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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="Witryna"> <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="Spis treści" 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">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</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">Początek</div> </a> </li> <li id="toc-Równania" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Równania"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Równania</span> </div> </a> <ul id="toc-Równania-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Własności" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Własności"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Własności</span> </div> </a> <ul id="toc-Własności-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trochoida" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trochoida"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Trochoida</span> </div> </a> <ul id="toc-Trochoida-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-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="Spis treści" 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="Przełącz stan spisu treści" > <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">Przełącz stan spisu treści</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">Cykloida</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="Przejdź do artykułu w innym języku. Treść dostępna w 48 językach" > <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-48" 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">48 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%88%D9%8A%D8%B1%D9%8A" title="دويري – arabski" lang="ar" hreflang="ar" data-title="دويري" data-language-autonym="العربية" data-language-local-name="arabski" 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%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – bułgarski" lang="bg" hreflang="bg" data-title="Циклоида" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cicloide" title="Cicloide – kataloński" lang="ca" hreflang="ca" data-title="Cicloide" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – czuwaski" lang="cv" hreflang="cv" data-title="Циклоида" data-language-autonym="Чӑвашла" data-language-local-name="czuwaski" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Cykloida" title="Cykloida – czeski" lang="cs" hreflang="cs" data-title="Cykloida" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Cykloide" title="Cykloide – duński" lang="da" hreflang="da" data-title="Cykloide" data-language-autonym="Dansk" data-language-local-name="duński" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zykloide" title="Zykloide – niemiecki" lang="de" hreflang="de" data-title="Zykloide" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ts%C3%BCkloid" title="Tsükloid – estoński" lang="et" hreflang="et" data-title="Tsükloid" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Cycloid" title="Cycloid – angielski" lang="en" hreflang="en" data-title="Cycloid" data-language-autonym="English" data-language-local-name="angielski" 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/Cicloide" title="Cicloide – hiszpański" lang="es" hreflang="es" data-title="Cicloide" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Cikloido" title="Cikloido – esperanto" lang="eo" hreflang="eo" data-title="Cikloido" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zikloide" title="Zikloide – baskijski" lang="eu" hreflang="eu" data-title="Zikloide" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%DA%86%D8%B1%D8%AE%E2%80%8C%D8%B2%D8%A7%D8%AF" title="چرخ‌زاد – perski" lang="fa" hreflang="fa" data-title="چرخ‌زاد" data-language-autonym="فارسی" data-language-local-name="perski" 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/Cyclo%C3%AFde" title="Cycloïde – francuski" lang="fr" hreflang="fr" data-title="Cycloïde" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ciogl%C3%B3ideach" title="Cioglóideach – irlandzki" lang="ga" hreflang="ga" data-title="Cioglóideach" data-language-autonym="Gaeilge" data-language-local-name="irlandzki" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Cicloide" title="Cicloide – galicyjski" lang="gl" hreflang="gl" data-title="Cicloide" data-language-autonym="Galego" data-language-local-name="galicyjski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%82%AC%EC%9D%B4%ED%81%B4%EB%A1%9C%EC%9D%B4%EB%93%9C" title="사이클로이드 – koreański" lang="ko" hreflang="ko" data-title="사이클로이드" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%9C" title="चक्रज – hindi" lang="hi" hreflang="hi" data-title="चक्रज" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Cikloida" title="Cikloida – chorwacki" lang="hr" hreflang="hr" data-title="Cikloida" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Cikloido" title="Cikloido – ido" lang="io" hreflang="io" data-title="Cikloido" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hj%C3%B3lferill" title="Hjólferill – islandzki" lang="is" hreflang="is" data-title="Hjólferill" data-language-autonym="Íslenska" data-language-local-name="islandzki" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Cicloide" title="Cicloide – włoski" lang="it" hreflang="it" data-title="Cicloide" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%90%D7%99%D7%93%D7%94" title="ציקלואידה – hebrajski" lang="he" hreflang="he" data-title="ציקלואידה" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – kazachski" lang="kk" hreflang="kk" data-title="Циклоида" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ciklo%C4%ABda" title="Cikloīda – łotewski" lang="lv" hreflang="lv" data-title="Cikloīda" data-language-autonym="Latviešu" data-language-local-name="łotewski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Cikloid%C4%97" title="Cikloidė – litewski" lang="lt" hreflang="lt" data-title="Cikloidė" data-language-autonym="Lietuvių" data-language-local-name="litewski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ciklois" title="Ciklois – węgierski" lang="hu" hreflang="hu" data-title="Ciklois" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%9A%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%BE%E0%B4%AD%E0%B4%82" title="ചക്രാഭം – malajalam" lang="ml" hreflang="ml" data-title="ചക്രാഭം" data-language-autonym="മലയാളം" data-language-local-name="malajalam" 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/Cyclo%C3%AFde" title="Cycloïde – niderlandzki" lang="nl" hreflang="nl" data-title="Cycloïde" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B5%E3%82%A4%E3%82%AF%E3%83%AD%E3%82%A4%E3%83%89" title="サイクロイド – japoński" lang="ja" hreflang="ja" data-title="サイクロイド" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Sykloide" title="Sykloide – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Sykloide" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Sykloide" title="Sykloide – norweski (nynorsk)" lang="nn" hreflang="nn" data-title="Sykloide" data-language-autonym="Norsk nynorsk" data-language-local-name="norweski (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sicl%C3%B2ida" title="Siclòida – piemoncki" lang="pms" hreflang="pms" data-title="Siclòida" data-language-autonym="Piemontèis" data-language-local-name="piemoncki" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Cicloide" title="Cicloide – portugalski" lang="pt" hreflang="pt" data-title="Cicloide" data-language-autonym="Português" data-language-local-name="portugalski" 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/Cicloid%C4%83" title="Cicloidă – rumuński" lang="ro" hreflang="ro" data-title="Cicloidă" data-language-autonym="Română" data-language-local-name="rumuński" 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%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – rosyjski" lang="ru" hreflang="ru" data-title="Циклоида" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – jakucki" lang="sah" hreflang="sah" data-title="Циклоида" data-language-autonym="Саха тыла" data-language-local-name="jakucki" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Cykloida" title="Cykloida – słowacki" lang="sk" hreflang="sk" data-title="Cykloida" data-language-autonym="Slovenčina" data-language-local-name="słowacki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Cikloida" title="Cikloida – słoweński" lang="sl" hreflang="sl" data-title="Cikloida" data-language-autonym="Slovenščina" data-language-local-name="słoweński" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – serbski" lang="sr" hreflang="sr" data-title="Циклоида" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Sykloidi" title="Sykloidi – fiński" lang="fi" hreflang="fi" data-title="Sykloidi" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Cykloid" title="Cykloid – szwedzki" lang="sv" hreflang="sv" data-title="Cykloid" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%B3%E0%AF%8D%E0%AE%B3%E0%AE%BF%E0%AE%AF%E0%AF%81%E0%AE%B0%E0%AF%81" title="வட்டப்புள்ளியுரு – tamilski" lang="ta" hreflang="ta" data-title="வட்டப்புள்ளியுரு" data-language-autonym="தமிழ்" data-language-local-name="tamilski" 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%84%E0%B8%8B%E0%B8%84%E0%B8%A5%E0%B8%AD%E0%B8%A2%E0%B8%94%E0%B9%8C" title="ไซคลอยด์ – tajski" lang="th" hreflang="th" data-title="ไซคลอยด์" data-language-autonym="ไทย" data-language-local-name="tajski" 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/Sikloid" title="Sikloid – turecki" lang="tr" hreflang="tr" data-title="Sikloid" data-language-autonym="Türkçe" data-language-local-name="turecki" 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%A6%D0%B8%D0%BA%D0%BB%D0%BE%D1%97%D0%B4%D0%B0" title="Циклоїда – ukraiński" lang="uk" hreflang="uk" data-title="Циклоїда" data-language-autonym="Українська" data-language-local-name="ukraiński" 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/%E6%91%86%E7%BA%BF" title="摆线 – chiński" lang="zh" hreflang="zh" data-title="摆线" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%A6%D0%B8%D0%BA%D0%BB%D0%BE%D0%B8%D0%B4%D0%B0" title="Циклоида – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Циклоида" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</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/Q204072#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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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">Narzędzia</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">Narzędzia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button 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mw-list-item"><a href="/w/index.php?title=Cykloida&action=history"><span>Wyświetl historię</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ogólne </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Specjalna:Linkuj%C4%85ce/Cykloida" title="Pokaż listę wszystkich stron linkujących do tej strony [j]" accesskey="j"><span>Linkujące</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Specjalna:Zmiany_w_linkowanych/Cykloida" rel="nofollow" title="Ostatnie zmiany w stronach, do których ta strona linkuje [k]" accesskey="k"><span>Zmiany w linkowanych</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pl.wikipedia.org/wiki/Wikipedia:Prześlij_plik" title="Prześlij pliki [u]" accesskey="u"><span>Prześlij plik</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Specjalna:Strony_specjalne" title="Lista wszystkich stron specjalnych [q]" accesskey="q"><span>Strony specjalne</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Cykloida&oldid=74201352" title="Stały link do tej wersji tej strony"><span>Link do tej wersji</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Cykloida&action=info" title="Więcej informacji na temat tej strony"><span>Informacje o tej stronie</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Cytuj&page=Cykloida&id=74201352&wpFormIdentifier=titleform" title="Informacja o tym jak należy cytować tę stronę"><span>Cytowanie tego artykułu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Skr%C3%B3%C4%87_adres_URL&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FCykloida"><span>Zobacz skrócony adres URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Kod_QR&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FCykloida"><span>Pobierz kod QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Drukuj lub eksportuj </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=Specjalna:Ksi%C4%85%C5%BCka&bookcmd=book_creator&referer=Cykloida"><span>Utwórz książkę</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Specjalna:DownloadAsPdf&page=Cykloida&action=show-download-screen"><span>Pobierz jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Cykloida&printable=yes" title="Wersja do wydruku [p]" accesskey="p"><span>Wersja do druku</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"> W innych projektach </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:Cycloid" hreflang="en"><span>Wikimedia 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/Q204072" title="Link do powiązanego elementu w repozytorium danych [g]" accesskey="g"><span>Element Wikidanych</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="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <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 vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Wygląd</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</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">Z Wikipedii, wolnej encyklopedii</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="pl" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Plik:Cycloid_f.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Cycloid_f.gif/240px-Cycloid_f.gif" decoding="async" width="240" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Cycloid_f.gif/360px-Cycloid_f.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/69/Cycloid_f.gif 2x" data-file-width="400" data-file-height="200" /></a><figcaption>Zakreślanie cykloidy</figcaption></figure> <p><b>Cykloida</b> – <a href="/wiki/Krzywa" title="Krzywa">krzywa</a>, jaką zakreśla punkt leżący na obwodzie koła, które toczy się bez poślizgu po prostej<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup>. Cykloidę można narysować za pomocą <a href="/w/index.php?title=Cykloidograf&action=edit&redlink=1" class="new" title="Cykloidograf (strona nie istnieje)">cykloidografu</a><sup id="cite_ref-epwn-cg_2-0" class="reference"><a href="#cite_note-epwn-cg-2">[2]</a></sup>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Równania"><span id="R.C3.B3wnania"></span>Równania</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cykloida&veaction=edit&section=1" title="Edytuj sekcję: Równania" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cykloida&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Równania"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Plik:Cycloid04.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cycloid04.png/240px-Cycloid04.png" decoding="async" width="240" height="61" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cycloid04.png/360px-Cycloid04.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Cycloid04.png/480px-Cycloid04.png 2x" data-file-width="743" data-file-height="190" /></a><figcaption>Cykloida</figcaption></figure> <p>Cykloida opisana jest <a href="/wiki/R%C3%B3wnanie_parametryczne" title="Równanie parametryczne">równaniami parametrycznymi</a> postaci<sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup>: </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=r{\begin{pmatrix}t-\sin t\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>t</mi> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=r{\begin{pmatrix}t-\sin t\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c99ea1b345cbb4e043114d5ca483dbd9ce770ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.8ex; height:2.843ex;" alt="{\displaystyle x=r{\begin{pmatrix}t-\sin t\end{pmatrix}}}"></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=r{\begin{pmatrix}1-\cos t\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=r{\begin{pmatrix}1-\cos t\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1994fdaf361ecce795f04672c52c994a3d427958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.851ex; height:2.843ex;" alt="{\displaystyle y=r{\begin{pmatrix}1-\cos t\end{pmatrix}},}"></span></dd></dl> <p>gdzie: </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 t\in \mathbb {R} ,\ r>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>r</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} ,\ r>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/351a823a8deb35456970ff44d7675c185e0f57b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.929ex; height:2.509ex;" alt="{\displaystyle t\in \mathbb {R} ,\ r>0.}"></span></dd></dl> <p>Rozwiązując równania ogólne dla <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}"></span> otrzymuje się: </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\pi rk\pm \left(r\,\arccos \left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>k</mi> <mo>±</mo> <mrow> <mo>(</mo> <mrow> <mi>r</mi> <mspace width="thinmathspace" /> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>r</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=2\pi rk\pm \left(r\,\arccos \left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5abe59be9f0b3514f3370ff9122ebe73ecaf30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.979ex; height:4.843ex;" alt="{\displaystyle x=2\pi rk\pm \left(r\,\arccos \left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}}\right),}"></span></dd></dl> <p>gdzie: </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 k\in \mathbb {Z} ,\ r>0,\ 0\leqslant y\leqslant 2r.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>r</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>⩽</mo> <mi>y</mi> <mo>⩽</mo> <mn>2</mn> <mi>r</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} ,\ r>0,\ 0\leqslant y\leqslant 2r.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d361ff16253e5e549f66b0f2014d49af7e0c877" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.514ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {Z} ,\ r>0,\ 0\leqslant y\leqslant 2r.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Własności"><span id="W.C5.82asno.C5.9Bci"></span>Własności</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cykloida&veaction=edit&section=2" title="Edytuj sekcję: Własności" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cykloida&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Własności"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cykloida jest też związana z zagadnieniem: </p> <ul><li>krzywej najkrótszego spadku (<a href="/wiki/Brachistochrona" title="Brachistochrona">brachistochrony</a>) będącej fragmentem łuku cykloidy,</li> <li>krzywej będącej odwróconą cykloidą (<a href="/wiki/Tautochrona_(fizyka)" title="Tautochrona (fizyka)">tautochroną</a>), po której masa punktowa stacza się do najniższego punktu krzywej w takim samym czasie, niezależnie od punktu startowego na tej krzywej.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Trochoida">Trochoida</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cykloida&veaction=edit&section=3" title="Edytuj sekcję: Trochoida" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cykloida&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Trochoida"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uogólnieniem zwykłej cykloidy jest trochoida (<a href="/wiki/Gr." class="mw-redirect" title="Gr.">gr.</a> <i>trochós</i> – koło, <i>eídos</i> – kształt)<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup>. </p><p>Równania ogólne postaci<sup id="cite_ref-skrócona_5-0" class="reference"><a href="#cite_note-skrócona-5">[5]</a></sup><sup id="cite_ref-wydłużona_6-0" class="reference"><a href="#cite_note-wydłużona-6">[6]</a></sup>: </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=rt-c\cdot \sin t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>r</mi> <mi>t</mi> <mo>−</mo> <mi>c</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=rt-c\cdot \sin t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f6eceb3f5b7021eee671f245a010f3ac18b89ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.925ex; height:2.343ex;" alt="{\displaystyle x=rt-c\cdot \sin t}"></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=r-c\cdot \cos t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mo>−</mo> <mi>c</mi> <mo>⋅</mo> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=r-c\cdot \cos t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80fce818bdd9d1c350ee5039013b9c22855f5bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.814ex; height:2.343ex;" alt="{\displaystyle y=r-c\cdot \cos t,}"></span></dd></dl> <p>gdzie: </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 t\in \mathbb {R} ,\ r>0,\ c>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>r</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mi>c</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {R} ,\ r>0,\ c>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a460c90cf436a0931114a43399afee86093c6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.812ex; height:2.509ex;" alt="{\displaystyle t\in \mathbb {R} ,\ r>0,\ c>0.}"></span></dd></dl> <p>Zależność odległości <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> punktu zakreślającego krzywą od środka toczącego się koła i promienia <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> tego koła jest następująca: </p> <ul><li>dla <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 c<r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo><</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c<r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689988b38e5bd11a9221ec0d2056352c3f490808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.154ex; height:1.843ex;" alt="{\displaystyle c<r}"></span> <b>trochoidę skróconą</b>, zakreślaną przez ustalony punkt leżący wewnątrz toczącego się koła<sup id="cite_ref-skrócona_5-1" class="reference"><a href="#cite_note-skrócona-5">[5]</a></sup> (linia czerwona na poniższym rysunku),</li> <li>dla <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 c>r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b40abf21c427fcd98b4d0d5a09d7d816f859900" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.154ex; height:1.843ex;" alt="{\displaystyle c>r}"></span> <b>trochoidę wydłużoną</b>, zakreślaną przez ustalony punkt leżący na zewnątrz koła<sup id="cite_ref-wydłużona_6-1" class="reference"><a href="#cite_note-wydłużona-6">[6]</a></sup> (linia niebieska).</li> <li>dla <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 c=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22b1b2e307ba7fe946df74832eaf2d7e1616527" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.154ex; height:1.676ex;" alt="{\displaystyle c=r}"></span> zwykłą <b>cykloidę,</b> zakreślaną przez punkt na brzegu koła (linia zielona).</li></ul> <style data-mw-deduplicate="TemplateStyles:r70384462">.mw-parser-output .szablon-duza-grafika{display:flex}@media(max-width:720px){body.skin-minerva .mw-parser-output .szablon-duza-grafika{display:block}}</style><div class="thumb szablon-duza-grafika" style="max-width:910px; margin-left: auto; margin-right:auto;"> <div class="thumbinner"><div class="noresize" style="overflow:auto; overflow-y:display; overflow-x:auto;"><span typeof="mw:File"><a href="/wiki/Plik:Cykloidy.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cykloidy.svg/900px-Cykloidy.svg.png" decoding="async" width="900" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cykloidy.svg/1350px-Cykloidy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cykloidy.svg/1800px-Cykloidy.svg.png 2x" data-file-width="1200" data-file-height="200" /></a></span></div> </div></div> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Plik:TrohoidH0,8.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/TrohoidH0%2C8.gif/240px-TrohoidH0%2C8.gif" decoding="async" width="240" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/TrohoidH0%2C8.gif/360px-TrohoidH0%2C8.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/TrohoidH0%2C8.gif/480px-TrohoidH0%2C8.gif 2x" data-file-width="640" data-file-height="320" /></a><figcaption>Zakreślanie trochoidy skróconej, c<r</figcaption></figure> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Plik:TrohoidH1,25.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/TrohoidH1%2C25.gif/240px-TrohoidH1%2C25.gif" decoding="async" width="240" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/TrohoidH1%2C25.gif/360px-TrohoidH1%2C25.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4b/TrohoidH1%2C25.gif 2x" data-file-width="400" data-file-height="200" /></a><figcaption>Zakreślanie trochoidy wydłużonej, c>r</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cykloida&veaction=edit&section=4" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cykloida&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint plainlinks"> <tbody><tr> <td style="text-align: center; font-weight: bold;">Informacje w <a href="/wiki/Wikipedia:Projekty_siostrzane" title="Wikipedia:Projekty siostrzane">projektach siostrzanych</a></td> </tr> <tr> <td><span style="margin:1px"><span typeof="mw:File"><a href="/wiki/Plik:Commons-logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/13px-Commons-logo.svg.png" decoding="async" width="13" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/26px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></span>  <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Cycloid?uselang=pl"><b>Multimedia</b></a></span> w <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></td> </tr><tr> <td><span typeof="mw:File"><a href="/wiki/Plik:WiktionaryPl_nodesc.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/15px-WiktionaryPl_nodesc.svg.png" decoding="async" width="15" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/23px-WiktionaryPl_nodesc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/30px-WiktionaryPl_nodesc.svg.png 2x" data-file-width="122" data-file-height="117" /></a></span>  <b><a href="https://pl.wiktionary.org/wiki/cykloida" class="extiw" title="wikt:cykloida">Definicje słownikowe</a></b> w <a href="/wiki/Wikis%C5%82ownik" title="Wikisłownik">Wikisłowniku</a></td> </tr> </tbody></table> <ul><li><a href="/wiki/Epicykloida" title="Epicykloida">epicykloida</a></li> <li><a href="/wiki/Hipocykloida" title="Hipocykloida">hipocykloida</a></li> <li><a href="/wiki/Traktrysa" title="Traktrysa">traktrysa</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Cykloida&veaction=edit&section=5" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Cykloida&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><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"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3888681"><i>Cykloida</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-07-29]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=Cykloida&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3888681" style="display:none"> </span>.</cite></span> </li> <li id="cite_note-epwn-cg-2"><span class="mw-cite-backlink"><a href="#cite_ref-epwn-cg_2-0">↑</a></span> <span class="reference-text"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3888682"><i>cykloidograf</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-10-16]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=cykloidograf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3888682" style="display:none"> </span>.</cite></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W." style="display:none">E.W.</span> </span><span class="cite-lastname">Weisstein</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W.">E.W.</span></span>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Cycloid.html"><i>Cycloid</i></a>, [w:] <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=bookitem&rft.aufirst=Eric+W.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.btitle=%5B%5BMathWorld%5D%5D&rft.atitle=Cycloid&rft.aulast=Weisstein&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCycloid.html" style="display:none"> </span> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>.</cite></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3989224"><i>trochoida</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2024-07-05]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=trochoida&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3989224" style="display:none"> </span>.</cite></span> </li> <li id="cite_note-skrócona-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-skrócona_5-0">a</a></sup> <sup><a href="#cite_ref-skrócona_5-1">b</a></sup></span> <span class="reference-text"><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W." style="display:none">E.W.</span> </span><span class="cite-lastname">Weisstein</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W.">E.W.</span></span>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/CurtateCycloid.html"><i>Curtate Cycloid</i></a>, [w:] <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=bookitem&rft.aufirst=Eric+W.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.btitle=%5B%5BMathWorld%5D%5D&rft.atitle=Curtate+Cycloid&rft.aulast=Weisstein&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCurtateCycloid.html" style="display:none"> </span> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>.</cite></span> </li> <li id="cite_note-wydłużona-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-wydłużona_6-0">a</a></sup> <sup><a href="#cite_ref-wydłużona_6-1">b</a></sup></span> <span class="reference-text"><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W." style="display:none">E.W.</span> </span><span class="cite-lastname">Weisstein</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Eric W.</span><span class="cite-name-initials" title="Eric W.">E.W.</span></span>, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ProlateCycloid.html"><i>Prolate Cycloid</i></a>, [w:] <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>, <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=bookitem&rft.aufirst=Eric+W.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.btitle=%5B%5BMathWorld%5D%5D&rft.atitle=Prolate+Cycloid&rft.aulast=Weisstein&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FProlateCycloid.html" style="display:none"> </span> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>.</cite></span> </li> </ol></div></div> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r74983602">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output table.navbox:not(.pionowy){width:100%}.mw-parser-output .navbox+.navbox{border-top:0;margin-top:0}.mw-parser-output .navbox.pionowy{width:250px;float:right;clear:right;margin:0 0 0.4em 1.4em}.mw-parser-output .navbox.pionowy 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Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Krzywe_cykliczne&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Krzywa_cykliczna" title="Krzywa cykliczna">Krzywe cykliczne</a> – utworzone <a href="/wiki/Toczenie" title="Toczenie">toczeniem</a></div><div class="mw-collapsible-content flex"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row">po <a href="/wiki/Prosta" title="Prosta">prostej</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a class="mw-selflink selflink">cykloida</a> <ul><li><a href="/wiki/Brachistochrona" title="Brachistochrona">brachistochrona</a></li> <li><a href="/wiki/Tautochrona_(fizyka)" title="Tautochrona (fizyka)">tautochrona</a></li></ul></li> <li><a class="mw-selflink-fragment" href="#Uogólnienia">cykloida skrócona</a></li> <li><a class="mw-selflink-fragment" href="#Uogólnienia">cykloida wydłużona</a></li></ul> </td></tr><tr class="a2"><th class="navbox-group opis" scope="row">po <a href="/wiki/Okr%C4%85g" title="Okrąg">okręgu</a> –<br /><a href="/wiki/Trochoida" title="Trochoida">trochoidy</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a2_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Epitrochoida" title="Epitrochoida">epitrochoidy</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Epicykloida" title="Epicykloida">epicykloidy</a> <ul><li><a href="/wiki/Kardioida" title="Kardioida">kardioida</a></li></ul></li></ul> </td></tr><tr class="a2_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Hipotrochoida" title="Hipotrochoida">hipotrochoidy</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Hipocykloida" title="Hipocykloida">hipocykloidy</a> <ul><li><a href="/wiki/Hipocykloida#Przykłady" title="Hipocykloida">deltoida</a></li> <li><a href="/wiki/Hipocykloida#Przykłady" title="Hipocykloida">asteroida</a></li></ul></li></ul> </td></tr></tbody></table></td></tr><tr class="a3"><th class="navbox-group opis" scope="row"><a href="/wiki/Twierdzenie" title="Twierdzenie">twierdzenia</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Twierdzenie_Kopernika" title="Twierdzenie Kopernika">Kopernika</a></li></ul> </td></tr><tr class="a4"><th class="navbox-group opis" scope="row"><a href="/wiki/Narz%C4%99dzie" title="Narzędzie">narzędzia</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/w/index.php?title=Cykloidograf&action=edit&redlink=1" class="new" title="Cykloidograf (strona nie istnieje)">cykloidograf</a></li> <li><a href="/wiki/Spirograf_(przyrz%C4%85d)" title="Spirograf (przyrząd)">spirograf</a></li></ul> </td></tr></tbody></table><div class="navbox-after after"> <p><span typeof="mw:File"><a href="/wiki/Plik:Tidens_naturl%C3%A6re_fig20.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tidens_naturl%C3%A6re_fig20.png/200px-Tidens_naturl%C3%A6re_fig20.png" decoding="async" width="200" height="54" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tidens_naturl%C3%A6re_fig20.png/300px-Tidens_naturl%C3%A6re_fig20.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Tidens_naturl%C3%A6re_fig20.png/400px-Tidens_naturl%C3%A6re_fig20.png 2x" data-file-width="1164" data-file-height="312" /></a></span> </p> </div></div></div> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r74983602"><ul class="tnavbar noprint plainlinks hlist"><li><a href="/wiki/Szablon:Funkcje_elementarne" title="Szablon:Funkcje elementarne"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/w/index.php?title=Dyskusja_szablonu:Funkcje_elementarne&action=edit&redlink=1" class="new" title="Dyskusja szablonu:Funkcje elementarne (strona nie istnieje)"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Funkcje_elementarne&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Funkcje_elementarne" title="Funkcje elementarne">Funkcje elementarne</a></div><div class="mw-collapsible-content"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row"><a href="/wiki/Funkcja_algebraiczna" title="Funkcja algebraiczna">algebraiczne</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Funkcja_wymierna" title="Funkcja wymierna">wymierne</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Funkcja_homograficzna" title="Funkcja homograficzna">homografie</a> <ul><li><a href="/wiki/Liczba_odwrotna" title="Liczba odwrotna">liczba odwrotna</a></li></ul></li> <li><a href="/wiki/Wielomian" title="Wielomian">wielomianowe</a> <ul><li><a href="/wiki/Funkcja_sta%C5%82a" title="Funkcja stała">stałe</a></li> <li><a href="/wiki/Funkcja_liniowa" title="Funkcja liniowa">liniowe</a></li> <li><a href="/wiki/Funkcja_kwadratowa" title="Funkcja kwadratowa">kwadratowe</a></li> <li><a href="/w/index.php?title=Wielomian_stopnia_trzeciego&action=edit&redlink=1" class="new" title="Wielomian stopnia trzeciego (strona nie istnieje)">stopnia trzeciego</a></li> <li><a href="/wiki/Wielomian_stopnia_czwartego" title="Wielomian stopnia czwartego">stopnia czwartego</a></li></ul></li></ul> </td></tr><tr class="a1_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Funkcja_pot%C4%99gowa" title="Funkcja potęgowa">potęgowe</a> o wykładniku<br /><a href="/wiki/Liczby_wymierne" title="Liczby wymierne">wymiernym</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Pierwiastkowanie" title="Pierwiastkowanie">pierwiastek arytmetyczny</a> <ul><li><a href="/wiki/Pierwiastek_kwadratowy" title="Pierwiastek kwadratowy">pierwiastek kwadratowy</a></li> <li><a href="/wiki/Pierwiastek_sze%C5%9Bcienny" title="Pierwiastek sześcienny">pierwiastek sześcienny</a></li></ul></li></ul> </td></tr><tr class="a1_3"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Warto%C5%9B%C4%87_bezwzgl%C4%99dna" title="Wartość bezwzględna">wartość bezwzględna</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a2"><th class="navbox-group opis" scope="row"><a href="/wiki/Funkcja_algebraiczna" title="Funkcja algebraiczna">przestępne</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a2_1"><th class="navbox-group opis" scope="row">definiowane<br /><a href="/wiki/Pot%C4%99gowanie" title="Potęgowanie">potęgowaniem</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Funkcja_pot%C4%99gowa" title="Funkcja potęgowa">potęgowe</a> o wykładniku <a href="/wiki/Liczby_niewymierne" title="Liczby niewymierne">niewymiernym</a></li> <li><a href="/wiki/Funkcja_wyk%C5%82adnicza" title="Funkcja wykładnicza">wykładnicze</a></li> <li><a href="/wiki/Logarytm" title="Logarytm">logarytmy</a> <ul><li><a href="/wiki/Funkcja_logarytmiczna" title="Funkcja logarytmiczna">funkcje logarytmiczne</a></li> <li><a href="/wiki/Logarytm_binarny" title="Logarytm binarny">logarytm binarny</a></li> <li><a href="/wiki/Logarytm_dziesi%C4%99tny" title="Logarytm dziesiętny">logarytm dziesiętny</a></li> <li><a href="/wiki/Logarytm_naturalny" title="Logarytm naturalny">logarytm naturalny</a></li></ul></li> <li><a href="/wiki/Funkcje_hiperboliczne" title="Funkcje hiperboliczne">hiperboliczne</a></li> <li><a href="/wiki/Funkcje_hiperboliczne_odwrotne" title="Funkcje hiperboliczne odwrotne">area (polowe)</a></li> <li><a href="/wiki/Tetracja" title="Tetracja">tetracja</a></li></ul> </td></tr><tr class="a2_2"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Funkcje_trygonometryczne" title="Funkcje trygonometryczne">trygonometryczne</a></li> <li><a href="/wiki/Funkcje_cyklometryczne" title="Funkcje cyklometryczne">cyklometryczne (kołowe)</a></li> <li><a href="/wiki/Funkcja_sinc" title="Funkcja sinc">funkcja sinc</a></li> <li><a href="/wiki/Funkcja_Gudermanna" title="Funkcja Gudermanna">funkcja Gudermanna</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a3"><th class="navbox-group opis" scope="row"><a href="/wiki/Krzywa" title="Krzywa">krzywe</a> tworzące<br /><a href="/wiki/Wykres_funkcji" title="Wykres funkcji">wykresy</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a3_1"><th class="navbox-group opis" scope="row">funkcji algebraicznych</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Prosta" title="Prosta">prosta</a></li> <li><a href="/wiki/Krzywa_sto%C5%BCkowa" title="Krzywa stożkowa">stożkowe</a> <ul><li><a href="/wiki/Okr%C4%85g" title="Okrąg">okrąg</a></li> <li><a href="/wiki/P%C3%B3%C5%82okr%C4%85g" title="Półokrąg">półokrąg</a></li> <li><a href="/wiki/Elipsa" title="Elipsa">elipsa</a></li> <li><a href="/wiki/Parabola_(matematyka)" title="Parabola (matematyka)">parabola</a></li> <li><a href="/wiki/Hiperbola_(matematyka)" title="Hiperbola (matematyka)">hiperbola</a></li></ul></li> <li><a href="/wiki/Parabola_sze%C5%9Bcienna" title="Parabola sześcienna">parabola sześcienna</a></li> <li><a href="/wiki/Lok_Agnesi" title="Lok Agnesi">lok Agnesi</a></li> <li><a href="/wiki/Parabola_semikubiczna" title="Parabola semikubiczna">parabola semikubiczna</a></li></ul> </td></tr><tr class="a3_2"><th class="navbox-group opis" scope="row">funkcji przestępnych</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Krzywa_%C5%82a%C5%84cuchowa" title="Krzywa łańcuchowa">krzywa łańcuchowa</a></li> <li><a class="mw-selflink selflink">cykloida</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a4"><th class="navbox-group opis" scope="row">powiązane tematy</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Arytmetyka_elementarna" title="Arytmetyka elementarna">działania arytmetyczne</a></li> <li><a href="/wiki/Z%C5%82o%C5%BCenie_funkcji" title="Złożenie funkcji">złożenie funkcji</a></li> <li><a href="/wiki/Funkcja_odwrotna" title="Funkcja odwrotna">funkcja odwrotna</a></li> <li><a href="/wiki/Notacja_strza%C5%82kowa" title="Notacja strzałkowa">notacja strzałkowa</a></li> <li><a href="/wiki/Trygonometria" title="Trygonometria">trygonometria</a></li></ul> </td></tr></tbody></table></div></div> <style data-mw-deduplicate="TemplateStyles:r74016753">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-fehlt"><div><a href="/wiki/Kontrola_autorytatywna" title="Kontrola autorytatywna">Kontrola autorytatywna</a> (<span class="description"><a href="/wiki/Trochoida" title="Trochoida">trochoida</a></span>):</div><ul><li><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://lccn.loc.gov/sh85035084">sh85035084</a></span></li><li><a href="/wiki/Biblioteka_Narodowa_Izraela" title="Biblioteka Narodowa Izraela">J9U</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007538314405171">987007538314405171</a></span></li><li><a href="/wiki/Biblioteka_Narodowa_%C5%81otwy" title="Biblioteka Narodowa Łotwy">LNB</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://kopkatalogs.lv/F?func=direct&local_base=lnc10&doc_number=000149396&P_CON_LNG=ENG">000149396</a></span></li></ul></div><div class="normdaten-andere"><div><a href="/wiki/Encyklopedia_internetowa" title="Encyklopedia internetowa">Encyklopedie internetowe</a>:</div> <ul><li><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">PWN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3888681.html">3888681</a></span></li> <li><a href="/wiki/Encyklopedia_Britannica" title="Encyklopedia Britannica">Britannica</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/cycloid">topic/cycloid</a></span></li> <li><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Treccani</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/cicloide">cicloide</a></span></li> <li><a href="/wiki/Wielka_Encyklopedia_Rosyjska" title="Wielka Encyklopedia Rosyjska">БРЭ</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://old.bigenc.ru/text/4677105">4677105</a></span></li> <li><a href="/wiki/Store_norske_leksikon" title="Store norske leksikon">SNL</a>: <span class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=4342&url_prefix=https://snl.no/&id=sykloide">sykloide</a></span></li> <li><a href="/wiki/Visuotin%C4%97_lietuvi%C5%B3_enciklopedija" title="Visuotinė lietuvių enciklopedija">VLE</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.vle.lt/straipsnis/cikloide/">cikloide</a></span></li> <li><a href="https://www.wikidata.org/wiki/Property:P1296" class="extiw" title="d:Property:P1296">Catalana</a>: <span class="uid"><a rel="nofollow" class="external text" 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