8D%E5%81%B6%E6%9B%B2%E7%B7%9A&action=edit&redlink=1" class="new" title="對偶曲線（页面不存在）">對偶曲線</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/dual_curve" class="extiw" title="en:dual curve"><span lang="en" dir="auto">dual curve</span></a></span>）</span></span>為 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-x^{2}-(3x+1)y^{2}=0,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-x^{2}-(3x+1)y^{2}=0,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78821c18deb634889f1efbcfa3afffc26dc236f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.263ex; height:3.176ex;" alt="{\displaystyle x^{3}-x^{2}-(3x+1)y^{2}=0,\,}"></span></dd></dl> <p>在原點有一個二重點，若進行一個虛軸上的旋轉y ↦ iy，曲線會變為下式，就可以看到其二重點 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-x^{2}+(3x+1)y^{2}=0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-x^{2}+(3x+1)y^{2}=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e089c4a429df20c867c531975bd77038b2466e39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.616ex; height:3.176ex;" alt="{\displaystyle x^{3}-x^{2}+(3x+1)y^{2}=0\,}"></span></dd></dl> <p>在實平面的原點上有二重點。 </p> <div class="mw-heading mw-heading2"><h2 id="面積及周長"><span id=".E9.9D.A2.E7.A9.8D.E5.8F.8A.E5.91.A8.E9.95.B7"></span>面積及周長</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=2" title="编辑章节：面積及周長"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>三尖瓣线的面積為<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa94947b47987b6c6f69c4760d936eea3efce08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.779ex; height:2.676ex;" alt="{\displaystyle 2\pi a^{2}}"></span>，其中<i>a</i>為小圓的半徑，其面積是小圓面積的兩倍<sup id="cite_ref-Weisstein_2-0" class="reference"><a href="#cite_note-Weisstein-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p><p>其周長為16<i>a</i><sup id="cite_ref-Weisstein_2-1" class="reference"><a href="#cite_note-Weisstein-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="歷史"><span id=".E6.AD.B7.E5.8F.B2"></span>歷史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=3" title="编辑章节：歷史"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>早在1599年時，<a href="/wiki/%E4%BC%BD%E5%88%A9%E7%95%A5%C2%B7%E4%BC%BD%E5%88%A9%E8%8E%B1" title="伽利略·伽利莱">伽利略·伽利莱</a>及<a href="/wiki/%E9%A9%AC%E5%85%B0%C2%B7%E6%A2%85%E6%A3%AE" title="马兰·梅森">马兰·梅森</a>就已開始研究常見的<a href="/wiki/%E6%91%86%E7%BA%BF" title="摆线">摆线</a>，而<a href="/wiki/%E5%A5%A7%E5%8B%92%C2%B7%E7%BE%85%E9%BB%98" title="奧勒·羅默">奧勒·羅默</a>在1674年研究齒輪的最佳外形時，也有用到摆线。<a href="/wiki/%E6%9D%8E%E6%98%82%E5%93%88%E5%BE%B7%C2%B7%E6%AD%90%E6%8B%89" class="mw-redirect" title="李昂哈德·歐拉">李昂哈德·歐拉</a>認為他是最早（1745年）將三尖瓣线應用在實際光學問題的人。 </p> <div class="mw-heading mw-heading2"><h2 id="應用"><span id=".E6.87.89.E7.94.A8"></span>應用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=4" title="编辑章节：應用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>三尖瓣线有應用在許多的數學領域中，舉例如下： </p> <ul><li>三維<span class="ilh-all" data-orig-title="unistochastic" data-lang-code="en" data-lang-name="英语" data-foreign-title="unistochastic"><span class="ilh-page"><a href="/w/index.php?title=Unistochastic&action=edit&redlink=1" class="new" title="Unistochastic（页面不存在）">unistochastic</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/unistochastic" class="extiw" title="en:unistochastic"><span lang="en" dir="auto">unistochastic</span></a></span>）</span></span>矩陣複數特徵值的集合即為三尖瓣线。</li> <li>SU(3)<a href="/wiki/%E7%BE%A4" title="群">群</a>裡所有<a href="/wiki/%E9%85%89%E7%9F%A9%E9%98%B5" title="酉矩阵">酉矩阵</a>的可能跡的集合會組成三尖瓣线。</li> <li>二個三尖瓣线的交集會形成一群6階的<span class="ilh-all" data-orig-title="複數Hadamard矩陣" data-lang-code="en" data-lang-name="英语" data-foreign-title="Complex Hadamard matrix"><span class="ilh-page"><a href="/w/index.php?title=%E8%A4%87%E6%95%B8Hadamard%E7%9F%A9%E9%99%A3&action=edit&redlink=1" class="new" title="複數Hadamard矩陣（页面不存在）">複數Hadamard矩陣</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Complex_Hadamard_matrix" class="extiw" title="en:Complex Hadamard matrix"><span lang="en" dir="auto">Complex Hadamard matrix</span></a></span>）</span></span>。</li> <li>三角形的所有<a href="/wiki/%E8%A5%BF%E5%A7%86%E6%9D%BE%E5%AE%9A%E7%90%86" title="西姆松定理">西姆松線</a>的集合，其<a href="/wiki/%E5%8C%85%E7%B5%A1%E7%B7%9A" title="包絡線">包絡線</a>會是三尖瓣线。因為<a href="/w/index.php?title=%E9%9B%85%E5%90%84%E5%B8%83%C2%B7%E6%96%BD%E6%B3%B0%E7%BA%B3&action=edit&redlink=1" class="new" title="雅各布·施泰纳（页面不存在）">雅各布·施泰纳</a>在1856年描述過此曲線的形狀及對稱性，因此此曲線稱為這稱為施泰纳三尖瓣线，<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。</li> <li>三角形<a href="/wiki/%E5%B9%B3%E5%88%86%E7%B7%9A" title="平分線">面積平分線</a>的<a href="/wiki/%E5%8C%85%E7%B5%A1%E7%B7%9A" title="包絡線">包絡線</a>會是三尖瓣线，三個頂點是<a href="/wiki/%E4%B8%AD%E7%B7%9A" title="中線">中線</a>的中點。三尖瓣线的三邊是<a href="/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线">双曲线</a>的弧<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-#1_1-1" class="reference"><a href="#cite_note-#1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。</li> <li>三尖瓣线屬於<span class="ilh-all" data-orig-title="掛谷集" data-lang-code="en" data-lang-name="英语" data-foreign-title="Kakeya set"><span class="ilh-page"><a href="/w/index.php?title=%E6%8E%9B%E8%B0%B7%E9%9B%86&action=edit&redlink=1" class="new" title="掛谷集（页面不存在）">掛谷集</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Kakeya_set" class="extiw" title="en:Kakeya set"><span lang="en" dir="auto">Kakeya set</span></a></span>）</span></span>中的掛谷針集合（Kakeya needle set，長度為1的針可以在其中旋轉360度），有科學家認為三尖瓣线是掛谷針問題（Kakeya needle problem，面積最小小的掛谷針集合）的解，後來發現不是。</li></ul> <div class="mw-heading mw-heading2"><h2 id="相關條目"><span id=".E7.9B.B8.E9.97.9C.E6.A2.9D.E7.9B.AE"></span>相關條目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=5" title="编辑章节：相關條目"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%98%9F%E5%BD%A2%E7%BA%BF" title="星形线">星形线</a>：有三個尖點的圆内螺线</li> <li><span class="ilh-all" data-orig-title="伪三角形" data-lang-code="en" data-lang-name="英语" data-foreign-title="Pseudotriangle"><span class="ilh-page"><a href="/w/index.php?title=%E4%BC%AA%E4%B8%89%E8%A7%92%E5%BD%A2&action=edit&redlink=1" class="new" title="伪三角形（页面不存在）">伪三角形</a></span><span class="noprint ilh-comment">（<span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Pseudotriangle" class="extiw" title="en:Pseudotriangle"><span lang="en" dir="auto">Pseudotriangle</span></a></span>）</span></span></li> <li><a href="/wiki/%E5%8B%92%E6%B4%9B%E4%B8%89%E8%A7%92%E5%BD%A2" title="勒洛三角形">勒洛三角形</a></li> <li><a href="/wiki/%E8%B6%85%E6%A9%A2%E5%9C%93" title="超橢圓">超橢圓</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考资料"><span id=".E5.8F.82.E8.80.83.E8.B5.84.E6.96.99"></span>参考资料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&action=edit&section=6" title="编辑章节：参考资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-#1-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-#1_1-0"><sup><b>1.0</b></sup></a> <a href="#cite_ref-#1_1-1"><sup><b>1.1</b></sup></a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20171121043036/http://www.se16.info/js/halfarea.htm">存档副本</a>. <span class="reference-accessdate"> [<span class="nowrap">2017-10-24</span>]</span>. （<a rel="nofollow" class="external text" href="http://www.se16.info/js/halfarea.htm">原始内容</a>存档于2017-11-21）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&rft.btitle=%E5%AD%98%E6%A1%A3%E5%89%AF%E6%9C%AC&rft.genre=unknown&rft_id=http%3A%2F%2Fwww.se16.info%2Fjs%2Fhalfarea.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Weisstein-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Weisstein_2-0"><sup><b>2.0</b></sup></a> <a href="#cite_ref-Weisstein_2-1"><sup><b>2.1</b></sup></a></span> <span class="reference-text">Weisstein, Eric W. "Deltoid." From <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>--A Wolfram Web Resource. <a rel="nofollow" class="external free" href="http://mathworld.wolfram.com/Deltoid.html">http://mathworld.wolfram.com/Deltoid.html</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20171025073410/http://mathworld.wolfram.com/Deltoid.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Lockwood</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Dunn, J. A., and Pretty, J. A., "Halving a triangle," <i><a href="/w/index.php?title=Mathematical_Gazette&action=edit&redlink=1" class="new" title="Mathematical Gazette（页面不存在）">Mathematical Gazette</a></i> 56, May 1972, 105-108.</span> </li> </ol></div> <ul><li><cite class="citation book">E. H. Lockwood. Chapter 8: The Deltoid. A Book of Curves. Cambridge University Press. 1961.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&rft.atitle=Chapter+8%3A+The+Deltoid&rft.au=E.+H.+Lockwood&rft.btitle=A+Book+of+Curves&rft.date=1961&rft.genre=bookitem&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">J. Dennis Lawrence. <a rel="nofollow" class="external text" href="https://archive.org/details/catalogspecialpl00lawr_291">A catalog of special plane curves</a>. Dover Publications. 1972: <a rel="nofollow" class="external text" href="https://archive.org/details/catalogspecialpl00lawr_291/page/n147">131</a>–134. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-486-60288-5" title="Special:网络书源/0-486-60288-5"><span title="国际标准书号">ISBN</span> 0-486-60288-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&rft.au=J.+Dennis+Lawrence&rft.btitle=A+catalog+of+special+plane+curves&rft.date=1972&rft.genre=book&rft.isbn=0-486-60288-5&rft.pages=131-134&rft.pub=Dover+Publications&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcatalogspecialpl00lawr_291&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation book">Wells D. The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. 1991: 52. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-14-011813-6" title="Special:网络书源/0-14-011813-6"><span title="国际标准书号">ISBN</span> 0-14-011813-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&rft.au=Wells+D&rft.btitle=The+Penguin+Dictionary+of+Curious+and+Interesting+Geometry&rft.date=1991&rft.genre=book&rft.isbn=0-14-011813-6&rft.pages=52&rft.place=New+York&rft.pub=Penguin+Books&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/Curves/Tricuspoid.html">"Tricuspoid" at MacTutor's Famous Curves Index</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20171121042222/http://www-history.mcs.st-andrews.ac.uk/history/Curves/Tricuspoid.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="http://www.mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml">"Deltoid" at MathCurve</a>（<a rel="nofollow" class="external text" href="//web.archive.org/web/20171022130536/http://www.mathcurve.com/courbes2d.gb/deltoid/deltoid.shtml">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><cite id="CITEREFSokolov2001" class="citation">Sokolov, D.D., <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=S/s087650">Steiner curve</a>, Hazewinkel, Michiel (编), <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4"><span title="国际标准书号">ISBN</span> 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E4%B8%89%E5%B0%96%E7%93%A3%E7%BA%BF&rft.atitle=Steiner+curve&rft.aufirst=D.D.&rft.aulast=Sokolov&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DS%2Fs087650&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" 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