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href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=cs.wikipedia.org&uselang=cs" class=""><span>Podpořte Wikipedii</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=Speci%C3%A1ln%C3%AD:Vytvo%C5%99it_%C3%BA%C4%8Det&returnto=Elipsa" title="Doporučujeme vytvořit si účet a přihlásit se, ovšem není to povinné" class=""><span>Vytvoření účtu</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=Speci%C3%A1ln%C3%AD:P%C5%99ihl%C3%A1sit&returnto=Elipsa" title="Doporučujeme vám přihlásit se, ovšem není to povinné. 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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Přihlášení</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"> Stránky pro odhlášené editory <a href="/wiki/N%C3%A1pov%C4%9Bda:%C3%9Avod" aria-label="Více informací o editování"><span>dozvědět se více</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/Speci%C3%A1ln%C3%AD:Moje_p%C5%99%C3%ADsp%C4%9Bvky" title="Seznam editací provedených z této IP adresy [y]" accesskey="y"><span>Příspěvky</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Moje_diskuse" title="Diskuse o editacích provedených z této IP adresy [n]" accesskey="n"><span>Diskuse</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="Projekt"> <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="Obsah" 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">Obsah</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</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">(úvod)</div> </a> </li> <li id="toc-Pojmosloví" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pojmosloví"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Pojmosloví</span> </div> </a> <ul id="toc-Pojmosloví-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rovnice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rovnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Rovnice</span> </div> </a> <button aria-controls="toc-Rovnice-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Rovnice</span> </button> <ul id="toc-Rovnice-sublist" class="vector-toc-list"> <li id="toc-Kanonický_tvar" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kanonický_tvar"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Kanonický tvar</span> </div> </a> <ul id="toc-Kanonický_tvar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vrcholová_rovnice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vrcholová_rovnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Vrcholová rovnice</span> </div> </a> <ul id="toc-Vrcholová_rovnice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rovnice_kuželosečky" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rovnice_kuželosečky"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Rovnice kuželosečky</span> </div> </a> <ul id="toc-Rovnice_kuželosečky-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametrické_rovnice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Parametrické_rovnice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Parametrické rovnice</span> </div> </a> <ul id="toc-Parametrické_rovnice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rovnice_v_polárních_souřadnicích" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rovnice_v_polárních_souřadnicích"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Rovnice v polárních souřadnicích</span> </div> </a> <ul id="toc-Rovnice_v_polárních_souřadnicích-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Geometrické_vlastnosti_elipsy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometrické_vlastnosti_elipsy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Geometrické vlastnosti elipsy</span> </div> </a> <ul id="toc-Geometrické_vlastnosti_elipsy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konstrukce_elipsy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konstrukce_elipsy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Konstrukce elipsy</span> </div> </a> <button aria-controls="toc-Konstrukce_elipsy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Konstrukce elipsy</span> </button> <ul id="toc-Konstrukce_elipsy-sublist" class="vector-toc-list"> <li id="toc-Zahradnická_konstrukce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zahradnická_konstrukce"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Zahradnická konstrukce</span> </div> </a> <ul id="toc-Zahradnická_konstrukce-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trojúhelníková_konstrukce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trojúhelníková_konstrukce"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Trojúhelníková konstrukce</span> </div> </a> <ul id="toc-Trojúhelníková_konstrukce-sublist" class="vector-toc-list"> <li id="toc-Postup" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Postup"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Postup</span> </div> </a> <ul id="toc-Postup-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proužková_konstrukce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proužková_konstrukce"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Proužková konstrukce</span> </div> </a> <ul id="toc-Proužková_konstrukce-sublist" class="vector-toc-list"> <li id="toc-Postup_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Postup_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Postup</span> </div> </a> <ul id="toc-Postup_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Příčková_konstrukce" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Příčková_konstrukce"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Příčková konstrukce</span> </div> </a> <ul id="toc-Příčková_konstrukce-sublist" class="vector-toc-list"> <li id="toc-Postup_3" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Postup_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Postup</span> </div> </a> <ul id="toc-Postup_3-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rytzova_konstrukce_os" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rytzova_konstrukce_os"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Rytzova konstrukce os</span> </div> </a> <ul id="toc-Rytzova_konstrukce_os-sublist" class="vector-toc-list"> <li id="toc-Postup_4" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Postup_4"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5.1</span> <span>Postup</span> </div> </a> <ul id="toc-Postup_4-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Elipsa_ve_fyzice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elipsa_ve_fyzice"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Elipsa ve fyzice</span> </div> </a> <ul id="toc-Elipsa_ve_fyzice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reference" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reference"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-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="Obsah" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Obsah" > <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="Přepnout obsah" > <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">Přepnout obsah</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">Elipsa</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="Přejděte k článku v jiném jazyce. Je dostupný v 97 jazycích" > <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-97" 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">97 jazyků</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/Ellips" title="Ellips – afrikánština" lang="af" hreflang="af" data-title="Ellips" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Ellipse" title="Ellipse – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Ellipse" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9E%E1%88%8B%E1%88%8B" title="ሞላላ – amharština" lang="am" hreflang="am" data-title="ሞላላ" data-language-autonym="አማርኛ" data-language-local-name="amharština" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%B7%D8%B9_%D9%86%D8%A7%D9%82%D8%B5" title="قطع ناقص – arabština" lang="ar" hreflang="ar" data-title="قطع ناقص" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Elipse" title="Elipse – asturština" lang="ast" hreflang="ast" data-title="Elipse" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ellips" title="Ellips – ázerbájdžánština" lang="az" hreflang="az" data-title="Ellips" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81" title="Эллипс – baškirština" lang="ba" hreflang="ba" data-title="Эллипс" data-language-autonym="Башҡортса" data-language-local-name="baškirština" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Elipsn" title="Elipsn – bavorština" lang="bar" hreflang="bar" data-title="Elipsn" data-language-autonym="Boarisch" data-language-local-name="bavorština" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D1%96%D0%BF%D1%81" title="Эліпс – běloruština" lang="be" hreflang="be" data-title="Эліпс" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%AD%D0%BB%D1%96%D0%BF%D1%81" title="Эліпс – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Эліпс" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%B0" title="Елипса – bulharština" lang="bg" hreflang="bg" data-title="Елипса" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="উপবৃত্ত – bengálština" lang="bn" hreflang="bn" data-title="উপবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Elipsa" title="Elipsa – bosenština" lang="bs" hreflang="bs" data-title="Elipsa" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/El%C2%B7lipse" title="El·lipse – katalánština" lang="ca" hreflang="ca" data-title="El·lipse" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cdo mw-list-item"><a href="https://cdo.wikipedia.org/wiki/Ti%C5%8D-u%C3%B2ng" title="Tiō-uòng – Mindong" lang="cdo" hreflang="cdo" data-title="Tiō-uòng" data-language-autonym="閩東語 / Mìng-dĕ̤ng-ngṳ̄" data-language-local-name="Mindong" class="interlanguage-link-target"><span>閩東語 / Mìng-dĕ̤ng-ngṳ̄</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DA%95%DA%AF%DB%95%DB%8C_%D9%86%D8%A7%D8%AA%DB%95%D9%88%D8%A7%D9%88" title="بڕگەی ناتەواو – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="بڕگەی ناتەواو" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81" title="Эллипс – čuvaština" lang="cv" hreflang="cv" data-title="Эллипс" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/El%C3%ADps" title="Elíps – velština" lang="cy" hreflang="cy" data-title="Elíps" data-language-autonym="Cymraeg" data-language-local-name="velština" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ellipse_(geometri)" title="Ellipse (geometri) – dánština" lang="da" hreflang="da" data-title="Ellipse (geometri)" data-language-autonym="Dansk" data-language-local-name="dánština" 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/Ellipse" title="Ellipse – němčina" lang="de" hreflang="de" data-title="Ellipse" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%88%CE%BB%CE%BB%CE%B5%CE%B9%CF%88%CE%B7_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Έλλειψη (γεωμετρία) – řečtina" lang="el" hreflang="el" data-title="Έλλειψη (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" 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/Ellipse" title="Ellipse – angličtina" lang="en" hreflang="en" data-title="Ellipse" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elipso_(matematiko)" title="Elipso (matematiko) – esperanto" lang="eo" hreflang="eo" data-title="Elipso (matematiko)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Elipse" title="Elipse – španělština" lang="es" hreflang="es" data-title="Elipse" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ellips" title="Ellips – estonština" lang="et" hreflang="et" data-title="Ellips" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Elipse" title="Elipse – baskičtina" lang="eu" hreflang="eu" data-title="Elipse" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="dobrý článek"><a href="https://fa.wikipedia.org/wiki/%D8%A8%DB%8C%D8%B6%DB%8C" title="بیضی – perština" lang="fa" hreflang="fa" data-title="بیضی" data-language-autonym="فارسی" data-language-local-name="perština" 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/Ellipsi" title="Ellipsi – finština" lang="fi" hreflang="fi" data-title="Ellipsi" data-language-autonym="Suomi" data-language-local-name="finština" 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/Ellipse_(math%C3%A9matiques)" title="Ellipse (mathématiques) – francouzština" lang="fr" hreflang="fr" data-title="Ellipse (mathématiques)" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Elips" title="Elips – fríština (severní)" lang="frr" hreflang="frr" data-title="Elips" data-language-autonym="Nordfriisk" data-language-local-name="fríština (severní)" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/%C3%89ilips" title="Éilips – irština" lang="ga" hreflang="ga" data-title="Éilips" data-language-autonym="Gaeilge" data-language-local-name="irština" 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/Elipse_(xeometr%C3%ADa)" title="Elipse (xeometría) – galicijština" lang="gl" hreflang="gl" data-title="Elipse (xeometría)" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%9C%D7%99%D7%A4%D7%A1%D7%94" title="אליפסה – hebrejština" lang="he" hreflang="he" data-title="אליפסה" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%80%E0%A4%B0%E0%A5%8D%E0%A4%98%E0%A4%B5%E0%A5%83%E0%A4%A4%E0%A5%8D%E0%A4%A4" title="दीर्घवृत्त – hindština" lang="hi" hreflang="hi" data-title="दीर्घवृत्त" data-language-autonym="हिन्दी" data-language-local-name="hindština" 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/Elipsa" title="Elipsa – chorvatština" lang="hr" hreflang="hr" data-title="Elipsa" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Elips" title="Elips – haitština" lang="ht" hreflang="ht" data-title="Elips" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitština" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ellipszis_(g%C3%B6rbe)" title="Ellipszis (görbe) – maďarština" lang="hu" hreflang="hu" data-title="Ellipszis (görbe)" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%AC%D5%AB%D5%BA%D5%BD" title="Էլիպս – arménština" lang="hy" hreflang="hy" data-title="Էլիպս" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Ellipse_(forma_geometric)" title="Ellipse (forma geometric) – interlingua" lang="ia" hreflang="ia" data-title="Ellipse (forma geometric)" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Elips" title="Elips – indonéština" lang="id" hreflang="id" data-title="Elips" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Elipso_(matematiko)" title="Elipso (matematiko) – ido" lang="io" hreflang="io" data-title="Elipso (matematiko)" 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/Sporbaugur" title="Sporbaugur – islandština" lang="is" hreflang="is" data-title="Sporbaugur" data-language-autonym="Íslenska" data-language-local-name="islandština" 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/Ellisse" title="Ellisse – italština" lang="it" hreflang="it" data-title="Ellisse" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A5%95%E5%86%86" title="楕円 – japonština" lang="ja" hreflang="ja" data-title="楕円" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Elips" title="Elips – javánština" lang="jv" hreflang="jv" data-title="Elips" data-language-autonym="Jawa" data-language-local-name="javánština" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%94%E1%83%9A%E1%83%98%E1%83%A4%E1%83%A1%E1%83%98" title="ელიფსი – gruzínština" lang="ka" hreflang="ka" data-title="ელიფსი" data-language-autonym="ქართული" data-language-local-name="gruzínština" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Aglay_(tusnakt)" title="Aglay (tusnakt) – kabylština" lang="kab" hreflang="kab" data-title="Aglay (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabylština" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81" title="Эллипс – kazaština" lang="kk" hreflang="kk" data-title="Эллипс" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9F%81%E1%9E%9B%E1%9E%B8%E1%9E%94" title="អេលីប – khmérština" lang="km" hreflang="km" data-title="អេលីប" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmérština" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B3%80%E0%B2%B0%E0%B3%8D%E0%B2%98%E0%B2%B5%E0%B3%83%E0%B2%A4%E0%B3%8D%E0%B2%A4" title="ದೀರ್ಘವೃತ್ತ – kannadština" lang="kn" hreflang="kn" data-title="ದೀರ್ಘವೃತ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannadština" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%83%80%EC%9B%90" title="타원 – korejština" lang="ko" hreflang="ko" data-title="타원" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81" title="Эллипс – kyrgyzština" lang="ky" hreflang="ky" data-title="Эллипс" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Ellipsis" title="Ellipsis – latina" lang="la" hreflang="la" data-title="Ellipsis" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ellips" title="Ellips – lucemburština" lang="lb" hreflang="lb" data-title="Ellips" data-language-autonym="Lëtzebuergesch" data-language-local-name="lucemburština" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Elips%C4%97" title="Elipsė – litevština" lang="lt" hreflang="lt" data-title="Elipsė" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elipse" title="Elipse – lotyština" lang="lv" hreflang="lv" data-title="Elipse" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%B0" title="Елипса – makedonština" lang="mk" hreflang="mk" data-title="Елипса" data-language-autonym="Македонски" data-language-local-name="makedonština" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A4%82%E0%A4%AC%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%81%E0%A4%B3" title="लंबवर्तुळ – maráthština" lang="mr" hreflang="mr" data-title="लंबवर्तुळ" data-language-autonym="मराठी" data-language-local-name="maráthština" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Elips" title="Elips – malajština" lang="ms" hreflang="ms" data-title="Elips" data-language-autonym="Bahasa Melayu" data-language-local-name="malajština" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ellips_(wiskunde)" title="Ellips (wiskunde) – nizozemština" lang="nl" hreflang="nl" data-title="Ellips (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ellipse" title="Ellipse – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Ellipse" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ellipse" title="Ellipse – norština (bokmål)" lang="nb" hreflang="nb" data-title="Ellipse" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Ellipsa" title="Ellipsa – okcitánština" lang="oc" hreflang="oc" data-title="Ellipsa" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Okokkee" title="Okokkee – oromština" lang="om" hreflang="om" data-title="Okokkee" data-language-autonym="Oromoo" data-language-local-name="oromština" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Elipsa" title="Elipsa – polština" lang="pl" hreflang="pl" data-title="Elipsa" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Eliss" title="Eliss – piemonština" lang="pms" hreflang="pms" data-title="Eliss" data-language-autonym="Piemontèis" data-language-local-name="piemonština" 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/Elipse" title="Elipse – portugalština" lang="pt" hreflang="pt" data-title="Elipse" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Lunp%27u" title="Lunp'u – kečuánština" lang="qu" hreflang="qu" data-title="Lunp'u" data-language-autonym="Runa Simi" data-language-local-name="kečuánština" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Elips%C4%83" title="Elipsă – rumunština" lang="ro" hreflang="ro" data-title="Elipsă" data-language-autonym="Română" data-language-local-name="rumunština" 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%BB%D0%BB%D0%B8%D0%BF%D1%81" title="Эллипс – ruština" lang="ru" hreflang="ru" data-title="Эллипс" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%B0" title="Елипса – rusínština" lang="rue" hreflang="rue" data-title="Елипса" data-language-autonym="Русиньскый" data-language-local-name="rusínština" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Ellissi" title="Ellissi – sicilština" lang="scn" hreflang="scn" data-title="Ellissi" data-language-autonym="Sicilianu" data-language-local-name="sicilština" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Elipsa" title="Elipsa – srbochorvatština" lang="sh" hreflang="sh" data-title="Elipsa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ellipse" title="Ellipse – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ellipse" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elipsa" title="Elipsa – slovenština" lang="sk" hreflang="sk" data-title="Elipsa" data-language-autonym="Slovenčina" data-language-local-name="slovenština" 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/Elipsa" title="Elipsa – slovinština" lang="sl" hreflang="sl" data-title="Elipsa" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Denderende" title="Denderende – šonština" lang="sn" hreflang="sn" data-title="Denderende" data-language-autonym="ChiShona" data-language-local-name="šonština" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Qabaal" title="Qabaal – somálština" lang="so" hreflang="so" data-title="Qabaal" data-language-autonym="Soomaaliga" data-language-local-name="somálština" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Elipsi" title="Elipsi – albánština" lang="sq" hreflang="sq" data-title="Elipsi" data-language-autonym="Shqip" data-language-local-name="albánština" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%B0" title="Елипса – srbština" lang="sr" hreflang="sr" data-title="Елипса" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ellips_(matematik)" title="Ellips (matematik) – švédština" lang="sv" hreflang="sv" data-title="Ellips (matematik)" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Duaradufu" title="Duaradufu – svahilština" lang="sw" hreflang="sw" data-title="Duaradufu" data-language-autonym="Kiswahili" data-language-local-name="svahilština" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%80%E0%AE%B3%E0%AF%8D%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="நீள்வட்டம் – tamilština" lang="ta" hreflang="ta" data-title="நீள்வட்டம்" data-language-autonym="தமிழ்" data-language-local-name="tamilština" 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%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B8%87%E0%B8%A3%E0%B8%B5" title="รูปวงรี – thajština" lang="th" hreflang="th" data-title="รูปวงรี" data-language-autonym="ไทย" data-language-local-name="thajština" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Elipse" title="Elipse – tagalog" lang="tl" hreflang="tl" data-title="Elipse" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elips" title="Elips – turečtina" lang="tr" hreflang="tr" data-title="Elips" data-language-autonym="Türkçe" data-language-local-name="turečtina" 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%BB%D1%96%D0%BF%D1%81" title="Еліпс – ukrajinština" lang="uk" hreflang="uk" data-title="Еліпс" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%DB%8C%D8%B6%DB%81" title="بیضہ – urdština" lang="ur" hreflang="ur" data-title="بیضہ" data-language-autonym="اردو" data-language-local-name="urdština" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ellips_(chiziq)" title="Ellips (chiziq) – uzbečtina" lang="uz" hreflang="uz" data-title="Ellips (chiziq)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/E%C5%82ise" title="Ełise – benátština" lang="vec" hreflang="vec" data-title="Ełise" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Elip" title="Elip – vietnamština" lang="vi" hreflang="vi" data-title="Elip" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Elipse" title="Elipse – warajština" lang="war" hreflang="war" data-title="Elipse" data-language-autonym="Winaray" data-language-local-name="warajština" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%A9%A2%E5%9C%93" title="橢圓 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="橢圓" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A4%AD%E5%9C%86" title="椭圆 – čínština" lang="zh" hreflang="zh" data-title="椭圆" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%A9%A2%E5%9C%93" title="橢圓 – čínština (klasická)" lang="lzh" hreflang="lzh" data-title="橢圓" data-language-autonym="文言" data-language-local-name="čínština (klasická)" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Tn%CC%82g-ko-%C3%AE%E2%81%BF" title="Tn̂g-ko-îⁿ – čínština (dialekty Minnan)" lang="nan" hreflang="nan" data-title="Tn̂g-ko-îⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="čínština (dialekty Minnan)" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A9%A2%E5%9C%93_(%E5%B9%BE%E4%BD%95)" title="橢圓 (幾何) – kantonština" lang="yue" hreflang="yue" data-title="橢圓 (幾何)" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</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/Q40112#sitelinks-wikipedia" title="Editovat mezijazykové odkazy" class="wbc-editpage">Upravit odkazy</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="Jmenné prostory"> <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/Elipsa" title="Zobrazit obsahovou stránku [c]" accesskey="c"><span>Článek</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskuse:Elipsa" rel="discussion" title="Diskuse ke stránce [t]" accesskey="t"><span>Diskuse</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="Změnit variantu jazyka" > <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">čeština</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="Zobrazení"> <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/Elipsa"><span>Číst</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elipsa&veaction=edit" title="Editovat tuto stránku [v]" accesskey="v"><span>Editovat</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elipsa&action=edit" title="Editovat zdrojový kód této stránky [e]" accesskey="e"><span>Editovat zdroj</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elipsa&action=history" title="Starší verze této stránky. [h]" accesskey="h"><span>Zobrazit historii</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Nástroje ke stránce"> <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="Nástroje" > <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">Nástroje</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">Nástroje</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skrýt</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Další možnosti" > <div class="vector-menu-heading"> Akce </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/Elipsa"><span>Číst</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Elipsa&veaction=edit" title="Editovat tuto stránku [v]" accesskey="v"><span>Editovat</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Elipsa&action=edit" title="Editovat zdrojový kód této stránky [e]" accesskey="e"><span>Editovat zdroj</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Elipsa&action=history"><span>Zobrazit historii</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Obecné </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Co_odkazuje_na/Elipsa" title="Seznam všech wikistránek, které sem odkazují [j]" accesskey="j"><span>Odkazuje sem</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speci%C3%A1ln%C3%AD:Souvisej%C3%ADc%C3%AD_zm%C4%9Bny/Elipsa" rel="nofollow" title="Nedávné změny stránek, na které je odkazováno [k]" accesskey="k"><span>Související změny</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=cs" title="Nahrát obrázky či jiná multimédia [u]" accesskey="u"><span>Načíst soubor</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Elipsa&oldid=24719595" title="Trvalý odkaz na současnou verzi této stránky"><span>Trvalý odkaz</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Elipsa&action=info" title="Více informací o této stránce"><span>Informace o stránce</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:Citovat&page=Elipsa&id=24719595&wpFormIdentifier=titleform" title="Informace o tom, jak citovat tuto stránku"><span>Citovat stránku</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:UrlShortener&url=https%3A%2F%2Fcs.wikipedia.org%2Fwiki%2FElipsa"><span>Získat zkrácené URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:QrCode&url=https%3A%2F%2Fcs.wikipedia.org%2Fwiki%2FElipsa"><span>Stáhnout QR kód</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"> Tisk/export </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=Speci%C3%A1ln%C3%AD:Kniha&bookcmd=book_creator&referer=Elipsa"><span>Vytvořit knihu</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1ln%C3%AD:DownloadAsPdf&page=Elipsa&action=show-download-screen"><span>Stáhnout jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Elipsa&printable=yes" title="Tato stránka v podobě vhodné k tisku [p]" accesskey="p"><span>Verze k tisku</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"> Na jiných projektech </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/Ellipses" 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/Q40112" title="Odkaz na propojenou položku datového úložiště [g]" accesskey="g"><span>Položka Wikidat</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="Nástroje ke stránce"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Vzhled"> <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">Vzhled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skrýt</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 Wikipedie, otevřené encyklopedie</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="cs" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Elipse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Elipse.svg/300px-Elipse.svg.png" decoding="async" width="300" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Elipse.svg/450px-Elipse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Elipse.svg/600px-Elipse.svg.png 2x" data-file-width="658" data-file-height="516" /></a><figcaption>Elipsa</figcaption></figure> <div class="uvodni-upozorneni hatnote noprint">Tento článek je o pojmu z geometrie. O lingvistickém pojmu pojednává článek <a href="/wiki/Elipsa_(lingvistika)" title="Elipsa (lingvistika)">Elipsa (lingvistika)</a>.</div> <p><b>Elipsa</b> je uzavřená <a href="/wiki/K%C5%99ivka" title="Křivka">křivka</a> v <a href="/wiki/Rovina" title="Rovina">rovině</a>. Elipsu lze definovat jako množinu všech <a href="/wiki/Bod" title="Bod">bodů</a> v rovině, které mají stálý <a href="/wiki/Sou%C4%8Det" class="mw-redirect" title="Součet">součet</a> <a href="/wiki/Vzd%C3%A1lenost" title="Vzdálenost">vzdáleností</a> <i>2a</i> od dvou pevně daných bodů, tzv. <b>ohnisek</b> (v obrázku označeny <i>F<sub>1</sub>, F<sub>2</sub></i>; <i>|F<sub>1</sub>F<sub>2</sub>| < 2a</i>). Elipsa patří mezi <a href="/wiki/Ku%C5%BEelose%C4%8Dka" title="Kuželosečka">kuželosečky</a>, je to algebraická křivka 2. stupně. Velký praktický význam má v konstruktivní <a href="/wiki/Geometrie" title="Geometrie">geometrii</a>, protože vzniká jako průmět kružnice, jiné kuželosečky anebo v <a href="/wiki/Astronomie" title="Astronomie">astronomii</a>, protože velmi přesně popisuje tvar dráhy těles v gravitačním poli centrálního tělesa. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Pojmosloví"><span id="Pojmoslov.C3.AD"></span>Pojmosloví</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=1" title="Editace sekce: Pojmosloví" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=1" title="Editovat zdrojový kód sekce Pojmosloví"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b>Průvodič</b> je <a href="/wiki/%C3%9Ase%C4%8Dka" title="Úsečka">úsečka</a>, spojující libovolný bod na elipse s ohniskem.</li> <li><b>Střed elipsy</b> je střed <a href="/wiki/%C3%9Ase%C4%8Dka" title="Úsečka">úsečky</a>, jejíž konce tvoří ohniska elipsy (značí se <i>S</i>).</li> <li><b>Hlavní osa</b> je přímka procházející ohnisky elipsy. Někdy se takto označuje pouze průměr (úsečka) procházející ohnisky. Je to nejdelší průměr elipsy (délky <i>2a</i>).</li> <li><b>Hlavní vrcholy</b> jsou průsečíky elipsy s hlavní osou (v obrázku označeny <i>A</i>, <i>B</i>).</li> <li><b>Hlavní poloosa</b> je úsečka spojující střed elipsy s hlavním vrcholem (délky <i>a=|AS|=|BS|</i>).</li> <li><b>Vedlejší osa</b> je kolmice na hlavní osu procházející středem elipsy. Někdy se takto označuje pouze průměr kolmý na hlavní osu elipsy. Jedná se o nejkratší průměr elipsy (délky <i>2b</i>).</li> <li><b>Vedlejší vrcholy</b> jsou průsečíky elipsy s vedlejší osou (v obrázku označeny <i>C</i>, <i>D</i>). Vzdálenost vedlejších vrcholů od ohnisek je rovna délce hlavní poloosy, tj. <i>|CF<sub>1</sub>|=|CF<sub>2</sub>|=|DF<sub>1</sub>|=|DF<sub>2</sub>|=a</i>.</li> <li><b>Vedlejší poloosa</b> je úsečka spojující střed elipsy s vedlejším vrcholem (délky <i>b=|CS|=|DS|</i>).</li> <li><b><a href="/wiki/Excentricita" class="mw-redirect" title="Excentricita">Excentricita</a></b> <b>(výstřednost),</b> podrobněji <b>lineární</b> nebo <b>délková excentricita,</b> je vzdálenost ohniska a středu elipsy (značí se <i>e</i>).</li> <li><b>Číselná</b> neboli <b>numerická excentricita</b> <i>ε</i> je <a href="/wiki/D%C4%9Blen%C3%AD" title="Dělení">podíl</a> excentricity a délky hlavní poloosy, tj. <i>ε=e/a</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Rovnice">Rovnice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=2" title="Editace sekce: Rovnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=2" title="Editovat zdrojový kód sekce Rovnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Soubor:Ellipse_parameters.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Ellipse_parameters.png/300px-Ellipse_parameters.png" decoding="async" width="300" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Ellipse_parameters.png/450px-Ellipse_parameters.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Ellipse_parameters.png/600px-Ellipse_parameters.png 2x" data-file-width="690" data-file-height="470" /></a><figcaption>Parametry elipsy:<br /> <i>a</i> – délka hlavní poloosy<br /> <i>b</i> – délka vedlejší poloosy<br /> <i>F<sub>1</sub></i>, <i>F<sub>2</sub></i> – ohniska<br /> <i>e</i> – excentricita<br /> <i>p</i> – parametr</figcaption></figure> <p>Rovnici elipsy lze zapsat v různých tvarech. </p> <div class="mw-heading mw-heading3"><h3 id="Kanonický_tvar"><span id="Kanonick.C3.BD_tvar"></span>Kanonický tvar</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=3" title="Editace sekce: Kanonický tvar" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=3" title="Editovat zdrojový kód sekce Kanonický tvar"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kanonický tvar rovnice elipsy v normální poloze (tzn. hlavní osa je <a href="/wiki/Rovnob%C4%9B%C5%BEky" title="Rovnoběžky">rovnoběžná</a> s osou <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> a střed má souřadnice <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_{0},y_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0},y_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d3caa55d3f5b0f9ed69a4352c251877397060fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.905ex; height:2.843ex;" alt="{\displaystyle [x_{0},y_{0}]}" /></span>) je </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 {\frac {{(x-x_{0})}^{2}}{a^{2}}}+{\frac {{(y-y_{0})}^{2}}{b^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{(x-x_{0})}^{2}}{a^{2}}}+{\frac {{(y-y_{0})}^{2}}{b^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33169ba5c4961d4848bdee460860eccb42548b28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.244ex; height:6.343ex;" alt="{\displaystyle {\frac {{(x-x_{0})}^{2}}{a^{2}}}+{\frac {{(y-y_{0})}^{2}}{b^{2}}}=1}" /></span></dd></dl> <p>V <a href="/wiki/Kart%C3%A9zsk%C3%A1_soustava_sou%C5%99adnic" title="Kartézská soustava souřadnic">kartézských souřadnicích</a> lze v normální poloze elipsu se středem v počátku vyjádřit <a href="/wiki/Rovnice" title="Rovnice">rovnicí</a> </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 \left({x \over a}\right)^{2}+\left({y \over b}\right)^{2}=1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>b</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({x \over a}\right)^{2}+\left({y \over b}\right)^{2}=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2128fc4473a63e622bcdfd3e80950f0f4c9929b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.305ex; height:5.509ex;" alt="{\displaystyle \left({x \over a}\right)^{2}+\left({y \over b}\right)^{2}=1\,}" /></span>,</dd></dl> <p>kde <i>a</i> je délka hlavní poloosy, <i>b</i> je délka vedlejší poloosy 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 [x,y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x,y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7bd6292c6023626c6358bfd3943a031b27d663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="{\displaystyle [x,y]}" /></span> jsou souřadnice libovolného bodu elipsy. Veličina <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 e={\sqrt {a^{2}-b^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {a^{2}-b^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd1c201b7b281f01e16123156b0a9ee320c9ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.682ex; height:3.509ex;" alt="{\displaystyle e={\sqrt {a^{2}-b^{2}}}}" /></span> se nazývá <b>excentricita elipsy</b> (výstřednost) a vyjadřuje <a href="/wiki/Vzd%C3%A1lenost" title="Vzdálenost">vzdálenost</a> ohniska od středu elipsy. </p> <div class="mw-heading mw-heading3"><h3 id="Vrcholová_rovnice"><span id="Vrcholov.C3.A1_rovnice"></span>Vrcholová rovnice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=4" title="Editace sekce: Vrcholová rovnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=4" title="Editovat zdrojový kód sekce Vrcholová rovnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vrcholová rovnice elipsy má tvar </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 y^{2}=2px-{\frac {p}{a}}x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>a</mi> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=2px-{\frac {p}{a}}x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83262c4179d5b3336e8b423f8a3f6341f10eaa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.265ex; height:4.843ex;" alt="{\displaystyle y^{2}=2px-{\frac {p}{a}}x^{2}}" /></span>,</dd></dl> <p>kde <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 p={\frac {b^{2}}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p={\frac {b^{2}}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ba3648fa608ef48d8e45a82cf9cd44c98a3944" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.089ex; width:7.245ex; height:5.676ex;" alt="{\displaystyle p={\frac {b^{2}}{a}}}" /></span> je tzv. <i>parametr elipsy</i>. Tato rovnice vyjadřuje elipsu, jejíž hlavní vrchol leží v počátku souřadné soustavy a hlavní osa je rovnoběžná s osou <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Rovnice_kuželosečky"><span id="Rovnice_ku.C5.BEelose.C4.8Dky"></span>Rovnice kuželosečky</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=5" title="Editace sekce: Rovnice kuželosečky" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=5" title="Editovat zdrojový kód sekce Rovnice kuželosečky"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ku%C5%BEelose%C4%8Dka#Algebraické_vyjádření" title="Kuželosečka">Obecná rovnice kuželosečky</a> </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 a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0822f0343f8a88bf180bda8e7fbdf6f1d9dcba75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:50.155ex; height:3.009ex;" alt="{\displaystyle a_{11}x^{2}+2a_{12}xy+a_{22}y^{2}+2a_{13}x+2a_{23}y+a_{33}=0}" /></span></dd></dl> <p>je rovnicí elipsy, pokud platí <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 a_{12}^{2}-4a_{11}a_{22}<0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo><</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{12}^{2}-4a_{11}a_{22}<0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57ad00b3a36b9f02edc565641ec14aafa5f8c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.229ex; height:3.176ex;" alt="{\displaystyle a_{12}^{2}-4a_{11}a_{22}<0,}" /></span> a rovnicí elipsy v normální poloze, pokud jsou splněny následující podmínky </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 \operatorname {sgn} a_{11}=\operatorname {sgn} a_{22}=\operatorname {sgn}(a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} a_{11}=\operatorname {sgn} a_{22}=\operatorname {sgn}(a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1c639ba25d94f7b8e7159b7d582270dc370b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:52.529ex; height:3.176ex;" alt="{\displaystyle \operatorname {sgn} a_{11}=\operatorname {sgn} a_{22}=\operatorname {sgn}(a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33})}" /></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 a_{12}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{12}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06c7d0226601a562cd8688b699bfbce6294ee70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.367ex; height:2.509ex;" alt="{\displaystyle a_{12}=0}" /></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 |a_{11}|<|a_{22}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{11}|<|a_{22}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28691010f700237cb98f20c22a62337e0d87d836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.898ex; height:2.843ex;" alt="{\displaystyle |a_{11}|<|a_{22}|}" /></span></dd></dl> <p>Elipsa zadaná <a href="/wiki/Rovnice" title="Rovnice">rovnicí</a> kuželosečky splňující tyto podmínky má hlavní poloosu o délce </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 a={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{11}^{2}a_{22}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{11}^{2}a_{22}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38ac6fc40fbade992729f9f83536560572c869b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:35.041ex; height:7.676ex;" alt="{\displaystyle a={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{11}^{2}a_{22}}}}}" /></span></dd></dl> <p>Délka vedlejší poloosy je </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 b={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{22}^{2}a_{11}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{22}^{2}a_{11}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/628f80806367a0dea5427f7e46dc104711bfe5e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.809ex; height:7.676ex;" alt="{\displaystyle b={\sqrt {\frac {a_{22}a_{13}^{2}+a_{11}a_{23}^{2}-a_{11}a_{22}a_{33}}{a_{22}^{2}a_{11}}}}}" /></span></dd></dl> <p>Střed elipsy má souřadnice </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 \left[-{\frac {a_{13}}{a_{11}}},-{\frac {a_{23}}{a_{22}}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[-{\frac {a_{13}}{a_{11}}},-{\frac {a_{23}}{a_{22}}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d0d0759766224dca68710c1b70c320680e31e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.99ex; height:6.176ex;" alt="{\displaystyle \left[-{\frac {a_{13}}{a_{11}}},-{\frac {a_{23}}{a_{22}}}\right]}" /></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Parametrické_rovnice"><span id="Parametrick.C3.A9_rovnice"></span>Parametrické rovnice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=6" title="Editace sekce: Parametrické rovnice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=6" title="Editovat zdrojový kód sekce Parametrické rovnice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elipsu lze vyjádřit <a href="/wiki/Parametrick%C3%A1_funkce" class="mw-redirect" title="Parametrická funkce">parametrickými rovnicemi</a> </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=a\,\cos t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace"></mspace> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=a\,\cos t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0f77fe1f78514578e8028f462731b206a78fc7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.77ex; height:2.009ex;" alt="{\displaystyle x=a\,\cos 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=b\,\sin t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b\,\sin t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2017f33630985af1f23aac6bfb4d70d1ec0a70bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.108ex; height:2.509ex;" alt="{\displaystyle y=b\,\sin t}" /></span></dd></dl> <p>kde <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 \langle 0,2\pi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>∈</mo> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\in \langle 0,2\pi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1376485597f883e1a5c466165fe546085a8c8397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.18ex; height:2.843ex;" alt="{\displaystyle t\in \langle 0,2\pi \rangle }" /></span> je tzv. <i>excentrická anomálie</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Rovnice_v_polárních_souřadnicích"><span id="Rovnice_v_pol.C3.A1rn.C3.ADch_sou.C5.99adnic.C3.ADch"></span>Rovnice v polárních souřadnicích</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=7" title="Editace sekce: Rovnice v polárních souřadnicích" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=7" title="Editovat zdrojový kód sekce Rovnice v polárních souřadnicích"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V <a href="/wiki/Pol%C3%A1rn%C3%AD_soustava_sou%C5%99adnic" title="Polární soustava souřadnic">polárních souřadnicích</a> lze v případě, že ohnisko <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 F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd17e0779153d765b40ebef91533489b87b2e37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{2}}" /></span> leží v počátku souřadnicové soustavy a <a href="/w/index.php?title=Pol%C3%A1rn%C3%AD_osa&action=edit&redlink=1" class="new" title="Polární osa (stránka neexistuje)">polární osou</a> je <a href="/wiki/Polop%C5%99%C3%ADmka" title="Polopřímka">polopřímka</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 F_{2}S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dadb8b8ffa6b000a0b04246c23d15aa156a0d3ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.028ex; height:2.509ex;" alt="{\displaystyle F_{2}S_{1}}" /></span> (tj. na obrázku výše osa <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /></span>, od které bude proti směru hodinových ručiček přibývat úhel <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /></span>) psát rovnici elipsy ve tvaru </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 \rho ={\frac {p}{1+\varepsilon \cos \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {p}{1+\varepsilon \cos \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef31242f30a3a1a2b56a1cc025c4abc4d249d295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.628ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {p}{1+\varepsilon \cos \varphi }}}" /></span>,</dd></dl> <p>kde <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 \varepsilon =e/a<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =e/a<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc808de60d07af35dc063e70821b421671ffaa63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.919ex; height:2.843ex;" alt="{\displaystyle \varepsilon =e/a<1}" /></span> je tzv. <b>číselná výstřednost</b> 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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /></span> je <i>parametr elipsy</i>. Číselná výstřednost vyjadřuje míru zploštění elipsy, míru odlišnosti od <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a>. Má smysl ji porovnávat i pro různě velké elipsy. Geometrický význam <i>parametru</i> je polovina délky <a href="/wiki/T%C4%9Btiva_(geometrie)" title="Tětiva (geometrie)">tětivy</a> vedené ohniskem <a href="/wiki/Ortogonalita" title="Ortogonalita">kolmo</a> na hlavní osu. </p><p>Pokud v počátku souřadnicové soustavy leží střed elipsy a polární osou je polopřímka <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 SS_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SS_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab219164856ea150d13603697669b567ebc9e32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.978ex; height:2.509ex;" alt="{\displaystyle SS_{1}}" /></span>, pak dostáváme rovnici </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 \rho ^{2}={\frac {b^{2}}{1-\varepsilon ^{2}\cos ^{2}\varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{2}={\frac {b^{2}}{1-\varepsilon ^{2}\cos ^{2}\varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2deff759b5fc5865549642bbfc18a51205f294a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.791ex; height:6.343ex;" alt="{\displaystyle \rho ^{2}={\frac {b^{2}}{1-\varepsilon ^{2}\cos ^{2}\varphi }}}" /></span></dd></dl> <p>pro <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 \varepsilon <1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon <1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d78d118d219da41212022a84a8af05fd137420e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon <1}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Geometrické_vlastnosti_elipsy"><span id="Geometrick.C3.A9_vlastnosti_elipsy"></span>Geometrické vlastnosti elipsy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=8" title="Editace sekce: Geometrické vlastnosti elipsy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=8" title="Editovat zdrojový kód sekce Geometrické vlastnosti elipsy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Conicas1.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Conicas1.PNG/220px-Conicas1.PNG" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/12/Conicas1.PNG 1.5x" data-file-width="300" data-file-height="300" /></a><figcaption>Konstrukce elipsy jako kuželosečky</figcaption></figure> <p>O elipse říkáme, že je v <i>normální poloze</i>, je-li její hlavní osa <a href="/wiki/Rovnob%C4%9B%C5%BEky" title="Rovnoběžky">rovnoběžná</a> s osou <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span> nebo <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>. </p><p>Elipsa patří mezi <b><a href="/wiki/Ku%C5%BEelose%C4%8Dka" title="Kuželosečka">kuželosečky</a></b>, protože ji lze zkonstruovat jako <a href="/wiki/Rovinn%C3%BD_%C5%99ez" class="mw-redirect" title="Rovinný řez">řez</a> <a href="/wiki/Ku%C5%BEel" title="Kužel">rotační kuželové plochy</a> rovinou. Rovina řezu není kolmá k ose kužele, neprochází jeho <a href="/wiki/Vrchol_(geometrie)" title="Vrchol (geometrie)">vrcholem</a> a rovina s ní <a href="/wiki/Rovnob%C4%9B%C5%BEn%C3%A9_roviny" class="mw-redirect" title="Rovnoběžné roviny">rovnoběžná</a> vedená vrcholem nemá s kuželem žádný společný bod. </p><p>Každá přímka procházející středem <i>S</i> elipsy je jejím <b>průměrem</b>. Ze dvou na sebe kolmých průměrů <i>m</i> a <i>n</i> s pomocí trojúhelníkové konstrukce (viz Konstrukce elipsy) můžeme sestrojit <b>sdružené průměry</b> <i>m<sub>a</sub></i> a <i>n<sub>a</sub></i>, v jejichž koncovými body prochází rovnoběžky s <i>n<sub>a</sub></i> a <i>m<sub>a</sub></i>, které jsou zároveň <a href="/wiki/Te%C4%8Dna" title="Tečna">tečnami</a> elipsy, a vytvářejí tak <a href="/wiki/Rovnob%C4%9B%C5%BEn%C3%ADk" title="Rovnoběžník">rovnoběžník</a>. Jediné 2 sdružené průměry, které jsou na sebe kolmé, jsou osy elipsy. Dva sdružené průměry omezené body náležícími elipse jednoznačně určují elipsu. </p><p><b>Odrazová vlastnost</b> elipsy: U <a href="/wiki/Zrcadlo" title="Zrcadlo">zrcadla</a> s vnitřní eliptickou odrazovou plochou a zdrojem <a href="/wiki/Sv%C4%9Btlo" title="Světlo">světla</a> v jednom ohnisku se všechny paprsky podle <a href="/w/index.php?title=Z%C3%A1kon_odrazu&action=edit&redlink=1" class="new" title="Zákon odrazu (stránka neexistuje)">zákona odrazu</a> odrazí do jediného bodu — druhého ohniska. Žádná jiná křivka nemá tuto vlastnost, takže ji lze použít jako alternativní definici elipsy. </p><p><b><a href="/wiki/Obsah" title="Obsah">Obsah</a></b> plochy ohraničené elipsou lze určit ze vzorce </p> <div align="center"><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 S=\pi ab\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>π</mi> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\pi ab\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e1bd0a6b9edc1e4efeac1f070b0e24a8cf43f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.191ex; height:2.509ex;" alt="{\displaystyle S=\pi ab\,,}" /></span></div> <p>kde <i>a,b</i> jsou poloosy 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 \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" /></span> je <a href="/wiki/P%C3%AD_(%C4%8D%C3%ADslo)" title="Pí (číslo)">Ludolfovo číslo</a>. </p><p><b><a href="/wiki/Obvod_(geometrie)" title="Obvod (geometrie)">Obvod</a></b> (délku elipsy) lze určit pomocí vztahu </p> <div align="center"><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 o=4aE(\varepsilon ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o=4aE(\varepsilon ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be314c8e5f11584b5c26ef04d6b92a22458961ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.934ex; height:2.843ex;" alt="{\displaystyle o=4aE(\varepsilon ),}" /></span></div> <p>kde <i>a</i> je hlavní poloosa, <i>ε</i> (číselná) excentricita a funkce <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" /></span> je úplný <a href="/w/index.php?title=Eliptick%C3%BD_integr%C3%A1l&action=edit&redlink=1" class="new" title="Eliptický integrál (stránka neexistuje)">eliptický integrál</a> druhého druhu, nebo <i>přibližných</i> vzorců </p> <div align="center">Peanův vzorec (odchylka menší než 0,02 % až pro <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 \varepsilon <0{,}8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon <0{,}8}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73a21f077f610257946d22657d938461fcd349a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.154ex; height:2.509ex;" alt="{\displaystyle \varepsilon <0{,}8}" /></span>): <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 o\approx \pi \left[{{3 \over 2}\left(a+b\right)-{\sqrt {ab}}}\right]\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>≈</mo> <mi>π</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mi>b</mi> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o\approx \pi \left[{{3 \over 2}\left(a+b\right)-{\sqrt {ab}}}\right]\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/556736fb5a1ec3a12f0df751bb35bc1d319e283a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.088ex; height:6.176ex;" alt="{\displaystyle o\approx \pi \left[{{3 \over 2}\left(a+b\right)-{\sqrt {ab}}}\right]\,,}" /></span></div> <div align="center">nebo např. <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 o\approx {\pi \over 2}\left[{a+b+{\sqrt {2\left(a^{2}+b^{2}\right)}}}\right]\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o\approx {\pi \over 2}\left[{a+b+{\sqrt {2\left(a^{2}+b^{2}\right)}}}\right]\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a10c017acf06c32f1af752ae9ffff9fb81f2801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.485ex; height:4.843ex;" alt="{\displaystyle o\approx {\pi \over 2}\left[{a+b+{\sqrt {2\left(a^{2}+b^{2}\right)}}}\right]\,,}" /></span></div> <p>či částečným součtem <a href="/wiki/Nekone%C4%8Dn%C3%A1_%C5%99ada" class="mw-redirect" title="Nekonečná řada">nekonečné řady</a>: </p> <div align="center"><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 o=2\pi a\left[{1-\left({1 \over 2}\right)^{2}\varepsilon ^{2}-\left({1\cdot 3 \over 2\cdot 4}\right)^{2}{\varepsilon ^{4} \over 3}-\left({1\cdot 3\cdot 5 \over 2\cdot 4\cdot 6}\right)^{2}{\varepsilon ^{6} \over 5}-\dots }\right]\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>a</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>5</mn> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> <mo>⋅</mo> <mn>6</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mn>5</mn> </mfrac> </mrow> <mo>−</mo> <mo>…</mo> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o=2\pi a\left[{1-\left({1 \over 2}\right)^{2}\varepsilon ^{2}-\left({1\cdot 3 \over 2\cdot 4}\right)^{2}{\varepsilon ^{4} \over 3}-\left({1\cdot 3\cdot 5 \over 2\cdot 4\cdot 6}\right)^{2}{\varepsilon ^{6} \over 5}-\dots }\right]\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f834308b867589ebbf6c653155d153215f1159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.748ex; height:7.509ex;" alt="{\displaystyle o=2\pi a\left[{1-\left({1 \over 2}\right)^{2}\varepsilon ^{2}-\left({1\cdot 3 \over 2\cdot 4}\right)^{2}{\varepsilon ^{4} \over 3}-\left({1\cdot 3\cdot 5 \over 2\cdot 4\cdot 6}\right)^{2}{\varepsilon ^{6} \over 5}-\dots }\right]\,.}" /></span></div> <p><b>Speciálním případem</b> elipsy je <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a>, u které obě ohniska splývají. Excentricita je pak <a href="/wiki/Nula" title="Nula">nulová</a>, obě poloosy stejně dlouhé a říkáme jim <i><a href="/wiki/Polom%C4%9Br" title="Poloměr">poloměr</a></i>. </p><p><b>Limitním případem</b> elipsy je <a href="/wiki/Parabola_(matematika)" title="Parabola (matematika)">parabola</a>, kterou lze chápat jako elipsu s jedním <a href="/wiki/Nevlastn%C3%AD_bod" title="Nevlastní bod">nevlastním</a> ohniskem. </p> <div class="mw-heading mw-heading2"><h2 id="Konstrukce_elipsy">Konstrukce elipsy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=9" title="Editace sekce: Konstrukce elipsy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=9" title="Editovat zdrojový kód sekce Konstrukce elipsy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Zahradnická_konstrukce"><span id="Zahradnick.C3.A1_konstrukce"></span>Zahradnická konstrukce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=10" title="Editace sekce: Zahradnická konstrukce" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=10" title="Editovat zdrojový kód sekce Zahradnická konstrukce"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Elliko-g.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Elliko-g.svg/300px-Elliko-g.svg.png" decoding="async" width="300" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Elliko-g.svg/450px-Elliko-g.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Elliko-g.svg/600px-Elliko-g.svg.png 2x" data-file-width="296" data-file-height="246" /></a><figcaption>Zahradnická konstrukce elipsy</figcaption></figure> <p>Zahradnická konstrukce elipsy vychází přímo z definice, že elipsa je množina takových bodů v rovině, které mají stejný součet vzdáleností od obou ohnisek: v ohniscích elipsy se zabodnou kolíky (při kreslení na papír špendlíky), ke každému z nich se připevní jeden konec provázku (nitě) větší délky než je jejich vzdálenost, a opíše se elipsa, tak aby byl provázek napnutý. Je zřejmé, že délka provázku se rovná dvojnásobku velikosti velké poloosy, a že vzdálenost vedlejšího vrcholu od ohniska je rovna délce velké poloosy <i>a</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Trojúhelníková_konstrukce"><span id="Troj.C3.BAheln.C3.ADkov.C3.A1_konstrukce"></span>Trojúhelníková konstrukce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=11" title="Editace sekce: Trojúhelníková konstrukce" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=11" title="Editovat zdrojový kód sekce Trojúhelníková konstrukce"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Ellipse_affinite3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Ellipse_affinite3.svg/langcs-300px-Ellipse_affinite3.svg.png" decoding="async" width="300" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Ellipse_affinite3.svg/langcs-450px-Ellipse_affinite3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Ellipse_affinite3.svg/langcs-600px-Ellipse_affinite3.svg.png 2x" data-file-width="440" data-file-height="382" /></a><figcaption>Sestrojení elipsy s využitím <a href="/wiki/Afinn%C3%AD_zobrazen%C3%AD" title="Afinní zobrazení">afinity</a>. <a href="/wiki/Polop%C5%99%C3%ADmka" title="Polopřímka">Polopřímka</a> <i>p</i> je znázorněna červeně, části přímek <i>p<sub>1</sub></i> a <i>p<sub>2</sub></i> zeleně.</figcaption></figure> <p>Trojúhelníková konstrukce elipsy vychází z faktu, že elipsa je <a href="/wiki/Afinn%C3%AD_zobrazen%C3%AD" title="Afinní zobrazení">afinním</a> obrazem <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>: </p><p>Je zadán střed <i>S</i>, osy <i>o<sub>1</sub> a o<sub>2</sub></i> a délky <a href="/w/index.php?title=Poloosa&action=edit&redlink=1" class="new" title="Poloosa (stránka neexistuje)">poloos</a>: <i>a</i> (hlavní), <i>b</i> (vedlejší). </p> <div class="mw-heading mw-heading4"><h4 id="Postup">Postup</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=12" title="Editace sekce: Postup" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=12" title="Editovat zdrojový kód sekce Postup"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Sestrojíme kružnici <i>k<sub>1</sub></i> se středem v bodě <i>S</i> a <a href="/wiki/Polom%C4%9Br" title="Poloměr">poloměrem</a> velikosti <i>b</i>.</li> <li>Sestrojíme kružnici <i>k<sub>2</sub></i> se středem v bodě <i>S</i> a <a href="/wiki/Polom%C4%9Br" title="Poloměr">poloměrem</a> velikosti <i>a</i>.</li> <li>Vedeme libovolnou <a href="/wiki/Polop%C5%99%C3%ADmka" title="Polopřímka">polopřímku</a> <i>p</i> vycházející z bodu <i>S</i>.</li> <li>Bod <i>M<sub>1</sub></i> je průsečík polopřímky <i>p</i> s kružnicí <i>k<sub>1</sub></i>.</li> <li>Bod <i>M<sub>2</sub></i> je průsečík polopřímky <i>p</i> s kružnicí <i>k<sub>2</sub></i>.</li> <li>Bodem <i>M<sub>1</sub></i> vedeme přímku <i>p<sub>1</sub></i> rovnoběžnou s hlavní osou elipsy <i>o<sub>1</sub></i>.</li> <li>Bodem <i>M<sub>2</sub></i> vedeme přímku <i>p<sub>2</sub></i> rovnoběžnou s vedlejší osou elipsy <i>o<sub>2</sub></i>.</li> <li>Průsečíkem přímek <i>p<sub>1</sub></i> a <i>p<sub>2</sub></i> je bod <i>M</i>, který je bodem elipsy</li> <li>Opakujeme body postupu 3-8; všechny takto sestrojené body elipsy se nachází mezi kružnicemi <i>k<sub>1</sub></i> a <i>k<sub>2</sub></i> nebo přímo na kružnicích (v případě hlavních a vedlejších vrcholů).</li></ol> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Prouzkova_rozdilova_konstrukce_elipsy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Prouzkova_rozdilova_konstrukce_elipsy.jpg/300px-Prouzkova_rozdilova_konstrukce_elipsy.jpg" decoding="async" width="300" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Prouzkova_rozdilova_konstrukce_elipsy.jpg/450px-Prouzkova_rozdilova_konstrukce_elipsy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/a/aa/Prouzkova_rozdilova_konstrukce_elipsy.jpg 2x" data-file-width="455" data-file-height="323" /></a><figcaption>Rozdílová: proužková konstrukce elipsy. Používáme rozdíl <i>a - b</i>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Prouzkova_souctova_konstrukce_elipsy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Prouzkova_souctova_konstrukce_elipsy.jpg/300px-Prouzkova_souctova_konstrukce_elipsy.jpg" decoding="async" width="300" height="298" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Prouzkova_souctova_konstrukce_elipsy.jpg/450px-Prouzkova_souctova_konstrukce_elipsy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3f/Prouzkova_souctova_konstrukce_elipsy.jpg 2x" data-file-width="484" data-file-height="481" /></a><figcaption>Součtová: proužková konstrukce elipsy. Používáme součet <i>a + b</i>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Proužková_konstrukce"><span id="Prou.C5.BEkov.C3.A1_konstrukce"></span>Proužková konstrukce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=13" title="Editace sekce: Proužková konstrukce" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=13" title="Editovat zdrojový kód sekce Proužková konstrukce"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elipsa je určena hlavní osou <i>o<sub>1</sub></i>, hlavními vrcholy <i>A</i> a <i>B</i> a bodem <i>M</i>, který bude ležet na elipse, ale není vrcholem elipsy. </p> <div class="mw-heading mw-heading4"><h4 id="Postup_2">Postup</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=14" title="Editace sekce: Postup" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=14" title="Editovat zdrojový kód sekce Postup"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rozdílová konstrukce (viz obrázek): První krok pro získání velikosti poloosy <i>b</i> je podobný s trojúhelníkovou konstrukcí: sestrojíme k hlavní ose <i>o<sub>1</sub></i> <a href="/wiki/Kolmice" title="Kolmice">kolmici</a> <i>m<sub>1</sub></i> a <a href="/wiki/Rovnob%C4%9B%C5%BEka" title="Rovnoběžka">rovnoběžku</a> <i>m<sub>2</sub></i> které procházejí známým bodem M. Kružnice <i>k<sub>1</sub></i> se středem v <i>S</i> má poloměr velikosti <i>a</i>. Vznikl nám tak bod <i>M<sub>1</sub></i> který je průsečíkem kolmice <i>m<sub>1</sub></i> a kružnice <i>k<sub>1</sub></i>. Bod <i>M<sub>1</sub></i> spojíme přímkou <i>m</i> se středem <i>S</i> a nyní je už zřejmý bod <i>M<sub>2</sub></i>, jenž je průsečíkem <i>m</i> s <i>m<sub>2</sub></i>. Vzdálenost <i>M<sub>2</sub></i> od <i>S</i> je hledaná velikost poloosy <i>b</i>.<br /> Bodem <i>M</i> povedeme rovnoběžku <i>m´</i>, <i>m</i> || <i>m´</i>. <br />Průsečík této přímky m´ s hlavní poloosou <i>o<sub>1</sub></i> nazveme <i>P<sub>1</sub></i>. Vzniklá úsečka <i>MP<sub>1</sub></i> má velikost <i>b</i>. <br /> Průsečík této přímky m´ s vedlejší poloosou <i>o<sub>2</sub></i> nazveme <i>P<sub>2</sub></i>. Vzniklá úsečka <i>MP<sub>2</sub></i> má velikost <i>a</i>. Nyní si na okraji pomocného pruhu papíru označíme vzdálenosti <i>MP<sub>2</sub></i> a <i>MP<sub>1</sub></i> do jedné přímky <i>M</i> - <i>P<sub>1</sub></i> - <i>P<sub>2</sub></i> (z tohoto plyne název proužková konstrukce). Poté přikládáme proužek papíru tak, aby značka bodu <i>P<sub>2</sub></i> byla na vedlejší poloose <i>o<sub>2</sub></i> a zároveň značka bodu <i>P<sub>1</sub></i> byla na hlavní poloose <i>o<sub>1</sub></i>. Značka bodu <i>M</i> nám tak bude opisovat hledanou elipsu. Tento typ konstrukce se nazývá rozdílová. Používá se i součtová konstrukce, kterou můžete vidět na obrázku vpravo. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Prickova_konstrukce_elipsy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Prickova_konstrukce_elipsy.jpg/300px-Prickova_konstrukce_elipsy.jpg" decoding="async" width="300" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Prickova_konstrukce_elipsy.jpg/450px-Prickova_konstrukce_elipsy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Prickova_konstrukce_elipsy.jpg/600px-Prickova_konstrukce_elipsy.jpg 2x" data-file-width="662" data-file-height="499" /></a><figcaption>Příčková konstrukce elipsy. Hledání průsečíků, které leží na elipse.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Příčková_konstrukce"><span id="P.C5.99.C3.AD.C4.8Dkov.C3.A1_konstrukce"></span>Příčková konstrukce</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=15" title="Editace sekce: Příčková konstrukce" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=15" title="Editovat zdrojový kód sekce Příčková konstrukce"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elipsa je v tomto případě dána dvojicí sdružených průměrů <i>RQ</i> a <i>MN</i>, které na sebe nejsou kolmé (viz Geometrické vlastnosti elipsy). </p> <div class="mw-heading mw-heading4"><h4 id="Postup_3">Postup</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=16" title="Editace sekce: Postup" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=16" title="Editovat zdrojový kód sekce Postup"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>V bodech <i>R</i> a <i>Q</i> sestrojíme rovnoběžky s druhým sdruženým průměrem <i>MN</i>. V bodech <i>M</i> a <i>N</i> sestrojíme rovnoběžky s druhým sdruženým průměrem <i>RQ</i>. Vznikne nám <a href="/wiki/Rovnob%C4%9B%C5%BEn%C3%ADk" title="Rovnoběžník">rovnoběžník</a> s vrcholy <i>M<sub>1</sub>N<sub>1</sub>R<sub>1</sub>Q<sub>1</sub></i>, které jsou průsečíky rovnoběžek k průměrům <i>MN</i> a <i>RQ</i>. <br /> Sdružené průměry se protínají v průsečíku <i>S</i>, který je středem elipsy. <br /> Vytvoříme si čtveřici soustředných (dle středu <i>S</i>) <a href="/wiki/Soustava_sou%C5%99adnic" title="Soustava souřadnic">souřadnicových systémů</a>, jejichž počátky budou vrcholy vzniklého rovnoběžníku <i>M<sub>1</sub>N<sub>1</sub>R<sub>1</sub>Q<sub>1</sub></i> a poloosy budou strany rovnoběžníku. Na všech poloosách si stejně vyznačíme vhodné jednotky (př. 1, 2, 3, 4). Poloosy souřadných systémů se stýkají v koncových bodech sdružených průměrů <i>MNRQ</i>. Jednotky máme již vyznačené na rovnoběžkách sdružených průměrů, tak je ještě stejně vyznačíme přímo na sdružených průměrech <i>MN</i> a <i>RQ</i>, a to tak, že střed S je počátkem.<br /> Nyní budeme hledat body náležící elipse. Začneme například body nad sdruženým průměrem <i>MN</i> v souřadnicovém systému určeným body <i>NSQ</i>. Proložíme přímku bodem <i>N</i> a souřadnicí (např. 2) na poloose rovnoběžné k <i>NM</i>, na které leží bod <i>Q</i>. Poté proložíme přímku bodem <i>M</i> a souřadnicí 2 na poloose určené body <i>SQ</i> (<i>SQ</i> náleží průměru <i>RQ</i>). Průsečík takto proložených přímek nám dává bod ležící na elipse.<br /> (Pozn. zmiňované poloosy nejsou poloosami elipsy, ale souřadných systémů.) </p><p>Tato elipsa nemá určené osy ani vrcholy, abychom je zjistili, musíme použít Rytzovu konstrukci os elipsy. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Soubor:Rytzova_konstrukce_elipsy.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Rytzova_konstrukce_elipsy.jpg/300px-Rytzova_konstrukce_elipsy.jpg" decoding="async" width="300" height="266" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Rytzova_konstrukce_elipsy.jpg/450px-Rytzova_konstrukce_elipsy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Rytzova_konstrukce_elipsy.jpg/600px-Rytzova_konstrukce_elipsy.jpg 2x" data-file-width="685" data-file-height="607" /></a><figcaption>Rytzova konstrukce os elipsy.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Rytzova_konstrukce_os">Rytzova konstrukce os</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=17" title="Editace sekce: Rytzova konstrukce os" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=17" title="Editovat zdrojový kód sekce Rytzova konstrukce os"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elipsa je dána dvojicí omezených sdružených průměrů <i>MN</i> a <i>RQ</i>. </p> <div class="mw-heading mw-heading4"><h4 id="Postup_4">Postup</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=18" title="Editace sekce: Postup" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=18" title="Editovat zdrojový kód sekce Postup"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sestrojíme přímku <i>p</i> kolmou k jednomu ze sdružených průměrů (např. <i>RQ</i>, tak aby procházela středem <i>S</i> (průsečík sdružených průměrů). Vzdálenost |<i>RS</i>| je shodná se vzdáleností |<i>SP</i>|, bod <i>P</i> leží na přímce <i>p</i>. Proložíme přímku body <i>PM</i>.<br />Najdeme střed <i>O</i> úsečky <i>PM</i>. Sestrojíme kružnici <i>k</i> o poloměru |OS|. Průsečíky kružnice <i>k</i> s přímkou určenou body <i>P</i>, <i>O</i> a <i>M</i> nazveme 1 a 2. Sestrojíme přímku procházející průsečíkem <i>1</i> a středem <i>S</i> a přímku procházející průsečíkem <i>2</i> a středem <i>S</i>. Tyto přímky jsou na sebe kolmé a leží na nich osy elipsy. Velikost hlavní a vedlejší poloosy získáme ze vzdáleností |<i>2M</i>| a |<i>M1</i>|. </p> <div class="mw-heading mw-heading2"><h2 id="Elipsa_ve_fyzice">Elipsa ve fyzice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=19" title="Editace sekce: Elipsa ve fyzice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=19" title="Editovat zdrojový kód sekce Elipsa ve fyzice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> objevil, že <a href="/wiki/Planeta" title="Planeta">planety</a> se kolem <a href="/wiki/Slunce" title="Slunce">Slunce</a> pohybují po elipsách s malou excentricitou. To je <a href="/wiki/Keplerovy_z%C3%A1kony#1._Keplerův_zákon" title="Keplerovy zákony">první Keplerův zákon</a>. Později <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> vysvětlil tento fakt jako důsledek <a href="/wiki/Newton%C5%AFv_gravita%C4%8Dn%C3%AD_z%C3%A1kon" title="Newtonův gravitační zákon">zákona gravitace</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Reference">Reference</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=20" title="Editace sekce: Reference" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=20" title="Editovat zdrojový kód sekce Reference"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <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"><i>Konstruktivní geometrie</i>: str. 88-94. Jaroslav Černý a Marie Kočandrlová. Vydavatelství ČVUT, Praha 1998. <span style="white-space:nowrap"><a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Speci%C3%A1ln%C3%AD:Zdroje_knih/80-01-01815-6" title="Speciální:Zdroje knih/80-01-01815-6"><span class="ISBN">80-01-01815-6</span></a></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=21" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=21" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ku%C5%BEelose%C4%8Dka" title="Kuželosečka">Kuželosečky</a></li> <li><a href="/wiki/Kru%C5%BEnice" title="Kružnice">Kružnice</a></li> <li><a href="/wiki/Rovinn%C3%A9_geometrick%C3%A9_%C3%BAtvary" class="mw-redirect" title="Rovinné geometrické útvary">Rovinné geometrické útvary</a></li> <li><a href="/wiki/Tissotova_indikatrix" title="Tissotova indikatrix">Tissotova indikatrix</a></li> <li><a href="/wiki/Hypocykloida" title="Hypocykloida">Hypocykloida</a></li></ul> <div class="navbox noprint" title="Kuželosečky"><table title="" class="navbox-navbox mw-collapsible autocollapse nowraplinks noprint" cellspacing="0" data-collapsetext="skrýt" data-expandtext="zobrazit"><style data-mw-deduplicate="TemplateStyles:r17064757">.mw-parser-output .navbox-title .mw-collapsible-toggle{font-weight:normal}</style><tbody><tr> <th class="navbox-title" colspan="2"> <span class="navbox-titletext"><a href="/wiki/Ku%C5%BEelose%C4%8Dka" title="Kuželosečka">Kuželosečky</a></span> </th> </tr><tr class="navbox-padding"><td></td></tr> <tr> <td class="navbox-list navbox-odd" colspan="2"> <div> <p><a href="/wiki/Parabola_(matematika)" title="Parabola (matematika)">parabola</a> • <a class="mw-selflink selflink">elipsa</a> • <a href="/wiki/Kru%C5%BEnice" title="Kružnice">kružnice</a> • <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> </p> </div> </td> </tr></tbody></table> </div> <div class="mw-heading mw-heading2"><h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=22" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=22" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://commons.wikimedia.org/wiki/Category:Ellipses" class="extiw" title="c:Category:Ellipses">elipsa</a></span> na Wikimedia Commons</span></span></span><i> </i></li> <li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-file-width="411" data-file-height="411" /></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://cs.wiktionary.org/wiki/elipsa" class="extiw" title="wikt:elipsa">elipsa</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Ellipse.html">Elipsa</a> v encyklopedii <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span style="cursor:help;" data-lang="en" title="Jazyk cíle odkazu: angličtina">(anglicky)</span></li> <li>Dva způsoby rýsování elipsy: <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=co7U0BfYvgQ">využitím hlavní a vedlejší poloosy</a> nebo <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=CaokHrXP8HM">ohnisek</a> (<a href="/wiki/YouTube" title="YouTube">YouTube</a>)</li> <li><a rel="nofollow" class="external text" href="http://www.beda.cz/~jirkaj/elipsa/index.html">Výpočet obvodu elipsy</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r23078045">.mw-parser-output .navbox2{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox2 .navbox2{margin-top:0}.mw-parser-output .navbox2+.navbox2{margin-top:-1px}.mw-parser-output .navbox2-inner,.mw-parser-output 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„<a dir="ltr" href="https://cs.wikipedia.org/w/index.php?title=Elipsa&oldid=24719595">https://cs.wikipedia.org/w/index.php?title=Elipsa&oldid=24719595</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/N%C3%A1pov%C4%9Bda:Kategorie" title="Nápověda:Kategorie">Kategorie</a>: <ul><li><a href="/wiki/Kategorie:Obrazce" title="Kategorie:Obrazce">Obrazce</a></li><li><a href="/wiki/Kategorie:Ku%C5%BEelose%C4%8Dky" title="Kategorie:Kuželosečky">Kuželosečky</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Skryté kategorie: <ul><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_BNF" title="Kategorie:Monitoring:Články s identifikátorem BNF">Monitoring:Články s identifikátorem BNF</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_GND" title="Kategorie:Monitoring:Články s identifikátorem GND">Monitoring:Články s identifikátorem GND</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_LCCN" title="Kategorie:Monitoring:Články s identifikátorem LCCN">Monitoring:Články s identifikátorem LCCN</a></li><li><a href="/wiki/Kategorie:Monitoring:%C4%8Cl%C3%A1nky_s_identifik%C3%A1torem_NLI" title="Kategorie:Monitoring:Články s identifikátorem NLI">Monitoring:Články s identifikátorem NLI</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"> Stránka byla naposledy editována 4. 3. 2025 v 14:21.</li> <li id="footer-info-copyright">Text je dostupný pod <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.cs">licencí Creative Commons Uveďte původ – Zachovejte licenci</a>, případně za dalších podmínek. 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