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class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">amaga</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegació </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visiteu la pàgina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Article_aleatori" title="Carrega una pàgina a l’atzar [x]" accesskey="x"><span>Article a l'atzar</span></a></li><li id="n-Articles-de-qualitat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Articles_de_qualitat"><span>Articles de qualitat</span></a></li> </ul> </div> </div> <div id="p-Comunitat" class="vector-menu mw-portlet mw-portlet-Comunitat" > <div class="vector-menu-heading"> Comunitat </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-portal" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Portal" title="Sobre el projecte, què podeu fer, on trobareu les coses"><span>Portal viquipedista</span></a></li><li id="n-Agenda-d'actes" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Trobades"><span>Agenda d'actes</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Canvis_recents" title="Una llista dels canvis recents al wiki [r]" accesskey="r"><span>Canvis recents</span></a></li><li id="n-La-taverna" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:La_taverna"><span>La taverna</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Contacte"><span>Contacte</span></a></li><li id="n-Xat" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Canals_IRC"><span>Xat</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Viquip%C3%A8dia:Ajuda" title="El lloc per a saber més coses"><span>Ajuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Viquipèdia" src="/static/images/mobile/copyright/wikipedia-wordmark-ca.svg" style="width: 7.5em; height: 1.4375em;"> <img class="mw-logo-tagline" alt="l'Enciclopèdia Lliure" src="/static/images/mobile/copyright/wikipedia-tagline-ca.svg" width="120" height="14" style="width: 7.5em; height: 0.875em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Cerca" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Cerca a la Viquipèdia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Cerca</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Cerca a Viquipèdia" aria-label="Cerca a Viquipèdia" autocapitalize="sentences" title="Cerca a la Viquipèdia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Cerca"> </div> <button class="cdx-button cdx-search-input__end-button">Cerca</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Eines personals"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparença" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparença</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca" class=""><span>Donatius</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crea_compte&returnto=Epitrocoide" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori" class=""><span>Crea un compte</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Epitrocoide" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o" class=""><span>Inicia la sessió</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Més opcions" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Eines personals" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Eines personals</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú d'usuari" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ca.wikipedia.org&uselang=ca"><span>Donatius</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crea_compte&returnto=Epitrocoide" title="Us animem a crear un compte i iniciar una sessió, encara que no és obligatori"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crea un compte</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Registre_i_entrada&returnto=Epitrocoide" title="Us animem a registrar-vos, però no és obligatori [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Inicia la sessió</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"> Pàgines per a editors no registrats <a href="/wiki/Ajuda:Introducci%C3%B3" aria-label="Vegeu més informació sobre l'edició"><span>més informació</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/Especial:Contribucions_pr%C3%B2pies" title="Una llista de les modificacions fetes des d'aquesta adreça IP [y]" accesskey="y"><span>Contribucions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:Discussi%C3%B3_personal" title="Discussió sobre les edicions per aquesta adreça ip. [n]" accesskey="n"><span>Discussió per aquest IP</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="Lloc"> <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="Contingut" 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">Contingut</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">amaga</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">Inici</div> </a> </li> <li id="toc-Equacions_paramètriques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Equacions_paramètriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Equacions paramètriques</span> </div> </a> <ul id="toc-Equacions_paramètriques-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Doble_generació" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Doble_generació"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Doble generació</span> </div> </a> <ul id="toc-Doble_generació-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formes_particulars" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formes_particulars"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Formes particulars</span> </div> </a> <ul id="toc-Formes_particulars-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Exemples</span> </div> </a> <ul id="toc-Exemples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referències" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referències"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Referències</span> </div> </a> <ul id="toc-Referències-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vegeu_també" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vegeu_també"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Vegeu també</span> </div> </a> <ul id="toc-Vegeu_també-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enllaços_externs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enllaços_externs"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Enllaços externs</span> </div> </a> <ul id="toc-Enllaços_externs-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="Contingut" 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="Commuta la taula de continguts." > <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">Commuta la taula de continguts.</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">Epitrocoide</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="Vés a un article en una altra llengua. Disponible en 26 llengües" > <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-26" 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">26 llengües</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Epitrogo%C3%AFed" title="Epitrogoïed - afrikaans" lang="af" hreflang="af" data-title="Epitrogoïed" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AC%D9%84%D9%8A_%D9%81%D9%88%D9%82%D9%8A" title="عجلي فوقي - àrab" lang="ar" hreflang="ar" data-title="عجلي فوقي" data-language-autonym="العربية" data-language-local-name="àrab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BF%D0%B8%D1%82%D1%80%D0%BE%D1%85%D0%BE%D0%B8%D0%B4%D0%B0" title="Епитрохоида - búlgar" lang="bg" hreflang="bg" data-title="Епитрохоида" data-language-autonym="Български" data-language-local-name="búlgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Epitrochoida" title="Epitrochoida - txec" lang="cs" hreflang="cs" data-title="Epitrochoida" data-language-autonym="Čeština" data-language-local-name="txec" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%B9%CF%84%CF%81%CE%BF%CF%87%CE%BF%CE%B5%CE%B9%CE%B4%CE%AE%CF%82" title="Επιτροχοειδής - grec" lang="el" hreflang="el" data-title="Επιτροχοειδής" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Epitrochoid" title="Epitrochoid - anglès" lang="en" hreflang="en" data-title="Epitrochoid" data-language-autonym="English" data-language-local-name="anglès" 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/Epitrocoide" title="Epitrocoide - espanyol" lang="es" hreflang="es" data-title="Epitrocoide" data-language-autonym="Español" data-language-local-name="espanyol" 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/Epitrokoide" title="Epitrokoide - basc" lang="eu" hreflang="eu" data-title="Epitrokoide" data-language-autonym="Euskara" data-language-local-name="basc" 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%A8%D8%B1%D9%88%D9%86%E2%80%8C%DA%86%D8%B1%D8%AE%D9%87%E2%80%8C%D8%B2%D8%A7%D8%AF" title="برون‌چرخه‌زاد - persa" lang="fa" hreflang="fa" data-title="برون‌چرخه‌زاد" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Epitrokoidi" title="Epitrokoidi - finès" lang="fi" hreflang="fi" data-title="Epitrokoidi" data-language-autonym="Suomi" data-language-local-name="finès" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89pitrocho%C3%AFde" title="Épitrochoïde - francès" lang="fr" hreflang="fr" data-title="Épitrochoïde" data-language-autonym="Français" data-language-local-name="francès" 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%90%D7%A4%D7%99%D7%98%D7%A8%D7%95%D7%9B%D7%95%D7%90%D7%99%D7%93" title="אפיטרוכואיד - hebreu" lang="he" hreflang="he" data-title="אפיטרוכואיד" data-language-autonym="עברית" data-language-local-name="hebreu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Epitrocoide" title="Epitrocoide - italià" lang="it" hreflang="it" data-title="Epitrocoide" data-language-autonym="Italiano" data-language-local-name="italià" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A7%E0%B4%BF%E0%B4%9A%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%9C%E0%B4%82" title="അധിചക്രജം - malaiàlam" lang="ml" hreflang="ml" data-title="അധിചക്രജം" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Epitrocho%C3%AFde" title="Epitrochoïde - neerlandès" lang="nl" hreflang="nl" data-title="Epitrochoïde" data-language-autonym="Nederlands" data-language-local-name="neerlandès" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Epitrokoide" title="Epitrokoide - noruec bokmål" lang="nb" hreflang="nb" data-title="Epitrokoide" data-language-autonym="Norsk bokmål" data-language-local-name="noruec bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Epitrochoida" title="Epitrochoida - polonès" lang="pl" hreflang="pl" data-title="Epitrochoida" data-language-autonym="Polski" data-language-local-name="polonès" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Epitrocoide" title="Epitrocoide - portuguès" lang="pt" hreflang="pt" data-title="Epitrocoide" data-language-autonym="Português" data-language-local-name="portuguès" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Epitrohoid%C4%83" title="Epitrohoidă - romanès" lang="ro" hreflang="ro" data-title="Epitrohoidă" data-language-autonym="Română" data-language-local-name="romanès" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BF%D0%B8%D1%82%D1%80%D0%BE%D1%85%D0%BE%D0%B8%D0%B4%D0%B0" title="Эпитрохоида - rus" lang="ru" hreflang="ru" data-title="Эпитрохоида" data-language-autonym="Русский" data-language-local-name="rus" 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/Epitrohoida" title="Epitrohoida - eslovè" lang="sl" hreflang="sl" data-title="Epitrohoida" data-language-autonym="Slovenščina" data-language-local-name="eslovè" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AE%BF%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%81%E0%AE%B0%E0%AF%81" title="வெளிச்சில்லுரு - tàmil" lang="ta" hreflang="ta" data-title="வெளிச்சில்லுரு" data-language-autonym="தமிழ்" data-language-local-name="tàmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%AD%E0%B8%9E%E0%B8%B4%E0%B9%82%E0%B8%97%E0%B8%A3%E0%B8%84%E0%B8%AD%E0%B8%A2%E0%B8%94%E0%B9%8C" title="เอพิโทรคอยด์ - tai" lang="th" hreflang="th" data-title="เอพิโทรคอยด์" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Epitrokoid" title="Epitrokoid - turc" lang="tr" hreflang="tr" data-title="Epitrokoid" data-language-autonym="Türkçe" data-language-local-name="turc" 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%95%D0%BF%D1%96%D1%82%D1%80%D0%BE%D1%85%D0%BE%D1%97%D0%B4%D0%B0" title="Епітрохоїда - ucraïnès" lang="uk" hreflang="uk" data-title="Епітрохоїда" data-language-autonym="Українська" data-language-local-name="ucraïnès" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%96%E6%97%8B%E8%BD%AE%E7%BA%BF" title="外旋轮线 - xinès" lang="zh" hreflang="zh" data-title="外旋轮线" data-language-autonym="中文" data-language-local-name="xinès" class="interlanguage-link-target"><span>中文</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/Q1088210#sitelinks-wikipedia" title="Modifica enllaços interlingües" class="wbc-editpage">Modifica els enllaços</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="Espais de noms"> <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/Epitrocoide" title="Vegeu el contingut de la pàgina [c]" accesskey="c"><span>Pàgina</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discussi%C3%B3:Epitrocoide&action=edit&redlink=1" rel="discussion" class="new" title="Discussió sobre el contingut d'aquesta pàgina (encara no existeix) [t]" accesskey="t"><span>Discussió</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Canvia la variant de llengua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">català</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistes"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Epitrocoide"><span>Mostra</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Epitrocoide&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Epitrocoide&action=history" title="Versions antigues d'aquesta pàgina [h]" accesskey="h"><span>Mostra l'historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eines" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eines</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eines</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mou a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">amaga</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Més opcions" > <div class="vector-menu-heading"> Accions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Epitrocoide"><span>Mostra</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Epitrocoide&action=edit" title="Modifica el codi font d'aquesta pàgina [e]" accesskey="e"><span>Modifica</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Epitrocoide&action=history"><span>Mostra l'historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:Enlla%C3%A7os/Epitrocoide" title="Una llista de totes les pàgines wiki que enllacen amb aquesta [j]" accesskey="j"><span>Què hi enllaça</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Seguiment/Epitrocoide" rel="nofollow" title="Canvis recents a pàgines enllaçades des d'aquesta pàgina [k]" accesskey="k"><span>Canvis relacionats</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A0gines_especials" title="Llista totes les pàgines especials [q]" accesskey="q"><span>Pàgines especials</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Epitrocoide&oldid=34222874" title="Enllaç permanent a aquesta revisió de la 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Imprimeix/exporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Llibre&bookcmd=book_creator&referer=Epitrocoide"><span>Crea un llibre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Epitrocoide&action=show-download-screen"><span>Baixa com a PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Epitrocoide&printable=yes" title="Versió per a impressió d'aquesta pàgina [p]" accesskey="p"><span>Versió per a impressora</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En altres projectes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Epitrochoid" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1088210" title="Enllaç a l'element del repositori de dades connectat [g]" accesskey="g"><span>Element a Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Eines de la pàgina"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparença"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ca" dir="ltr"><p>Una <b>epitrocoide</b> és una <a href="/wiki/Ruleta_(geometria)" title="Ruleta (geometria)">ruleta</a> traçada per un punt unit a un <a href="/wiki/Cercle" title="Cercle">cercle</a> de <a href="/wiki/Radi_(geometria)" title="Radi (geometria)">radi</a> <i>r</i> que gira entorn de l'exterior d'un cercle fix de radi <i>R</i>, on el punt es troba a una distància <i>d</i> del centre del cercle exterior. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 300.66666666667px"> <div class="thumb" style="width: 298.66666666667px;"><span typeof="mw:File"><a href="/wiki/Fitxer:EpitrochoidIn3.gif" class="mw-file-description" title="La corba vermella és una epitrocoide dibuixada per un cercle negre que rodola sense lliscar al voltant d'un cercle blau (els paràmetres són R = 3, r = 1 i d = 1/2)"><img alt="La corba vermella és una epitrocoide dibuixada per un cercle negre que rodola sense lliscar al voltant d'un cercle blau (els paràmetres són R = 3, r = 1 i d = 1/2)" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/20/EpitrochoidIn3.gif/448px-EpitrochoidIn3.gif" decoding="async" width="299" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/20/EpitrochoidIn3.gif 1.5x" data-file-width="452" data-file-height="454" /></a></span></div> <div class="gallerytext">La corba vermella és una <i>epitrocoide</i> dibuixada per un cercle negre que rodola sense lliscar al voltant d'un cercle blau (els paràmetres són R = 3, r = 1 i d = 1/2)</div> </li> </ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Equacions_paramètriques"><span id="Equacions_param.C3.A8triques"></span>Equacions paramètriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=1" title="Modifica la secció: Equacions paramètriques"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les <a href="/wiki/Equaci%C3%B3_param%C3%A8trica" title="Equació paramètrica">equacions paramètriques</a> per a una epitrocoide són: </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(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>d</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \right),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7943b4c6d13640ab4bde6c4cc7f9258845736e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.355ex; height:6.176ex;" alt="{\displaystyle x(\theta )=(R+r)\cos \theta -d\cos \left({R+r \over r}\theta \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(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>d</mi> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1d5e7519b6c9df06a345773868cbabe9cb77ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.67ex; height:6.176ex;" alt="{\displaystyle y(\theta )=(R+r)\sin \theta -d\sin \left({R+r \over r}\theta \right).\,}"></span></dd></dl> <p>on <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> és un <a href="/wiki/Par%C3%A0metre" title="Paràmetre">paràmetre</a> (no l'<a href="/wiki/Coordenades_polars" title="Coordenades polars">angle polar</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Doble_generació"><span id="Doble_generaci.C3.B3"></span>Doble generació</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=2" title="Modifica la secció: Doble generació"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Qualsevol <a href="/wiki/Epicicloide" title="Epicicloide">epicicloide</a> de paràmetres <i>R</i>, <i>r</i>, <i>d</i> equival a un <a href="/w/index.php?title=Peritrocoide&action=edit&redlink=1" class="new" title="Peritrocoide (encara no existeix)">peritrocoide</a> de paràmetres <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 {\begin{array}{lll}R'={d \over r}R,&r'={d \over r}(R+r),&d'=R+r\end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="left left left" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>R</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>r</mi> </mfrac> </mrow> <mi>R</mi> <mo>,</mo> </mtd> <mtd> <msup> <mi>r</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mi>r</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{lll}R'={d \over r}R,&r'={d \over r}(R+r),&d'=R+r\end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa48231a253f0502a8e8a69a69b8d4d0612502c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.926ex; margin-bottom: -0.246ex; width:40.342ex; height:3.509ex;" alt="{\displaystyle {\begin{array}{lll}R'={d \over r}R,&r'={d \over r}(R+r),&d'=R+r\end{array}}}"></span>. </p><p>«Peritrocoide» significa la <a href="/wiki/Corba" title="Corba">corba</a> obtinguda mitjançant un punt lligat a un cercle mòbil que es roda sense lliscar al voltant d'un cercle de direcció que conté, una «<a href="/wiki/Hipotrocoide" title="Hipotrocoide">hipotrocoide</a>» per al qual <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>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8971c9610113faec012a76ec2d47fa6235e16d2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r>R}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Formes_particulars">Formes particulars</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=3" title="Modifica la secció: Formes particulars"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Quan el punt està situat sobre cercle en moviment (<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 d=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21cf89a5a7cb251bc64932aa6bf549ede7e10ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle d=r}"></span>), s'obté un <a href="/wiki/Epicicloide" title="Epicicloide">epicicloide</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>Nota 1<span class="cite-bracket">]</span></a></sup></li> <li>Quan els dos cercles tenen el mateix diàmetre (<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=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed97a934a158af5ecdf7587f66ec24b3234f52f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle R=r}"></span>), l'epitrocoide representa un <a href="/wiki/Cargol_de_Pascal" title="Cargol de Pascal">caragol de Pascal</a>; es tracta d'una <a href="/wiki/Cardioide" title="Cardioide">cardioide</a> si <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 d=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21cf89a5a7cb251bc64932aa6bf549ede7e10ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle d=r}"></span>.</li> <li>Per a <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 d=R+r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mi>R</mi> <mo>+</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=R+r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3976c94c0b3757e834461e0ce7a80a3080ba3dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.967ex; height:2.343ex;" alt="{\displaystyle d=R+r}"></span>, s'obté una <a href="/wiki/Rosa_(matem%C3%A0tiques)" title="Rosa (matemàtiques)">rosa</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=4" title="Modifica la secció: Exemples"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Amb la clàssica joguina <a href="/wiki/Espir%C3%B2graf" title="Espirògraf">espirògraf</a> es poden dibuixar corbes epitrocoides i <a href="/wiki/Hipotrocoide" title="Hipotrocoide">hipotrocoides</a>. </p><p>Les òrbites dels planetes segons el <a href="/wiki/Claudi_Ptolemeu" title="Claudi Ptolemeu">Sistema ptolemaic</a> de la <a href="/wiki/Teoria_geoc%C3%A8ntrica" title="Teoria geocèntrica">teoria geocèntrica</a> són epitrocoides. </p><p>La <a href="/wiki/Cambra_de_combusti%C3%B3" title="Cambra de combustió">cambra de combustió</a> del <a href="/wiki/Motor_Wankel" title="Motor Wankel">motor Wankel</a> és un epitrocoide. </p> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 203.33333333333px"> <div class="thumb" style="width: 201.33333333333px;"><span typeof="mw:File"><a href="/wiki/Fitxer:Spiograph_Animation.gif" class="mw-file-description" title="Animació d'un espirògraf"><img alt="Animació d'un espirògraf" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spiograph_Animation.gif/302px-Spiograph_Animation.gif" decoding="async" width="202" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Spiograph_Animation.gif/452px-Spiograph_Animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c1/Spiograph_Animation.gif 2x" data-file-width="562" data-file-height="559" /></a></span></div> <div class="gallerytext">Animació d'un <a href="/wiki/Espir%C3%B2graf" title="Espirògraf">espirògraf</a></div> </li> <li class="gallerybox" style="width: 202px"> <div class="thumb" style="width: 200px;"><span typeof="mw:File"><a href="/wiki/Fitxer:Ptolemaic_elements.svg" class="mw-file-description" title="Segons el Sistema ptolemaic, el planeta es mou sobre l'epicicle (línia de punts petita), que al seu torn es mou sobre el deferent (línia de punts gran). El centre del deferent és X, però el moviment angular de l'epicicle és uniforme només respecte al punt •, que és l'equant"><img alt="Segons el Sistema ptolemaic, el planeta es mou sobre l'epicicle (línia de punts petita), que al seu torn es mou sobre el deferent (línia de punts gran). El centre del deferent és X, però el moviment angular de l'epicicle és uniforme només respecte al punt •, que és l'equant" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ptolemaic_elements.svg/300px-Ptolemaic_elements.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ptolemaic_elements.svg/450px-Ptolemaic_elements.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Ptolemaic_elements.svg/600px-Ptolemaic_elements.svg.png 2x" data-file-width="168" data-file-height="168" /></a></span></div> <div class="gallerytext">Segons el <a href="/wiki/Teoria_geoc%C3%A8ntrica" title="Teoria geocèntrica">Sistema ptolemaic</a>, el planeta es mou sobre l'<a href="/wiki/Epicicle" title="Epicicle">epicicle</a> (línia de punts petita), que al seu torn es mou sobre el <a href="/wiki/Deferent" title="Deferent">deferent</a> (línia de punts gran). El centre del deferent és X, però el moviment angular de l'epicicle és uniforme només respecte al punt •, que és l'<a href="/wiki/Equant" title="Equant">equant</a></div> </li> <li class="gallerybox" style="width: 152px"> <div class="thumb" style="width: 150px;"><span typeof="mw:File"><a href="/wiki/Fitxer:Wankel_Cycle_anim2_ca.gif" class="mw-file-description" title="Animació del funcionament del motor Wankel"><img alt="Animació del funcionament del motor Wankel" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Wankel_Cycle_anim2_ca.gif/225px-Wankel_Cycle_anim2_ca.gif" decoding="async" width="150" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/18/Wankel_Cycle_anim2_ca.gif 1.5x" data-file-width="240" data-file-height="320" /></a></span></div> <div class="gallerytext">Animació del funcionament del <a href="/wiki/Motor_Wankel" title="Motor Wankel">motor Wankel</a></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=5" title="Modifica la secció: Notes"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist {{#if: | references-column-count references-column-count-{{{col}}}" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Per a <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 d<r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo><</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d<r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7736b56a7efaa6856ffdb2cb1b8b82361db639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle d<r}"></span> i <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 d>r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>></mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d>r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cec536c81a7994e4e257932a37384ca59530625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle d>r}"></span>, es parla d'epicicloides «escurçats» i «allargats».</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Referències"><span id="Refer.C3.A8ncies"></span>Referències</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=6" title="Modifica la secció: Referències"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREFLawrence1972"><span style="font-variant: small-caps;">Lawrence</span>, J. Dennis. <i>A catalog of special plane curves</i> (en anglès).  Dover Publications, 1972, p. 160-164. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Fonts_bibliogr%C3%A0fiques/0-486-60288-5" title="Especial:Fonts bibliogràfiques/0-486-60288-5">ISBN 0-486-60288-5</a></span>.</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+catalog+of+special+plane+curves&rft.aulast=Lawrence&rft.aufirst=J.+Dennis&rft.date=1972&rft.pub=Dover+Publications&rft.pages=160-164&rft.isbn=0-486-60288-5"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vegeu_també"><span id="Vegeu_tamb.C3.A9"></span>Vegeu també</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=7" title="Modifica la secció: Vegeu també"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cicloide" title="Cicloide">Cicloide</a></li> <li><a href="/wiki/Cicl%C3%B2gon" title="Ciclògon">Ciclògon</a></li> <li><a href="/wiki/Epicicloide" title="Epicicloide">Epicicloide</a></li> <li><a href="/wiki/Espir%C3%B2graf" title="Espirògraf">Espirògraf</a></li> <li><a href="/wiki/Hipocicloide" title="Hipocicloide">Hipocicloide</a></li> <li><a href="/wiki/Hipotrocoide" title="Hipotrocoide">Hipotrocoide</a></li> <li><a href="/wiki/Rosetta_(%C3%B2rbita)" title="Rosetta (òrbita)">Rosetta (òrbita)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Enllaços_externs"><span id="Enlla.C3.A7os_externs"></span>Enllaços externs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Epitrocoide&action=edit&section=8" title="Modifica la secció: Enllaços externs"><span>modifica</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.cs.aau.dk/~marius/gallery/generator.html">Epitrochoid generator</a></li> <li><span class="citació mathworld" id="Referència-Mathworld-Epitrochoid"><a href="/wiki/Eric_W._Weisstein" class="mw-redirect" title="Eric W. Weisstein"><span style="font-variant:small-caps; font-variant-caps: small-caps;">Weisstein</span>, Eric W.</a>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Epitrochoid.html">«Epitrochoid»</a> a <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> (en anglès).</span></li> <li><a rel="nofollow" class="external text" href="http://gerdbreitenbach.de/planet/planet.html">Interactive simulation of the geocentric graphical representation of planet paths </a></li> <li><span class="citation" style="font-style:normal"><span style="font-variant: small-caps;">O'Connor</span>, John J.; <span style="font-variant:small-caps; font-variant-caps: small-caps;">Robertson</span>, Edmund F. «<a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/Curves/Epitrochoid.html">Epitrochoid</a>» (en anglès). <i>MacTutor History of Mathematics archive</i>.  School of Mathematics and Statistics, University of St Andrews, Scotland.</span></li> <li><a rel="nofollow" class="external text" href="http://sourceforge.net/p/geofun/wiki/Home/">Plot Epitrochoid -- GeoFun</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtingut de «<a dir="ltr" href="https://ca.wikipedia.org/w/index.php?title=Epitrocoide&oldid=34222874">https://ca.wikipedia.org/w/index.php?title=Epitrocoide&oldid=34222874</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:Corbes" title="Categoria:Corbes">Corbes</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categoria oculta: <ul><li><a href="/wiki/Categoria:Control_d%27autoritats" title="Categoria:Control d'autoritats">Control d'autoritats</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> La pàgina va ser modificada per darrera vegada el 11 nov 2024 a les 11:00.</li> <li id="footer-info-copyright">El text està disponible sota la <a href="/wiki/Viquip%C3%A8dia:Text_de_la_llic%C3%A8ncia_de_Creative_Commons_Reconeixement-Compartir_Igual_4.0_No_adaptada" title="Viquipèdia:Text de la llicència de Creative Commons Reconeixement-Compartir Igual 4.0 No adaptada"> Llicència de Creative Commons Reconeixement i Compartir-Igual</a>; es poden aplicar termes addicionals. 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