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Elipsa - Wikipedia
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međutim, to nije obavezno"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Napravi korisnički račun</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Posebno:Korisni%C4%8Dka_prijava&returnto=Elipsa" title="Predlažemo vam da se prijavite, iako to nije obavezno [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Prijavi me</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"> Stranice za odjavljene urednike <a href="/wiki/Pomo%C4%87:Uvod" aria-label="Saznajte više o uređivanju"><span>detaljnije</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/Posebno:Moji_doprinosi" title="Spisak izmjena napravljenih s ove IP-adrese [y]" accesskey="y"><span>Doprinosi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Posebno:Moj_razgovor" title="Rasprava o izmjenama s ove IP-adrese [n]" accesskey="n"><span>Razgovor</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 id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Ezatvori\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"sitenotice\" lang=\"bs\" dir=\"ltr\"\u003E\u003Ctable class=\"fmbox fmbox-system\" role=\"presentation\" style=\"border:1px solid #EAAD3B; 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text-align:center;\"\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd class=\"mbox-image\"\u003E\u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/Datoteka:Wiki_Loves_Ramadan_Logomark_01.svg\" class=\"mw-file-description\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wiki_Loves_Ramadan_Logomark_01.svg/40px-Wiki_Loves_Ramadan_Logomark_01.svg.png\" decoding=\"async\" width=\"40\" height=\"40\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wiki_Loves_Ramadan_Logomark_01.svg/60px-Wiki_Loves_Ramadan_Logomark_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Wiki_Loves_Ramadan_Logomark_01.svg/120px-Wiki_Loves_Ramadan_Logomark_01.svg.png 2x\" data-file-width=\"1080\" data-file-height=\"1080\" /\u003E\u003C/a\u003E\u003C/span\u003E\u003C/td\u003E\u003Ctd class=\"mbox-text\"\u003EPridružite se uređivačkom maratonu \u003Ca href=\"/wiki/Wikipedia:Wikipedia_voli_ramazan_2025.\" title=\"Wikipedia:Wikipedia voli ramazan 2025.\"\u003EWikipedia voli ramazan 2025\u003C/a\u003E, koji traje od 25. 2. do 15. 4. Detaljnije na \u003Ca href=\"/wiki/Wikipedia:Wikipedia_voli_ramazan_2025.\" title=\"Wikipedia:Wikipedia voli ramazan 2025.\"\u003Estranici projekta\u003C/a\u003E.\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\n\u003Ctable class=\"fmbox fmbox-system\" role=\"presentation\" style=\"border:1px solid #409262; text-align:center;\"\u003E\u003Ctbody\u003E\u003Ctr\u003E\u003Ctd class=\"mbox-image\"\u003E\u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/Datoteka:Flag_of_the_Romani_people.svg\" class=\"mw-file-description\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Flag_of_the_Romani_people.svg/40px-Flag_of_the_Romani_people.svg.png\" decoding=\"async\" width=\"40\" height=\"27\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Flag_of_the_Romani_people.svg/60px-Flag_of_the_Romani_people.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Flag_of_the_Romani_people.svg/120px-Flag_of_the_Romani_people.svg.png 2x\" data-file-width=\"900\" data-file-height=\"600\" /\u003E\u003C/a\u003E\u003C/span\u003E\u003C/td\u003E\u003Ctd class=\"mbox-text\"\u003EPridružite se uređivačkom maratonu \u003Ca href=\"/wiki/Wikipedia:Ure%C4%91iva%C4%8Dki_maraton_povodom_Svjetskog_dana_Roma_2025.\" title=\"Wikipedia:Uređivački maraton povodom Svjetskog dana Roma 2025.\"\u003ESvjetski dan Roma 2025\u003C/a\u003E, koji traje od 1. 4. do 15. 4. Detaljnije na \u003Ca href=\"/wiki/Wikipedia:Ure%C4%91iva%C4%8Dki_maraton_povodom_Svjetskog_dana_Roma_2025.\" title=\"Wikipedia:Uređivački maraton povodom Svjetskog dana Roma 2025.\"\u003Estranici projekta\u003C/a\u003E.\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="Sajt"> <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="Sadržaj" 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">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Dijelovi_elipse" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dijelovi_elipse"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Dijelovi elipse</span> </div> </a> <ul id="toc-Dijelovi_elipse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formule" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formule"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Formule</span> </div> </a> <ul id="toc-Formule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Osobine" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Osobine"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Osobine</span> </div> </a> <ul id="toc-Osobine-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ekscentricitet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ekscentricitet"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ekscentricitet</span> </div> </a> <ul id="toc-Ekscentricitet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jednačina" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jednačina"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Jednačina</span> </div> </a> <ul id="toc-Jednačina-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Površina_i_obim" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Površina_i_obim"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Površina i obim</span> </div> </a> <ul id="toc-Površina_i_obim-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zbir_rastojanja_tačke_elipse_od_žiža" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zbir_rastojanja_tačke_elipse_od_žiža"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Zbir rastojanja tačke elipse od žiža</span> </div> </a> <ul id="toc-Zbir_rastojanja_tačke_elipse_od_žiža-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Također_pogledajte" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Također_pogledajte"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Također pogledajte</span> </div> </a> <ul id="toc-Također_pogledajte-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-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">10</span> <span>Reference</span> </div> </a> <ul id="toc-Reference-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sadržaj" > <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="Uključi/isključi sadržaj" > <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">Uključi/isključi sadržaj</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="Idi na članak na drugom jeziku. Dostupno na 97 jezika" > <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 jezika</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 – afrikans" lang="af" hreflang="af" data-title="Ellips" data-language-autonym="Afrikaans" data-language-local-name="afrikans" 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 – njemački (Švicarska)" lang="gsw" hreflang="gsw" data-title="Ellipse" data-language-autonym="Alemannisch" data-language-local-name="njemački (Švicarska)" 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 – amharski" lang="am" hreflang="am" data-title="ሞላላ" data-language-autonym="አማርኛ" data-language-local-name="amharski" 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 – arapski" lang="ar" hreflang="ar" data-title="قطع ناقص" data-language-autonym="العربية" data-language-local-name="arapski" 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 – asturijski" lang="ast" hreflang="ast" data-title="Elipse" data-language-autonym="Asturianu" data-language-local-name="asturijski" 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 – azerbejdžanski" lang="az" hreflang="az" data-title="Ellips" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdžanski" 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 – baškirski" lang="ba" hreflang="ba" data-title="Эллипс" data-language-autonym="Башҡортса" data-language-local-name="baškirski" 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 – Bavarian" lang="bar" hreflang="bar" data-title="Elipsn" data-language-autonym="Boarisch" data-language-local-name="Bavarian" 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 – bjeloruski" lang="be" hreflang="be" data-title="Эліпс" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" 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 – bugarski" lang="bg" hreflang="bg" data-title="Елипса" data-language-autonym="Български" data-language-local-name="bugarski" 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 – bengalski" lang="bn" hreflang="bn" data-title="উপবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</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 – katalonski" lang="ca" hreflang="ca" data-title="El·lipse" data-language-autonym="Català" data-language-local-name="katalonski" 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 – centralnokurdski" lang="ckb" hreflang="ckb" data-title="بڕگەی ناتەواو" data-language-autonym="کوردی" data-language-local-name="centralnokurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Elipsa" title="Elipsa – češki" lang="cs" hreflang="cs" data-title="Elipsa" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</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 – čuvaški" lang="cv" hreflang="cv" data-title="Эллипс" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" 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 – velški" lang="cy" hreflang="cy" data-title="Elíps" data-language-autonym="Cymraeg" data-language-local-name="velški" 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) – danski" lang="da" hreflang="da" data-title="Ellipse (geometri)" data-language-autonym="Dansk" data-language-local-name="danski" 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 – njemački" lang="de" hreflang="de" data-title="Ellipse" data-language-autonym="Deutsch" data-language-local-name="njemački" 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) – grčki" lang="el" hreflang="el" data-title="Έλλειψη (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="grčki" 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 – engleski" lang="en" hreflang="en" data-title="Ellipse" data-language-autonym="English" data-language-local-name="engleski" 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 – španski" lang="es" hreflang="es" data-title="Elipse" data-language-autonym="Español" data-language-local-name="španski" 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 – estonski" lang="et" hreflang="et" data-title="Ellips" data-language-autonym="Eesti" data-language-local-name="estonski" 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 – baskijski" lang="eu" hreflang="eu" data-title="Elipse" data-language-autonym="Euskara" data-language-local-name="baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa badge-Q17437798 badge-goodarticle mw-list-item" title="dobar članak"><a href="https://fa.wikipedia.org/wiki/%D8%A8%DB%8C%D8%B6%DB%8C" title="بیضی – perzijski" lang="fa" hreflang="fa" data-title="بیضی" data-language-autonym="فارسی" data-language-local-name="perzijski" 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 – finski" lang="fi" hreflang="fi" data-title="Ellipsi" data-language-autonym="Suomi" data-language-local-name="finski" 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) – francuski" lang="fr" hreflang="fr" data-title="Ellipse (mathématiques)" data-language-autonym="Français" data-language-local-name="francuski" 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 – sjeverni frizijski" lang="frr" hreflang="frr" data-title="Elips" data-language-autonym="Nordfriisk" data-language-local-name="sjeverni frizijski" 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 – irski" lang="ga" hreflang="ga" data-title="Éilips" data-language-autonym="Gaeilge" data-language-local-name="irski" 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) – galicijski" lang="gl" hreflang="gl" data-title="Elipse (xeometría)" data-language-autonym="Galego" data-language-local-name="galicijski" 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 – hebrejski" lang="he" hreflang="he" data-title="אליפסה" data-language-autonym="עברית" data-language-local-name="hebrejski" 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 – hindi" lang="hi" hreflang="hi" data-title="दीर्घवृत्त" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elipsa" title="Elipsa – hrvatski" lang="hr" hreflang="hr" data-title="Elipsa" data-language-autonym="Hrvatski" data-language-local-name="hrvatski" 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 – haićanski kreolski" lang="ht" hreflang="ht" data-title="Elips" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haićanski kreolski" 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) – mađarski" lang="hu" hreflang="hu" data-title="Ellipszis (görbe)" data-language-autonym="Magyar" data-language-local-name="mađarski" 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 – armenski" lang="hy" hreflang="hy" data-title="Էլիպս" data-language-autonym="Հայերեն" data-language-local-name="armenski" 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) – interlingva" lang="ia" hreflang="ia" data-title="Ellipse (forma geometric)" data-language-autonym="Interlingua" data-language-local-name="interlingva" 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 – indonezijski" lang="id" hreflang="id" data-title="Elips" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" 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 – islandski" lang="is" hreflang="is" data-title="Sporbaugur" data-language-autonym="Íslenska" data-language-local-name="islandski" 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 – italijanski" lang="it" hreflang="it" data-title="Ellisse" data-language-autonym="Italiano" data-language-local-name="italijanski" 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 – japanski" lang="ja" hreflang="ja" data-title="楕円" data-language-autonym="日本語" data-language-local-name="japanski" 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 – javanski" lang="jv" hreflang="jv" data-title="Elips" data-language-autonym="Jawa" data-language-local-name="javanski" 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 – gruzijski" lang="ka" hreflang="ka" data-title="ელიფსი" data-language-autonym="ქართული" data-language-local-name="gruzijski" 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) – kabile" lang="kab" hreflang="kab" data-title="Aglay (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabile" 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 – kazaški" lang="kk" hreflang="kk" data-title="Эллипс" data-language-autonym="Қазақша" data-language-local-name="kazaški" 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 – kmerski" lang="km" hreflang="km" data-title="អេលីប" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kmerski" 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 – kanada" lang="kn" hreflang="kn" data-title="ದೀರ್ಘವೃತ್ತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kanada" 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 – korejski" lang="ko" hreflang="ko" data-title="타원" data-language-autonym="한국어" data-language-local-name="korejski" 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 – kirgiški" lang="ky" hreflang="ky" data-title="Эллипс" data-language-autonym="Кыргызча" data-language-local-name="kirgiški" 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 – latinski" lang="la" hreflang="la" data-title="Ellipsis" data-language-autonym="Latina" data-language-local-name="latinski" 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 – luksemburški" lang="lb" hreflang="lb" data-title="Ellips" data-language-autonym="Lëtzebuergesch" data-language-local-name="luksemburški" 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 – litvanski" lang="lt" hreflang="lt" data-title="Elipsė" data-language-autonym="Lietuvių" data-language-local-name="litvanski" 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 – latvijski" lang="lv" hreflang="lv" data-title="Elipse" data-language-autonym="Latviešu" data-language-local-name="latvijski" 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 – makedonski" lang="mk" hreflang="mk" data-title="Елипса" data-language-autonym="Македонски" data-language-local-name="makedonski" 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 – marati" lang="mr" hreflang="mr" data-title="लंबवर्तुळ" data-language-autonym="मराठी" data-language-local-name="marati" 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 – malajski" lang="ms" hreflang="ms" data-title="Elips" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" 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) – nizozemski" lang="nl" hreflang="nl" data-title="Ellips (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="nizozemski" 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 – norveški (Nynorsk)" lang="nn" hreflang="nn" data-title="Ellipse" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški (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 – norveški (Bokmal)" lang="nb" hreflang="nb" data-title="Ellipse" data-language-autonym="Norsk bokmål" data-language-local-name="norveški (Bokmal)" 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 – oksitanski" lang="oc" hreflang="oc" data-title="Ellipsa" data-language-autonym="Occitan" data-language-local-name="oksitanski" 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 – oromo" lang="om" hreflang="om" data-title="Okokkee" data-language-autonym="Oromoo" data-language-local-name="oromo" 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 – poljski" lang="pl" hreflang="pl" data-title="Elipsa" data-language-autonym="Polski" data-language-local-name="poljski" 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 – Piedmontese" lang="pms" hreflang="pms" data-title="Eliss" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" 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 – portugalski" lang="pt" hreflang="pt" data-title="Elipse" data-language-autonym="Português" data-language-local-name="portugalski" 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 – kečua" lang="qu" hreflang="qu" data-title="Lunp'u" data-language-autonym="Runa Simi" data-language-local-name="kečua" 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 – rumunski" lang="ro" hreflang="ro" data-title="Elipsă" data-language-autonym="Română" data-language-local-name="rumunski" 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 – ruski" lang="ru" hreflang="ru" data-title="Эллипс" data-language-autonym="Русский" data-language-local-name="ruski" 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 – Rusyn" lang="rue" hreflang="rue" data-title="Елипса" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" 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 – sicilijanski" lang="scn" hreflang="scn" data-title="Ellissi" data-language-autonym="Sicilianu" data-language-local-name="sicilijanski" 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 – srpskohrvatski" lang="sh" hreflang="sh" data-title="Elipsa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpskohrvatski" 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 – slovački" lang="sk" hreflang="sk" data-title="Elipsa" data-language-autonym="Slovenčina" data-language-local-name="slovački" 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 – slovenski" lang="sl" hreflang="sl" data-title="Elipsa" data-language-autonym="Slovenščina" data-language-local-name="slovenski" 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 – šona" lang="sn" hreflang="sn" data-title="Denderende" data-language-autonym="ChiShona" data-language-local-name="šona" 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 – somalski" lang="so" hreflang="so" data-title="Qabaal" data-language-autonym="Soomaaliga" data-language-local-name="somalski" 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 – albanski" lang="sq" hreflang="sq" data-title="Elipsi" data-language-autonym="Shqip" data-language-local-name="albanski" 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 – srpski" lang="sr" hreflang="sr" data-title="Елипса" data-language-autonym="Српски / srpski" data-language-local-name="srpski" 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) – švedski" lang="sv" hreflang="sv" data-title="Ellips (matematik)" data-language-autonym="Svenska" data-language-local-name="švedski" 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 – svahili" lang="sw" hreflang="sw" data-title="Duaradufu" data-language-autonym="Kiswahili" data-language-local-name="svahili" 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 – tamilski" lang="ta" hreflang="ta" data-title="நீள்வட்டம்" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B9%E0%B8%9B%E0%B8%A7%E0%B8%87%E0%B8%A3%E0%B8%B5" title="รูปวงรี – tajlandski" lang="th" hreflang="th" data-title="รูปวงรี" data-language-autonym="ไทย" data-language-local-name="tajlandski" 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 – turski" lang="tr" hreflang="tr" data-title="Elips" data-language-autonym="Türkçe" data-language-local-name="turski" 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 – ukrajinski" lang="uk" hreflang="uk" data-title="Еліпс" data-language-autonym="Українська" data-language-local-name="ukrajinski" 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 – urdu" lang="ur" hreflang="ur" data-title="بیضہ" data-language-autonym="اردو" data-language-local-name="urdu" 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) – uzbečki" lang="uz" hreflang="uz" data-title="Ellips (chiziq)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" 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 – venecijanski" lang="vec" hreflang="vec" data-title="Ełise" data-language-autonym="Vèneto" data-language-local-name="venecijanski" 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 – vijetnamski" lang="vi" hreflang="vi" data-title="Elip" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" 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 – varej" lang="war" hreflang="war" data-title="Elipse" data-language-autonym="Winaray" data-language-local-name="varej" 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 – Wu kineski" lang="wuu" hreflang="wuu" data-title="橢圓" data-language-autonym="吴语" data-language-local-name="Wu kineski" 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 – kineski" lang="zh" hreflang="zh" data-title="椭圆" data-language-autonym="中文" data-language-local-name="kineski" 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 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="橢圓" data-language-autonym="文言" data-language-local-name="Literary Chinese" 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 – Minnan" lang="nan" hreflang="nan" data-title="Tn̂g-ko-îⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="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) – kantonski" lang="yue" hreflang="yue" data-title="橢圓 (幾何)" data-language-autonym="粵語" data-language-local-name="kantonski" 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="Uredi međujezičke linkove" class="wbc-editpage">Uredi veze</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="Imenski prostori"> <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="Prikaži stranicu sa sadržajem [c]" accesskey="c"><span>Stranica</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Razgovor:Elipsa" rel="discussion" title="Razgovor o sadržaju [r]" accesskey="r"><span>Razgovor</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="Promijeni varijantu jezika" > <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">bosanski</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="Pregledi"> <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>Čitaj</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="Uredite ovu stranicu [v]" accesskey="v"><span>Uredi</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="Uredite izvorni kod ove stranice [e]" accesskey="e"><span>Uredi izvor</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Elipsa&action=history" title="Prethodne izmjene ove stranice [h]" accesskey="h"><span>Historija</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <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="Alati" > <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">Alati</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">Alati</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">sakrij</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Radnje </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>Čitaj</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="Uredite ovu stranicu [v]" accesskey="v"><span>Uredi</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="Uredite izvorni kod ove stranice [e]" accesskey="e"><span>Uredi izvor</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>Historija</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Općenito </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Posebno:%C5%A0ta_vodi_ovamo/Elipsa" title="Spisak svih članaka koji su povezani s ovim [j]" accesskey="j"><span>Šta vodi ovamo</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Posebno:Srodne_izmjene/Elipsa" rel="nofollow" title="Nedavne izmjene na stranicama koje su povezane sa ovom [k]" accesskey="k"><span>Srodne izmjene</span></a></li><li id="t-upload" class="mw-list-item"><a href="//bs.wikipedia.org/wiki/Wikipedia:Upload" title="Postavi slike i druge medije [u]" accesskey="u"><span>Postavi datoteku</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Elipsa&oldid=3483360" title="Trajni link ove verzije stranice"><span>Trajni link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Elipsa&action=info" title="Više informacija o ovoj stranici"><span>Informacije o stranici</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Posebno:Citiraj&page=Elipsa&id=3483360&wpFormIdentifier=titleform" title="Informacije kako citirati ovu stranicu"><span>Citiraj ovu stranicu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Posebno:Skra%C4%87iva%C4%8D_adresa&url=https%3A%2F%2Fbs.wikipedia.org%2Fwiki%2FElipsa"><span>Skraćeni link</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Posebno:QrCode&url=https%3A%2F%2Fbs.wikipedia.org%2Fwiki%2FElipsa"><span>Preuzmi QR kod</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"> Štampanje / izvoz </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=Posebno:Kolekcija&bookcmd=book_creator&referer=Elipsa"><span>Napravi knjigu</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Posebno:DownloadAsPdf&page=Elipsa&action=show-download-screen"><span>Preuzmi kao PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Elipsa&printable=yes" title="Verzija ove stranice za štampanje [p]" accesskey="p"><span>Za štampanje</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 drugim projektima </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="Link na povezane podatke spremišnih stavki [g]" accesskey="g"><span>Stavka na Wikipodacima</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Izgled"> <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">Izgled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">premjesti na bočnu traku</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sakrij</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">S Wikipedije, slobodne enciklopedije</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="bs" dir="ltr"><table class="plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Datoteka:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span"><b>Ovaj članak ili neki od njegovih odlomaka nije dovoljno potkrijepljen <a href="/wiki/Wikipedia:Provjerljivost" title="Wikipedia:Provjerljivost">izvorima</a></b> (literatura, veb-sajtovi ili drugi izvori).<span class="hide-when-compact"> Ako se pravilno ne potkrijepe <a href="/wiki/Wikipedia:Pouzdani_izvori" title="Wikipedia:Pouzdani izvori">pouzdanim</a> izvorima, sporne rečenice i navodi mogli bi biti izbrisani. Pomozite Wikipediji tako što ćete navesti validne izvore putem <a href="/wiki/Wikipedia:Citiranje_izvora" class="mw-redirect" title="Wikipedia:Citiranje izvora">referenci</a> te nakon toga možete ukloniti ovaj šablon.</span></div></td></tr></tbody></table> <div role="note" class="hatnote navigation-not-searchable">Za druga značenja, pogledajte <a href="/w/index.php?title=Elipsa_(%C4%8Dvor)&action=edit&redlink=1" class="new" title="Elipsa (čvor) (stranica ne postoji)">Elipsa (čvor)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Elipse.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Elipse.png/300px-Elipse.png" decoding="async" width="300" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Elipse.png/450px-Elipse.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d3/Elipse.png 2x" data-file-width="510" data-file-height="310" /></a><figcaption>Elipsa:<br /><b>a</b> = velika poluosa<br /><b>b</b> = mala poluosa</figcaption></figure> <p><b>Elipsa</b> je zatvorena <a href="/wiki/Kriva" title="Kriva">kriva</a> koja je određena sa dvije <a href="/w/index.php?title=Poluosa&action=edit&redlink=1" class="new" title="Poluosa (stranica ne postoji)">poluose</a>: velikom (oznaka: <i>a</i>) i malom (oznaka: <i>b</i>). Oblik elipse definiše se njenim <a href="/wiki/Ekscentricitet" title="Ekscentricitet">ekscentricitetom</a> (ili eliptičnošću, oznaka: <i>e</i>). Elipsa se može također predstaviti kao kosi presjek <a href="/wiki/Ravan" title="Ravan">ravni</a> i <a href="/wiki/Valjak" title="Valjak">valjka</a>. Tačke F<sub>1</sub> i <i>F<sub>2</sub></i> nazivaju se <a href="/wiki/Fokus" title="Fokus">fokus</a>. Osobine tački <i>F<sub>1</sub></i> i <i>F<sub>2</sub></i> i promjenljive tačke <i>X </i> je da je suma dužina duži <i>F<sub>1</sub>X</i> i <i>F<sub>2</sub>X</i> uvijek jednaka. </p><p>Površina elipse se računa formulom: </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 P=a*b*\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>a</mi> <mo>∗<!-- ∗ --></mo> <mi>b</mi> <mo>∗<!-- ∗ --></mo> <mi>π<!-- π --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=a*b*\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa8fb41b961ebdb8f32f2a888b433fab8d47f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.793ex; height:2.176ex;" alt="{\displaystyle P=a*b*\pi }" /></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Dijelovi_elipse">Dijelovi elipse</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=1" title="Uredi odlomak "Dijelovi elipse"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=1" title="Edit section's source code: Dijelovi elipse"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Tačke <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" 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_{1}}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> zovu se žarišta ili fokusi elipse.</li> <li>Glavna osa elipse je prava kroz žarišta (prava određena tačkama <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" 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_{1}}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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>)</li> <li>Sredina duži <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_{1}F_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}F_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301185d21c44dfcc8b0ab5de2a3adb1c0a23a21c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.098ex; height:2.509ex;" alt="{\displaystyle F_{1}F_{2}}" /></span> je središte elipse.</li> <li>Sporedna osa je normala na glavnu osu kroz središte elipse.</li> <li>Tačke u kojima ose sijeku elipsu zovu se tjemena.</li> <li>Velika osa je duž <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 AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}" /></span>.</li> <li>Mala osa je duž <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 CD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9669379a3a9b8c55e7876c2371ccbc6e21b654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.691ex; height:2.176ex;" alt="{\displaystyle CD}" /></span>.</li> <li>Velike poluose su duži <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 AS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/281bec8604dc8fa466840d0282d451caf46bf1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.242ex; height:2.176ex;" alt="{\displaystyle AS}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eed49d7d5ad2e0394d219b63cf5c889614f7f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.263ex; height:2.176ex;" alt="{\displaystyle SB}" /></span>.</li> <li>Male poluose su duži <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 CS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126ed8ee23011971fd075553baca0dfefc83bfa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle CS}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af326440837e67870d4bee5bf63f49c37c9d89c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.423ex; height:2.176ex;" alt="{\displaystyle SD}" /></span>.</li> <li>Linearni ekscentricitet udaljenost žarišta od središta elipse.</li> <li>Radijus vektori tačke na elipsi su dužine <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_{1}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca3ea5fe4b5f8407aaf487263e397bda56cbf5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.185ex; height:2.509ex;" alt="{\displaystyle F_{1}T}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{2}T}"> <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> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cddd09a7104f7e981a80eb9be307a7cc35e340e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.185ex; height:2.509ex;" alt="{\displaystyle F_{2}T}" /></span>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Formule">Formule</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=2" title="Uredi odlomak "Formule"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=2" title="Edit section's source code: Formule"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <tbody><tr> <th>Tekst zaglavlja</th> <th>Tekst zaglavlja </th></tr> <tr> <td>osna jednačina</td> <td><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^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b70fe8f10c25fff24eaf47eb917fcc7335a932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.209ex; height:3.009ex;" alt="{\displaystyle b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}}" /></span> </td></tr> <tr> <td>segmentni oblik</td> <td><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^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <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> <mi>y</mi> <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^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863293f65080f512375d767be3a54281b1b5c2b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.019ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}" /></span> </td></tr> <tr> <td>poluparametar</td> <td><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> </td></tr> <tr> <td>numerički ekscentricitet</td> <td><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 ={\frac {e}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε<!-- ε --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>e</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon ={\frac {e}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d639c38faf38c6c0379cd49559e0af916cc2cf19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.248ex; height:4.676ex;" alt="{\displaystyle \varepsilon ={\frac {e}{a}}}" /></span> </td></tr> <tr> <td>linearni ekscentricitet</td> <td><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^{2}=a^{2}-b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}=a^{2}-b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243aa2c0e90de262c378dcdef878aa8449e51c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.412ex; height:2.843ex;" alt="{\displaystyle e^{2}=a^{2}-b^{2}}" /></span> </td></tr> <tr> <td>radijus vektor</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}+r_{2}=2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}+r_{2}=2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57fa3c9045123c9ec4ed90bdc1f408b6ee9f14dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.537ex; height:2.509ex;" alt="{\displaystyle r_{1}+r_{2}=2a}" /></span><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> </td></tr> <tr> <td>jednačina tangente</td> <td><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^{2}x_{1}x+a^{2}y_{1}y=a^{2}b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>y</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}x_{1}x+a^{2}y_{1}y=a^{2}b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c0a32bcaaf77327c1f355e926d2601b9ff2316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.673ex; height:3.009ex;" alt="{\displaystyle b^{2}x_{1}x+a^{2}y_{1}y=a^{2}b^{2}}" /></span> </td></tr> <tr> <td>uslov da prava <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=kx+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=kx+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48522d7314ca34218f43cb5a2fafd3722c3e369a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.03ex; height:2.509ex;" alt="{\displaystyle y=kx+n}" /></span> bude tangenta</td> <td><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^{2}k^{2}+b^{2}=n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</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> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}k^{2}+b^{2}=n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba441f7b3a33bd714747fde04ab86aac6e3f1c5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.989ex; height:2.843ex;" alt="{\displaystyle a^{2}k^{2}+b^{2}=n^{2}}" /></span> </td></tr> <tr> <td>koordinate tjemena</td> <td><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 (\pm a,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>±<!-- ± --></mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pm a,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f43d4d49576077a52ce7939e40b34e831b5fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.044ex; height:2.843ex;" alt="{\displaystyle (\pm a,0)}" /></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 (0,\pm b,)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mo>±<!-- ± --></mo> <mi>b</mi> <mo>,</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\pm b,)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd68be551f9443299d03087279d834bd01add0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.845ex; height:2.843ex;" alt="{\displaystyle (0,\pm b,)}" /></span>, </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Osobine">Osobine</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=3" title="Uredi odlomak "Osobine"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=3" title="Edit section's source code: Osobine"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>algebarska zatvorena kriva jedna je od konika</li> <li>skup tačaka ravni kojima je zbir udaljenosti od dvije čvrste tačke, žarišta, konstantan.</li> <li>Elipsa je simetrična s obzirom na dvije ose (glavne osi) i njihovo sjecište, središte simetrije.</li> <li>Žarišta elipse smještena su na velikoj osi simetrično s obzirom na središte, a apscisa im je <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /></span></li> <li>Svaka tetiva koja prolazi kroz središte elipse njezin je prečnik.</li> <li>Tangenta na elipsu u tački T elipse zatvara jednake uglove sa spojnicama <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /></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 r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /></span> tačke T sa žarištima elipse</li> <li><a href="/wiki/Planete" class="mw-redirect" title="Planete">Planete</a> se kreću po elipsama kojima se u jednom žarištu nalazi <a href="/wiki/Sunce" title="Sunce">Sunce</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ekscentricitet">Ekscentricitet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=4" title="Uredi odlomak "Ekscentricitet"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=4" title="Edit section's source code: Ekscentricitet"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ekscentricitet je konstanta karakteristična za svaku elipsu. Predstavlja minimalno rastojanje fokusne tačke elipse od elipse, duž ose. Označava se sa <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /></span>. i izračunava se kao: </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 e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <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> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f9893df3274d3b63537988b4158ae5f4e671e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.629ex; height:7.509ex;" alt="{\displaystyle e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}" /></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span> su dužine poluprečnika elipse. Ako sa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /></span> označimo rastojanje između fokusnih tačaka elipse dobićemo </p><p><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={\frac {c}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\frac {c}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4234fac6da52e8286a96a8e4d138204d85a4844b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.248ex; height:4.676ex;" alt="{\displaystyle e={\frac {c}{a}}}" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Jednačina"><span id="Jedna.C4.8Dina"></span>Jednačina</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=5" title="Uredi odlomak "Jednačina"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=5" title="Edit section's source code: Jednačina"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Neka se ose elipse poklapaju sa koordinantnim osama. <a href="/wiki/Jedna%C4%8Dina" title="Jednačina">Jednačina</a> elipse je. </p><p><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 \displaystyle {\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <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> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle {\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24c8170be42578ccded9d069eec2e5f2a1ace77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.918ex; height:5.509ex;" alt="{\displaystyle \displaystyle {\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}}=1}" /></span> </p> <dl><dt>Dokaz</dt></dl> <p>Neka nezavisni parametar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span> raste od 0 do <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<!-- 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> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=a^{2}\cos ^{2}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</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>θ<!-- θ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=a^{2}\cos ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd575e6a28eeb9abed31e192def924e59983d46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.443ex; height:2.676ex;" alt="{\displaystyle x^{2}=a^{2}\cos ^{2}\theta .}" /></span> </p><p><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}=b^{2}\sin ^{2}\theta .}"> <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> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=b^{2}\sin ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7f45d4ef2518db10f4ab6f4333bc5604e30cd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.786ex; height:3.009ex;" alt="{\displaystyle y^{2}=b^{2}\sin ^{2}\theta .}" /></span> </p><p><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^{2}}{a^{2}}}=\cos ^{2}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <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> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}=\cos ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d039646f728d209b2ff3f7ffad8cd493c6507dfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.608ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}=\cos ^{2}\theta .}" /></span> </p><p><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 {y^{2}}{b^{2}}}=\sin ^{2}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>y</mi> <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> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {y^{2}}{b^{2}}}=\sin ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09c6c7c0304d97a17444a67f968a5ee31d90bfd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.184ex; height:6.009ex;" alt="{\displaystyle {\frac {y^{2}}{b^{2}}}=\sin ^{2}\theta .}" /></span> </p><p><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^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=\cos ^{2}\theta +\sin ^{2}\theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <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> <mi>y</mi> <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> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mo>+</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=\cos ^{2}\theta +\sin ^{2}\theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7739522a40840ebc819bf4fd51c3cbce758ef4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.727ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=\cos ^{2}\theta +\sin ^{2}\theta .}" /></span> </p><p><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^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <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> <mi>y</mi> <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^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863293f65080f512375d767be3a54281b1b5c2b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.019ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}" /></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Površina_i_obim"><span id="Povr.C5.A1ina_i_obim"></span>Površina i obim</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=6" title="Uredi odlomak "Površina i obim"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=6" title="Edit section's source code: Površina i obim"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Povr%C5%A1ina_elipse&action=edit&redlink=1" class="new" title="Površina elipse (stranica ne postoji)">Površina</a> zatvorena elipsom je :<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=ab\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>π<!-- π --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=ab\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15c7769fd1f4d833528c1061dc5cd69eecc8206f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.403ex; height:2.176ex;" alt="{\displaystyle P=ab\pi }" /></span>, gdje su <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /></span> polovine velike i male ose, 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 =3,14159...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π<!-- π --></mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>14159...</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =3,14159...}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/002c1704d0631b32d027b728f923d1a3504d7068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.38ex; height:2.509ex;" alt="{\displaystyle \pi =3,14159...}" /></span> matematička konstanta. Do formule za površinu se dolazi izračunavanjem pomoću integrala. </p> <dl><dt>Dokaz.</dt></dl> <p>Površina elipse je </p><p><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=4\int _{0}^{a}ydx=4\int _{0}^{a}b\cdot {\sqrt {1-{\frac {x^{2}}{a^{2}}}}}dx=[x=a\sin t,\ dx=a\cos tdt]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>4</mn> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>y</mi> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>4</mn> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>b</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <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> </msqrt> </mrow> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>t</mi> <mo>,</mo> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>t</mi> <mi>d</mi> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=4\int _{0}^{a}ydx=4\int _{0}^{a}b\cdot {\sqrt {1-{\frac {x^{2}}{a^{2}}}}}dx=[x=a\sin t,\ dx=a\cos tdt]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e823743a50214588593f9e05a4eaa60e6e16d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:67.206ex; height:7.509ex;" alt="{\displaystyle P=4\int _{0}^{a}ydx=4\int _{0}^{a}b\cdot {\sqrt {1-{\frac {x^{2}}{a^{2}}}}}dx=[x=a\sin t,\ dx=a\cos tdt]}" /></span> </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 =4\int _{0}^{\frac {\pi }{2}}b\cdot {\sqrt {\cos ^{2}t}}\cdot \cos tdt=4ab\int _{0}^{\frac {\pi }{2}}\cos ^{2}tdt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>4</mn> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mi>b</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>t</mi> </msqrt> </mrow> <mo>⋅<!-- ⋅ --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>t</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡<!-- --></mo> <mi>t</mi> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =4\int _{0}^{\frac {\pi }{2}}b\cdot {\sqrt {\cos ^{2}t}}\cdot \cos tdt=4ab\int _{0}^{\frac {\pi }{2}}\cos ^{2}tdt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea72d5f6c3162f34f7c556d7b5731a4a4c3b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.159ex; height:6.509ex;" alt="{\displaystyle =4\int _{0}^{\frac {\pi }{2}}b\cdot {\sqrt {\cos ^{2}t}}\cdot \cos tdt=4ab\int _{0}^{\frac {\pi }{2}}\cos ^{2}tdt}" /></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 =4ab\int _{0}^{\frac {\pi }{2}}{\frac {1+\cos 2t}{2}}dt=4ab\int _{0}^{\frac {\pi }{2}}{\frac {dt}{2}}+ab\int _{0}^{\frac {\pi }{2}}\cos 2td2t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mn>2</mn> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>a</mi> <mi>b</mi> <msubsup> <mo>∫<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mi>cos</mi> <mo>⁡<!-- --></mo> <mn>2</mn> <mi>t</mi> <mi>d</mi> <mn>2</mn> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =4ab\int _{0}^{\frac {\pi }{2}}{\frac {1+\cos 2t}{2}}dt=4ab\int _{0}^{\frac {\pi }{2}}{\frac {dt}{2}}+ab\int _{0}^{\frac {\pi }{2}}\cos 2td2t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae47fe93cea9f96098310860013ed8d67e67466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:56.618ex; height:6.509ex;" alt="{\displaystyle =4ab\int _{0}^{\frac {\pi }{2}}{\frac {1+\cos 2t}{2}}dt=4ab\int _{0}^{\frac {\pi }{2}}{\frac {dt}{2}}+ab\int _{0}^{\frac {\pi }{2}}\cos 2td2t}" /></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 =4ab\cdot {\frac {\pi }{2}}{\Bigg |}^{\frac {\pi }{2}}+ab\sin 2t{\Bigg |}_{0}^{\frac {\pi }{2}}=ab\pi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mn>2</mn> <mi>t</mi> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </msubsup> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mi>π<!-- π --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =4ab\cdot {\frac {\pi }{2}}{\Bigg |}^{\frac {\pi }{2}}+ab\sin 2t{\Bigg |}_{0}^{\frac {\pi }{2}}=ab\pi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654dacf5a5b89d3e5895d614ccea5b48124f9e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.654ex; height:7.176ex;" alt="{\displaystyle =4ab\cdot {\frac {\pi }{2}}{\Bigg |}^{\frac {\pi }{2}}+ab\sin 2t{\Bigg |}_{0}^{\frac {\pi }{2}}=ab\pi .}" /></span></dd></dl> <p><a href="/w/index.php?title=Obim&action=edit&redlink=1" class="new" title="Obim (stranica ne postoji)">Obim</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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" /></span> elipse je <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 4aE(\varepsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>a</mi> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ε<!-- ε --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4aE(\varepsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d078d375b6aa0ec58b08888f27a246049a0965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.061ex; height:2.843ex;" alt="{\displaystyle 4aE(\varepsilon )}" /></span>, gdje je funkcija <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> totalni <a href="/w/index.php?title=Elipti%C4%8Dni_integral&action=edit&redlink=1" class="new" title="Eliptični integral (stranica ne postoji)">eliptični integral</a> <a href="/w/index.php?title=Elipti%C4%8Dni_integral&action=edit&redlink=1" class="new" title="Eliptični integral (stranica ne postoji)">druge vrste</a>. </p><p>Tačan <a href="/w/index.php?title=Beskona%C4%8Dan_red&action=edit&redlink=1" class="new" title="Beskonačan red (stranica ne postoji)">beskonačan red</a> glasi: </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 C=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>C</mi> <mo>=</mo> <mn>2</mn> <mi>π<!-- 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<!-- 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<!-- 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<!-- 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> <mo>;</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=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/9856e7323531776bfbf85f9e6dea584d29ec7708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:64ex; height:7.509ex;" alt="{\displaystyle C=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></dd></dl> <p>ili </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 C=2\pi a\sum _{n=0}^{\infty }{\left\lbrace -\left[\prod _{m=1}^{n}\left({2m-1 \over 2m}\right)\right]^{2}{\varepsilon ^{2n} \over 2n-1}\right\rbrace };\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>2</mn> <mi>π<!-- π --></mi> <mi>a</mi> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mrow> <mo>−<!-- − --></mo> <msup> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∏<!-- ∏ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ε<!-- ε --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mrow> <mn>2</mn> <mi>n</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>;</mo> <mspace width="thinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=2\pi a\sum _{n=0}^{\infty }{\left\lbrace -\left[\prod _{m=1}^{n}\left({2m-1 \over 2m}\right)\right]^{2}{\varepsilon ^{2n} \over 2n-1}\right\rbrace };\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a1598070e9de15ce43de0b668d4fdcc4803843c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; margin-right: -0.387ex; width:46.246ex; height:8.509ex;" alt="{\displaystyle C=2\pi a\sum _{n=0}^{\infty }{\left\lbrace -\left[\prod _{m=1}^{n}\left({2m-1 \over 2m}\right)\right]^{2}{\varepsilon ^{2n} \over 2n-1}\right\rbrace };\,\!}" /></span></dd></dl> <p>Dobra <a href="/w/index.php?title=Aproksimacija&action=edit&redlink=1" class="new" title="Aproksimacija (stranica ne postoji)">aproksimacija</a> <a href="/wiki/Srinivasa_Ramanujan" class="mw-redirect" title="Srinivasa Ramanujan">Ramanujanova</a>, a koja glasi: </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 C\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>≈<!-- ≈ --></mo> <mi>π<!-- π --></mi> <mrow> <mo>[</mo> <mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mrow> <mo>]</mo> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd246503ae5e29912d4149f118c5d7145d637085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.061ex; height:4.843ex;" alt="{\displaystyle C\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]\!\,}" /></span></dd></dl> <p>ili bolja <a href="/w/index.php?title=Aproksimacija&action=edit&redlink=1" class="new" title="Aproksimacija (stranica ne postoji)">aproksimacija</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 C\approx \pi \left(a+b\right)\left(1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right);\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>≈<!-- ≈ --></mo> <mi>π<!-- π --></mi> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>10</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>4</mn> <mo>−<!-- − --></mo> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−<!-- − --></mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>;</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\approx \pi \left(a+b\right)\left(1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right);\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb0c69145d1a2b7be89b69328a73116a95c5fd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:43.962ex; height:13.176ex;" alt="{\displaystyle C\approx \pi \left(a+b\right)\left(1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right);\!\,}" /></span></dd></dl> <p>Za posebni slučaj, gdje je mala osa polovine velike ose, možemo koristiti: </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 C\approx {\frac {\pi a(9-{\sqrt {35}})}{2}};\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>≈<!-- ≈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π<!-- π --></mi> <mi>a</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>35</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>;</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\approx {\frac {\pi a(9-{\sqrt {35}})}{2}};\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a490ca946d8d6ba7bb97808ae829e3e25b88b2ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.983ex; height:6.009ex;" alt="{\displaystyle C\approx {\frac {\pi a(9-{\sqrt {35}})}{2}};\!\,}" /></span></dd></dl> <p>ili </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\approx {\frac {a}{2}}{\sqrt {93+{\frac {1}{2}}{\sqrt {3}}}};\!\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>≈<!-- ≈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>93</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </msqrt> </mrow> <mo>;</mo> <mspace width="negativethinmathspace"></mspace> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\approx {\frac {a}{2}}{\sqrt {93+{\frac {1}{2}}{\sqrt {3}}}};\!\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02b47ad47c39c057b9772a8c01e3c5a4feef1b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.164ex; height:6.176ex;" alt="{\displaystyle C\approx {\frac {a}{2}}{\sqrt {93+{\frac {1}{2}}{\sqrt {3}}}};\!\,}" /></span> (better approximation). </p><p>Općenitije, <a href="/w/index.php?title=Du%C5%BEina_luka&action=edit&redlink=1" class="new" title="Dužina luka (stranica ne postoji)">dužina luka</a> dijela obima, kao funkcija obuhvatnog ugla, data je nepotpunim <a href="/w/index.php?title=Elipti%C4%8Dki_integral&action=edit&redlink=1" class="new" title="Eliptički integral (stranica ne postoji)">eliptičkim integralom</a>. <a href="/wiki/Inverzna_funkcija" title="Inverzna funkcija">Inverzna funkcija</a>, obuhvatni ugao kao funkcija dužine luka, je data preko <a href="/w/index.php?title=Elipti%C4%8Dka_funkcija&action=edit&redlink=1" class="new" title="Eliptička funkcija (stranica ne postoji)">eliptičkih funkcija</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Zbir_rastojanja_tačke_elipse_od_žiža"><span id="Zbir_rastojanja_ta.C4.8Dke_elipse_od_.C5.BEi.C5.BEa"></span>Zbir rastojanja tačke elipse od žiža</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=7" title="Uredi odlomak "Zbir rastojanja tačke elipse od žiža"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=7" title="Edit section's source code: Zbir rastojanja tačke elipse od žiža"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zbir rastojanja ma koje tačke elipse od njenih žiža, fokusa <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100c7fbf174fe8b06eacc2a6b0bb2e1badd1c7ce" 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_{1}}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> je konstantan i iznosi <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 2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d325c24be7d760207674a169b078892bdd5cbc76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.392ex; height:2.176ex;" alt="{\displaystyle 2a}" /></span> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Datoteka:Angular_eccentricity_and_linear_eccentricity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Angular_eccentricity_and_linear_eccentricity.svg/220px-Angular_eccentricity_and_linear_eccentricity.svg.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Angular_eccentricity_and_linear_eccentricity.svg/330px-Angular_eccentricity_and_linear_eccentricity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Angular_eccentricity_and_linear_eccentricity.svg/440px-Angular_eccentricity_and_linear_eccentricity.svg.png 2x" data-file-width="287" data-file-height="213" /></a><figcaption></figcaption></figure> <dl><dt>Dokaz</dt></dl> <p>Ako je <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 M(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b167f1545a525a774235a75c4e78ef8eb8b728d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.771ex; height:2.843ex;" alt="{\displaystyle M(x,y)}" /></span> proizvoljna tačka elipse, <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> podnožje normale iz te tačke na direktrisu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span>, 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 N^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd09b0bd0629c65e2e9bee09b39da7f07a1a32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.808ex; height:2.509ex;" alt="{\displaystyle N^{\prime }}" /></span> podnožje normale na direktrisu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8576984e010350846c33d12b7ba84a8fff60f9fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d^{\prime }}" /></span> , onda je </p><p><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 {\overline {FM}}=e\cdot {\overline {MN}},\ {\overline {F^{\prime }M}}=e\cdot {\overline {MN^{\prime }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>=</mo> <mi>e</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>N</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>=</mo> <mi>e</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {FM}}=e\cdot {\overline {MN}},\ {\overline {F^{\prime }M}}=e\cdot {\overline {MN^{\prime }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9754dc6c8387826ce81ee65bf9944bd1acaa7ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.697ex; height:3.509ex;" alt="{\displaystyle {\overline {FM}}=e\cdot {\overline {MN}},\ {\overline {F^{\prime }M}}=e\cdot {\overline {MN^{\prime }}},}" /></span> </p><p><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\in (0,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈<!-- ∈ --></mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in (0,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa4cf9072b2c926b2f975b321ca4376669e9b3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.092ex; height:2.843ex;" alt="{\displaystyle e\in (0,1)}" /></span> je ekscentricitet elipse </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 {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot ({\overline {MN}}+{\overline {MN^{\prime }}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>=</mo> <mi>e</mi> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>N</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot ({\overline {MN}}+{\overline {MN^{\prime }}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb1ec6a841e3f6d4fccb8747e99919cdb3b2dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.064ex; height:3.676ex;" alt="{\displaystyle {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot ({\overline {MN}}+{\overline {MN^{\prime }}})}" /></span>.</dd></dl> <p>Zbir u zagradi desno je rastojanje između direktrisa, koje iznosi <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*2*(a/e)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∗<!-- ∗ --></mo> <mn>2</mn> <mo>∗<!-- ∗ --></mo> <mo stretchy="false">(</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e*2*(a/e)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d36a673d1a45d096f098263568dc1dc1ed9b798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle e*2*(a/e)}" /></span>, pa 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 {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot {\frac {a}{e}}=2a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mi>M</mi> </mrow> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mo>=</mo> <mi>e</mi> <mo>⋅<!-- ⋅ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>e</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot {\frac {a}{e}}=2a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fdba58046343a909d381174515fdac65305c594" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.504ex; height:4.676ex;" alt="{\displaystyle {\overline {FM}}+{\overline {F^{\prime }M}}=e\cdot {\frac {a}{e}}=2a,}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Također_pogledajte"><span id="Tako.C4.91er_pogledajte"></span>Također pogledajte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=8" title="Uredi odlomak "Također pogledajte"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=8" title="Edit section's source code: Također pogledajte"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ekscentricitet" title="Ekscentricitet">Ekscentricitet</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elipsa&veaction=edit&section=9" title="Uredi odlomak "Izvori"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=9" title="Edit section's source code: Izvori"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a rel="nofollow" class="external text" href="http://www.mathopenref.com/tocs/ellipsetoc.html">Ellipses</a> </p><p><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Ellipse.html">Ellipse</a> </p><p><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org//index.php?title=Ellipse&oldid=11394">Ellipse, 7 February 2011 by 127.0.0.1</a> </p><p><a rel="nofollow" class="external text" href="http://tkrajina.github.io/Salabahteri/analiticka_geometrija.pdf">Analitička geometrija ravnine – osnovne formule</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=10" title="Uredi odlomak "Reference"" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elipsa&action=edit&section=10" title="Edit section's source code: Reference"><span>uredi izvor</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r3627399">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output 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href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>97 jezika</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Dodaj temu</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-65585cc8dc-56jhf","wgBackendResponseTime":160,"wgPageParseReport":{"limitreport":{"cputime":"0.117","walltime":"0.348","ppvisitednodes":{"value":577,"limit":1000000},"postexpandincludesize":{"value":4661,"limit":2097152},"templateargumentsize":{"value":301,"limit":2097152},"expansiondepth":{"value":11,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":4234,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 88.700 1 -total"," 48.71% 43.202 1 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