Hyperbel – Wikipedia

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class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_no.wikipedia.org&uselang=nb" class=""><span>Doner</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Spesial:Opprett_konto&returnto=Hyperbel" title="Du oppfordres til å opprette en konto og logge inn, men det er ikke obligatorisk" class=""><span>Opprett konto</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Spesial:Logg_inn&returnto=Hyperbel" title="Du oppfordres til å logge inn, men det er ikke obligatorisk [o]" accesskey="o" class=""><span>Logg inn</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out user-links-collapsible-item" title="Flere alternativer" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personlig" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personlig</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Brukermeny" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_no.wikipedia.org&uselang=nb"><span>Doner</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Spesial:Opprett_konto&returnto=Hyperbel" title="Du oppfordres til å opprette en konto og logge inn, men det er ikke obligatorisk"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Opprett konto</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Spesial:Logg_inn&returnto=Hyperbel" title="Du oppfordres til å logge inn, men det er ikke obligatorisk [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Logg inn</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Nettsted"> <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="Innhold" 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">Innhold</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skjul</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">(Til toppen)</div> </a> </li> <li id="toc-Geometrisk_definisjon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometrisk_definisjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Geometrisk definisjon</span> </div> </a> <ul id="toc-Geometrisk_definisjon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polarform" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polarform"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Polarform</span> </div> </a> <button aria-controls="toc-Polarform-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Polarform</span> </button> <ul id="toc-Polarform-sublist" class="vector-toc-list"> <li id="toc-Dobbelt_sett_av_brennpunkt_og_styrelinje" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dobbelt_sett_av_brennpunkt_og_styrelinje"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Dobbelt sett av brennpunkt og styrelinje</span> </div> </a> <ul id="toc-Dobbelt_sett_av_brennpunkt_og_styrelinje-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effekt_av_eksentrisiteten" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effekt_av_eksentrisiteten"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Effekt av eksentrisiteten</span> </div> </a> <ul id="toc-Effekt_av_eksentrisiteten-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sammenheng_mellom_geometriske_definisjoner" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sammenheng_mellom_geometriske_definisjoner"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Sammenheng mellom geometriske definisjoner</span> </div> </a> <ul id="toc-Sammenheng_mellom_geometriske_definisjoner-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Standardformer_i_kartesiske_koordinater" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Standardformer_i_kartesiske_koordinater"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Standardformer i kartesiske koordinater</span> </div> </a> <ul id="toc-Standardformer_i_kartesiske_koordinater-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Parametrisk_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Parametrisk_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Parametrisk form</span> </div> </a> <ul id="toc-Parametrisk_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konjugerte_hyperbler" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konjugerte_hyperbler"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Konjugerte hyperbler</span> </div> </a> <ul id="toc-Konjugerte_hyperbler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generell_kvadratisk_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generell_kvadratisk_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generell kvadratisk form</span> </div> </a> <ul id="toc-Generell_kvadratisk_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Degenerert_hyperbel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Degenerert_hyperbel"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Degenerert hyperbel</span> </div> </a> <ul id="toc-Degenerert_hyperbel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Egenskaper" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Egenskaper"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Egenskaper</span> </div> </a> <button aria-controls="toc-Egenskaper-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Vis/skjul underseksjonen Egenskaper</span> </button> <ul id="toc-Egenskaper-sublist" class="vector-toc-list"> <li id="toc-Symmetri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetri"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Symmetri</span> </div> </a> <ul id="toc-Symmetri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangentlinjer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangentlinjer"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Tangentlinjer</span> </div> </a> <ul id="toc-Tangentlinjer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konjungerte_diametre" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konjungerte_diametre"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Konjungerte diametre</span> </div> </a> <ul id="toc-Konjungerte_diametre-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Anvendelser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Anvendelser"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Anvendelser</span> </div> </a> <ul id="toc-Anvendelser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Litteratur" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Litteratur"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Litteratur</span> </div> </a> <ul id="toc-Litteratur-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="Innhold" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Hyperbel</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="Gå til en artikkel på et annet språk. Tilgjengelig på 74 språk" > <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-74" 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">74 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hyperbel" title="Hyperbel – norsk nynorsk" lang="nn" hreflang="nn" data-title="Hyperbel" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hyperbel" title="Hyperbel – dansk" lang="da" hreflang="da" data-title="Hyperbel" data-language-autonym="Dansk" data-language-local-name="dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hyperbel" title="Hyperbel – svensk" lang="sv" hreflang="sv" data-title="Hyperbel" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Brei%C3%B0bogi" title="Breiðbogi – islandsk" lang="is" hreflang="is" data-title="Breiðbogi" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Hiperbool" title="Hiperbool – afrikaans" lang="af" hreflang="af" data-title="Hiperbool" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%B7%D8%B9_%D8%B2%D8%A7%D8%A6%D8%AF" title="قطع زائد – arabisk" lang="ar" hreflang="ar" data-title="قطع زائد" data-language-autonym="العربية" data-language-local-name="arabisk" 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/Hip%C3%A9rbola" title="Hipérbola – asturisk" lang="ast" hreflang="ast" data-title="Hipérbola" data-language-autonym="Asturianu" data-language-local-name="asturisk" 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/Hiperbola_(riyaziyyat)" title="Hiperbola (riyaziyyat) – aserbajdsjansk" lang="az" hreflang="az" data-title="Hiperbola (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A7%E0%A6%BF%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="অধিবৃত্ত – bengali" lang="bn" hreflang="bn" data-title="অধিবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипербола (математика) – basjkirsk" lang="ba" hreflang="ba" data-title="Гипербола (математика)" data-language-autonym="Башҡортса" data-language-local-name="basjkirsk" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%96%D0%BF%D0%B5%D1%80%D0%B1%D0%B0%D0%BB%D0%B0_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Гіпербала (матэматыка) – belarusisk" lang="be" hreflang="be" data-title="Гіпербала (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarusisk" 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%93%D1%96%D0%BF%D1%8D%D1%80%D0%B1%D0%B0%D0%BB%D0%B0_(%D0%B3%D0%B5%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F)" title="Гіпэрбала (геамэтрыя) – belarusisk (klassisk ortografi)" lang="be-tarask" hreflang="be-tarask" data-title="Гіпэрбала (геамэтрыя)" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusisk (klassisk ortografi)" 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%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола – bulgarsk" lang="bg" hreflang="bg" data-title="Хипербола" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Hiperbola" title="Hiperbola – bosnisk" lang="bs" hreflang="bs" data-title="Hiperbola" data-language-autonym="Bosanski" data-language-local-name="bosnisk" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Hip%C3%A8rbola" title="Hipèrbola – katalansk" lang="ca" hreflang="ca" data-title="Hipèrbola" data-language-autonym="Català" data-language-local-name="katalansk" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипербола (математика) – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Гипербола (математика)" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hyperbola" title="Hyperbola – tsjekkisk" lang="cs" hreflang="cs" data-title="Hyperbola" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Hyperbola" title="Hyperbola – walisisk" lang="cy" hreflang="cy" data-title="Hyperbola" data-language-autonym="Cymraeg" data-language-local-name="walisisk" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik) – tysk" lang="de" hreflang="de" data-title="Hyperbel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/H%C3%BCperbool" title="Hüperbool – estisk" lang="et" hreflang="et" data-title="Hüperbool" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B2%CE%BF%CE%BB%CE%AE_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Υπερβολή (γεωμετρία) – gresk" lang="el" hreflang="el" data-title="Υπερβολή (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="gresk" 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/Hyperbola" title="Hyperbola – engelsk" lang="en" hreflang="en" data-title="Hyperbola" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola" title="Hipérbola – spansk" lang="es" hreflang="es" data-title="Hipérbola" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Hiperbolo" title="Hiperbolo – esperanto" lang="eo" hreflang="eo" data-title="Hiperbolo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hiperbola" title="Hiperbola – baskisk" lang="eu" hreflang="eu" data-title="Hiperbola" data-language-autonym="Euskara" data-language-local-name="baskisk" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D8%B0%D9%84%D9%88%D9%84%DB%8C" title="هذلولی – persisk" lang="fa" hreflang="fa" data-title="هذلولی" data-language-autonym="فارسی" data-language-local-name="persisk" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Hyperbole_(math%C3%A9matiques)" title="Hyperbole (mathématiques) – fransk" lang="fr" hreflang="fr" data-title="Hyperbole (mathématiques)" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Hipearb%C3%B3il" title="Hipearbóil – irsk" lang="ga" hreflang="ga" data-title="Hipearbóil" data-language-autonym="Gaeilge" data-language-local-name="irsk" 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/Hip%C3%A9rbole_(xeometr%C3%ADa)" title="Hipérbole (xeometría) – galisisk" lang="gl" hreflang="gl" data-title="Hipérbole (xeometría)" data-language-autonym="Galego" data-language-local-name="galisisk" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B3%A1%EC%84%A0" title="쌍곡선 – koreansk" lang="ko" hreflang="ko" data-title="쌍곡선" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%BA%D5%A5%D6%80%D5%A2%D5%B8%D5%AC" title="Հիպերբոլ – armensk" lang="hy" hreflang="hy" data-title="Հիպերբոլ" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A4%E0%A4%BF_%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A4%B2%E0%A4%AF" title="अति परवलय – hindi" lang="hi" hreflang="hi" data-title="अति परवलय" 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/Hiperbola_(krivulja)" title="Hiperbola (krivulja) – kroatisk" lang="hr" hreflang="hr" data-title="Hiperbola (krivulja)" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hiperbola" title="Hiperbola – indonesisk" lang="id" hreflang="id" data-title="Hiperbola" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Iperbole_(geometria)" title="Iperbole (geometria) – italiensk" lang="it" hreflang="it" data-title="Iperbole (geometria)" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%99%D7%A4%D7%A8%D7%91%D7%95%D7%9C%D7%94" title="היפרבולה – hebraisk" lang="he" hreflang="he" data-title="היפרבולה" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%B0%E1%83%98%E1%83%9E%E1%83%94%E1%83%A0%E1%83%91%E1%83%9D%E1%83%9A%E1%83%90" title="ჰიპერბოლა – georgisk" lang="ka" hreflang="ka" data-title="ჰიპერბოლა" data-language-autonym="ქართული" data-language-local-name="georgisk" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%B4%D0%B0" title="Гипербола математикада – kirgisisk" lang="ky" hreflang="ky" data-title="Гипербола математикада" data-language-autonym="Кыргызча" data-language-local-name="kirgisisk" 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/Hyperbola" title="Hyperbola – latin" lang="la" hreflang="la" data-title="Hyperbola" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Hiperbola" title="Hiperbola – latvisk" lang="lv" hreflang="lv" data-title="Hiperbola" data-language-autonym="Latviešu" data-language-local-name="latvisk" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hiperbol%C4%97_(matematika)" title="Hiperbolė (matematika) – litauisk" lang="lt" hreflang="lt" data-title="Hiperbolė (matematika)" data-language-autonym="Lietuvių" data-language-local-name="litauisk" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hiperbola_(matematika)" title="Hiperbola (matematika) – ungarsk" lang="hu" hreflang="hu" data-title="Hiperbola (matematika)" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола – makedonsk" lang="mk" hreflang="mk" data-title="Хипербола" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A7%E0%B4%BF%E0%B4%B5%E0%B4%B2%E0%B4%AF%E0%B4%82" title="അധിവലയം – malayalam" lang="ml" hreflang="ml" data-title="അധിവലയം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hyperbool_(meetkunde)" title="Hyperbool (meetkunde) – nederlandsk" lang="nl" hreflang="nl" data-title="Hyperbool (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A" title="双曲線 – japansk" lang="ja" hreflang="ja" data-title="双曲線" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Iperb%C3%B2la_(matematicas)" title="Iperbòla (matematicas) – oksitansk" lang="oc" hreflang="oc" data-title="Iperbòla (matematicas)" data-language-autonym="Occitan" data-language-local-name="oksitansk" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Giperbola" title="Giperbola – usbekisk" lang="uz" hreflang="uz" data-title="Giperbola" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbekisk" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9F%8A%E1%9E%B8%E1%9E%96%E1%9F%82%E1%9E%94%E1%9E%BC%E1%9E%9B" title="អ៊ីពែបូល – khmer" lang="km" hreflang="km" data-title="អ៊ីពែបូល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Ip%C3%A9rbol" title="Ipérbol – piemontesisk" lang="pms" hreflang="pms" data-title="Ipérbol" data-language-autonym="Piemontèis" data-language-local-name="piemontesisk" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Hiperbola_(matematyka)" title="Hiperbola (matematyka) – polsk" lang="pl" hreflang="pl" data-title="Hiperbola (matematyka)" data-language-autonym="Polski" data-language-local-name="polsk" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Hip%C3%A9rbole" title="Hipérbole – portugisisk" lang="pt" hreflang="pt" data-title="Hipérbole" data-language-autonym="Português" data-language-local-name="portugisisk" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Hiperbol%C4%83" title="Hiperbolă – rumensk" lang="ro" hreflang="ro" data-title="Hiperbolă" data-language-autonym="Română" data-language-local-name="rumensk" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Гипербола – rusinsk" lang="rue" hreflang="rue" data-title="Гипербола" data-language-autonym="Русиньскый" data-language-local-name="rusinsk" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гипербола (математика) – russisk" lang="ru" hreflang="ru" data-title="Гипербола (математика)" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Hyperbola" title="Hyperbola – skotsk" lang="sco" hreflang="sco" data-title="Hyperbola" data-language-autonym="Scots" data-language-local-name="skotsk" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Hiperbola_(matematik%C3%AB)" title="Hiperbola (matematikë) – albansk" lang="sq" hreflang="sq" data-title="Hiperbola (matematikë)" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Ip%C3%A8rbuli_(matim%C3%A0tica)" title="Ipèrbuli (matimàtica) – siciliansk" lang="scn" hreflang="scn" data-title="Ipèrbuli (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="siciliansk" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hyperbola" title="Hyperbola – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Hyperbola" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" 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/Hyperbola_(matematika)" title="Hyperbola (matematika) – slovakisk" lang="sk" hreflang="sk" data-title="Hyperbola (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" 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/Hiperbola" title="Hiperbola – slovensk" lang="sl" hreflang="sl" data-title="Hiperbola" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</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_%D8%B2%DB%8C%D8%A7%D8%AF" title="بڕگەی زیاد – sentralkurdisk" lang="ckb" hreflang="ckb" data-title="بڕگەی زیاد" data-language-autonym="کوردی" data-language-local-name="sentralkurdisk" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола – serbisk" lang="sr" hreflang="sr" data-title="Хипербола" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hiperbola" title="Hiperbola – serbokroatisk" lang="sh" hreflang="sh" data-title="Hiperbola" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hyperbeli" title="Hyperbeli – finsk" lang="fi" hreflang="fi" data-title="Hyperbeli" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%B5%E0%AF%81" title="அதிபரவளைவு – tamil" lang="ta" hreflang="ta" data-title="அதிபரவளைவு" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%84%E0%B8%AE%E0%B9%80%E0%B8%9E%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B9%82%E0%B8%9A%E0%B8%A5%E0%B8%B2" title="ไฮเพอร์โบลา – thai" lang="th" hreflang="th" data-title="ไฮเพอร์โบลา" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hiperbol" title="Hiperbol – tyrkisk" lang="tr" hreflang="tr" data-title="Hiperbol" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" 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%93%D1%96%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Гіпербола (математика) – ukrainsk" lang="uk" hreflang="uk" data-title="Гіпербола (математика)" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Hyperbol" title="Hyperbol – vietnamesisk" lang="vi" hreflang="vi" data-title="Hyperbol" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9B%99%E6%9B%B2%E7%B7%9A" title="雙曲線 – klassisk kinesisk" lang="lzh" hreflang="lzh" data-title="雙曲線" data-language-autonym="文言" data-language-local-name="klassisk kinesisk" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线 – wu" lang="wuu" hreflang="wuu" data-title="双曲线" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%99%E6%9B%B2%E7%B6%AB" title="雙曲綫 – kantonesisk" lang="yue" hreflang="yue" data-title="雙曲綫" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线 – kinesisk" lang="zh" hreflang="zh" data-title="双曲线" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q165301#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</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="Navnerom"> <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/Hyperbel" title="Vis innholdssiden [c]" accesskey="c"><span>Artikkel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskusjon:Hyperbel" rel="discussion" title="Diskusjon om innholdssiden [t]" accesskey="t"><span>Diskusjon</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="Bytt språkvariant" > <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">norsk bokmål</span> 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data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</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">Fra Wikipedia, den frie encyklopedi</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="nb" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbola_(PSF).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Hyperbola_%28PSF%29.svg/210px-Hyperbola_%28PSF%29.svg.png" decoding="async" width="210" height="330" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Hyperbola_%28PSF%29.svg/315px-Hyperbola_%28PSF%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Hyperbola_%28PSF%29.svg/420px-Hyperbola_%28PSF%29.svg.png 2x" data-file-width="854" data-file-height="1341" /></a><figcaption>To hyperbelgrener dannet ved et snitt mellom et plan og en kjegleflate</figcaption></figure> <p>En <b>hyperbel</b> er i <a href="/wiki/Matematikk" title="Matematikk">matematikk</a> en type <a href="/wiki/Kjeglesnitt" title="Kjeglesnitt">kjeglesnitt</a>, en plan <a href="/wiki/Kurve" title="Kurve">kurve</a> dannet som skjæringslinjen mellom et <a href="/wiki/Plan_(matematikk)" title="Plan (matematikk)">plan</a> og en <a href="/wiki/Kjegle" title="Kjegle">kjegleflate</a>.<sup id="cite_ref-THOMAS1_1-0" class="reference"><a href="#cite_note-THOMAS1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Andre typer kjeglesnitt er <a href="/wiki/Ellipse" title="Ellipse">ellipser</a> og <a href="/wiki/Parabel" title="Parabel">parabler</a>. </p><p>En hyperbel kan defineres geometrisk som en samling av punkt der avstanden til et gitt punkt og avstanden til en gitt rett linje har et konstant proporsjonalitetsforhold, og der proporsjonalitetskonstanten er større enn 1. Proporsjonalitetskonstanten kalles <i>eksentrisiteten</i>, og hyperbelen er et kjeglesnitt med eksentrisitet større enn 1. </p><p>Alternativt kan en hyperbel defineres som en kurve der avstanden til to gitte punkt har en konstant differens. </p><p>Hyperbel består av to <i>grener</i>, separate kurver som ligger symmetrisk om et punkt kalt sentrum. Begrepet «hyperbel» brukes både om en enkelt gren og om samhørende par av grener. Kurven har to <a href="/wiki/Asymptote" title="Asymptote">asymptoter</a>, som skjærer hverandre i hyperbelens sentrum. </p><p>Analytisk kan en hyperbel beskrives ved hjelp av en andregradsligning i to variable. For at ligningen skal framstille en hyperbel må <i>diskriminanten</i> definert ved ligningskoeffisientene være positiv. I visse tilfeller kan hyperbelen degenerere til to kryssende linjer. Standardformen for en hyperbel med sentrum i origo og halvakser <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> og <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> er </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x^{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/a24e3784b3cc27be20faa8b06c0c64e08dcabf7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.372ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Geometrisk_definisjon">Geometrisk definisjon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=1" title="Rediger avsnitt: Geometrisk definisjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=1" title="Rediger kildekoden til seksjonen Geometrisk definisjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En hyperbel kan defineres som det <a href="/wiki/Geometrisk_sted" title="Geometrisk sted">geometriske sted</a> for et punkt der avstanden til et gitt punkt og avstanden til en gitt rett linje har et konstant proporsjonalitetsforhold, og der proporsjonalitetskonstanten er større enn 1. Punktet kalles for <i>brennpunktet</i> eller <i>fokus</i>, og linjen kalles <i>styrelinje</i> eller <i>direktrise</i>. Generelt er et kjeglesnitt det geometriske sted for et punkt der avstanden fra brennpunktet er proporsjonal med avstanden til styrelinjen, og proprosjonaliteteskonstanten kalles <i>eksentrisiteten</i>. En hyperbel er altså et kjeglesnitt med eksentrisitet større enn 1. </p><p>Et plan som skjærer en rett kjegleflate med sirkulær basis vil framstille en hyperbel dersom toppvinkelen i kjeglen er <i>større</i> enn vinkelen som planet danner med kjegleaksen. Når kjegleflaten består av to kapper med et felles toppunkt, vil skjæringskurven ha to adskilte hyperbelgrener. </p><p>De to hyperbelgrenene dannes av hvert sitt sett av brennpunkt og styrelinjer, som ligger symmetrisk om sentrum. En hyperbel har altså <i>to</i> brennpunkt. Linjen gjennom disse to brennpunktene kalles <i>hyperbelaksen</i> eller <i>hovedaksen</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Polarform">Polarform</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=2" title="Rediger avsnitt: Polarform" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=2" title="Rediger kildekoden til seksjonen Polarform"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbel_def_norsk.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Hyperbel_def_norsk.png/400px-Hyperbel_def_norsk.png" decoding="async" width="400" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Hyperbel_def_norsk.png/600px-Hyperbel_def_norsk.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/Hyperbel_def_norsk.png/800px-Hyperbel_def_norsk.png 2x" data-file-width="1067" data-file-height="704" /></a><figcaption>Terminologi knyttet til hyperbelen</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbel_params.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hyperbel_params.png/400px-Hyperbel_params.png" decoding="async" width="400" height="267" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hyperbel_params.png/600px-Hyperbel_params.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Hyperbel_params.png/800px-Hyperbel_params.png 2x" data-file-width="1019" data-file-height="681" /></a><figcaption>Parametre for en hyperbel</figcaption></figure> <p>Gitt en styrelinje og et brennpunkt <i>F</i>, og la avstanden mellom disse være <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>. For et vilkårlig punkt på hyperbelgrenen <i>P</i> er avstanden til styrelinjen alltid proporsjonal med avstanden til brennpunktet: </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 |FP|=e|SP|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>F</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |FP|=e|SP|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3496fca82115e12e38fa33ff763ae21ad313402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.5ex; height:2.843ex;" alt="{\displaystyle |FP|=e|SP|}"></span></dd></dl> <p>Proporsjonalitetsfaktoren <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> kalles <i>eksentrisiteten</i>, og for en hyperbel er denne større enn 1. Linjen normalt på styrelinjen gjennom brennpunktet kalles <i>aksen</i> til hyperbelen. I <a href="/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem">polarkoordinater</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 (r,\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8396fdc359fb06c93722137c959e7496e47ed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.982ex; height:2.843ex;" alt="{\displaystyle (r,\theta )}"></span>, med polen definert i brennpunktet og akse langs hyperbelaksen, kan dette skrives som </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 {\begin{alignedat}{2}r&=e(h+r\cos \theta )\\&={\frac {eh}{1-e\cos \theta }}\qquad |\theta |<\theta ^{\ast }\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>r</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>h</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}r&=e(h+r\cos \theta )\\&={\frac {eh}{1-e\cos \theta }}\qquad |\theta |<\theta ^{\ast }\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b05b4dae150213e14b397554d66b4cc525a0de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.52ex; margin-bottom: -0.318ex; width:28.069ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{2}r&=e(h+r\cos \theta )\\&={\frac {eh}{1-e\cos \theta }}\qquad |\theta |<\theta ^{\ast }\end{alignedat}}}"></span></dd></dl> <p>Vinkelen <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 ^{\ast }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{\ast }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc305ec80889f8f96a755bb856fa5dc6481a3256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.343ex;" alt="{\displaystyle \theta ^{\ast }}"></span> er definert ved <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\cos \theta ^{\ast }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cos \theta ^{\ast }=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6c44032c1baba2c973003f58f3f4dea9e6509d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.374ex; height:2.343ex;" alt="{\displaystyle e\cos \theta ^{\ast }=1}"></span>. Dette er lik vinkelen en asymptote danner med hyperbelaksen. Hyperbelgrenen skjærer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-aksen i ett <i>toppunkt</i>, for <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 =180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfe2a9d00d19df1cda947de4bd87632ffa66a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.731ex; height:2.343ex;" alt="{\displaystyle \theta =180^{\circ }}"></span>. Avstanden fra dette toppunktene til brennpunktet <i>F</i> er </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 {\begin{alignedat}{2}g=r(\theta =180^{\circ })={\frac {eh}{1+e}}\\\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>g</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>h</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}g=r(\theta =180^{\circ })={\frac {eh}{1+e}}\\\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b17ae5f4852e584e6c6d4ca4c4b6264c2f4f5f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.906ex; margin-bottom: -0.266ex; width:25.575ex; height:5.509ex;" alt="{\displaystyle {\begin{alignedat}{2}g=r(\theta =180^{\circ })={\frac {eh}{1+e}}\\\end{alignedat}}}"></span></dd></dl> <p>Korden mellom to punkt på hyperbelen, parallelt med styrelinjen og gjennom brennpunktet, kalles <i>latus rectum</i>. Lengden <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> av denne er </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 l=2r(\theta =90^{\circ })=2eh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> <mo>=</mo> <mn>2</mn> <mi>r</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>e</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l=2r(\theta =90^{\circ })=2eh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be3bc15ed8d826b36c3a70b1a8a93ac9ba41f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.064ex; height:2.843ex;" alt="{\displaystyle l=2r(\theta =90^{\circ })=2eh}"></span></dd></dl> <p>Halve korden kalles <a href="/wiki/Semi_latus_rectum" title="Semi latus rectum">semi-latus rectum</a>, med lengde <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=l/2=eh}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <mi>e</mi> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=l/2=eh}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70ee07e27b842d68df10b235643d5d08b325bf95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:12.896ex; height:2.843ex;" alt="{\displaystyle p=l/2=eh}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Dobbelt_sett_av_brennpunkt_og_styrelinje">Dobbelt sett av brennpunkt og styrelinje</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=3" title="Rediger avsnitt: Dobbelt sett av brennpunkt og styrelinje" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=3" title="Rediger kildekoden til seksjonen Dobbelt sett av brennpunkt og styrelinje"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png/250px-Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png" decoding="async" width="250" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png/375px-Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png/500px-Hyperbelgren_med_to_brennpunkt_og_styrelinjer.png 2x" data-file-width="934" data-file-height="824" /></a><figcaption>Hyperbelgren med to sett av brennpunkt og styrelinjer</figcaption></figure> <p>Polarformen av hyperbelen kan brukes til å vise at kurven har et alternativt sett av brennpunkt og styrelinje. La et valgt brennpunkt være <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 en avstand <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> fra styrelinjen <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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span>, og la <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> være en valgt eksentrisitet. Definer et punkt <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> på hyperbelaksen, i avstanden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> fra brennpunktet <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>, der </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 s={\frac {2e^{2}h}{e^{2}-1}}\qquad (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> </mrow> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s={\frac {2e^{2}h}{e^{2}-1}}\qquad (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f279a7e79ff7384c78b46744aecefa90fa2a810" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.363ex; height:6.176ex;" alt="{\displaystyle s={\frac {2e^{2}h}{e^{2}-1}}\qquad (A)}"></span></dd></dl> <p>Punktet <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> skal velges slik at styrelinjen ligger <i>mellom</i> punktene <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> og <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>. En linje <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 L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"></span> legges parallelt med <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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span>, mellom de to punktene <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> og <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>, i en avstand <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> fra <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>. </p><p>La <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> være et vilkårlig punkt på hyperbelen, og la <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> og <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> være avstandene fra punktet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> til de to punktene <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> og <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>. Linjestykkene <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}P}"> <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>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{1}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9ced3e510d6a6f85ca230b5d438814eaab54cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.294ex; height:2.509ex;" alt="{\displaystyle F_{1}P}"></span> og <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}P}"> <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>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{2}P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13693159d1a5c3a5328ae51f92b48978dd1cc008" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.294ex; height:2.509ex;" alt="{\displaystyle F_{2}P}"></span> danner vinklene <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 _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f84b9443d095623e02fd287cd095123d70b0278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{1}}"></span> og <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 _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ed6ea624b20b153403979ffaf5434fc36de2990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{2}}"></span> med hyperbelaksen. Fra den geometriske definisjonen er </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}=e(h+r_{1}\cos \theta _{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> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}=e(h+r_{1}\cos \theta _{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c62ed20327e199e1b81bfec9c27ede41dcf38823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.406ex; height:2.843ex;" alt="{\displaystyle r_{1}=e(h+r_{1}\cos \theta _{1})}"></span></dd></dl> <p>Fra <a href="/wiki/Pythagoras%E2%80%99_l%C3%A6resetning#Consinussetningen" class="mw-redirect" title="Pythagoras’ læresetning">cosinussetningen</a> er </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 {\begin{alignedat}{2}r_{1}^{2}&=r_{2}^{2}+s^{2}-2r_{2}s\cos \theta _{2}\\r_{2}^{2}&=r_{1}^{2}+s^{2}-2r_{1}s\cos(\pi -\theta _{1})=r_{1}^{2}+s^{2}+2r_{1}s\cos \theta _{1}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>s</mi> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}r_{1}^{2}&=r_{2}^{2}+s^{2}-2r_{2}s\cos \theta _{2}\\r_{2}^{2}&=r_{1}^{2}+s^{2}-2r_{1}s\cos(\pi -\theta _{1})=r_{1}^{2}+s^{2}+2r_{1}s\cos \theta _{1}\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b135e82599d6211ac259619dea01b097083ca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.274ex; height:6.843ex;" alt="{\displaystyle {\begin{alignedat}{2}r_{1}^{2}&=r_{2}^{2}+s^{2}-2r_{2}s\cos \theta _{2}\\r_{2}^{2}&=r_{1}^{2}+s^{2}-2r_{1}s\cos(\pi -\theta _{1})=r_{1}^{2}+s^{2}+2r_{1}s\cos \theta _{1}\end{alignedat}}}"></span></dd></dl> <p>Ved å kombinere disse to ligningene finner en </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\cos \theta _{1}=-r_{2}\cos \theta _{2}+{\frac {r_{2}^{2}-r_{1}^{2}}{s}}\qquad (B)}"> <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> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mi>s</mi> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}\cos \theta _{1}=-r_{2}\cos \theta _{2}+{\frac {r_{2}^{2}-r_{1}^{2}}{s}}\qquad (B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fd16a894550383414fd3592c63eb8d1e17437f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.114ex; height:5.843ex;" alt="{\displaystyle r_{1}\cos \theta _{1}=-r_{2}\cos \theta _{2}+{\frac {r_{2}^{2}-r_{1}^{2}}{s}}\qquad (B)}"></span></dd></dl> <p>Avstanden fra punktet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> til hyperbelaksen gir ligningen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\sin \theta _{1}=r_{2}\sin \theta _{2}\qquad (C)}"> <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> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</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> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}\sin \theta _{1}=r_{2}\sin \theta _{2}\qquad (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631c2a2684b2104bd1cffe646366aafaf9ecec82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.074ex; height:2.843ex;" alt="{\displaystyle r_{1}\sin \theta _{1}=r_{2}\sin \theta _{2}\qquad (C)}"></span></dd></dl> <p>Fra kvadratet av den geometriske definisjonen finner en </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}^{2}=e^{2}(h^{2}+2hr_{1}\cos \theta _{1}+r_{1}^{2}\cos ^{2}\theta _{1})\qquad (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>h</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}^{2}=e^{2}(h^{2}+2hr_{1}\cos \theta _{1}+r_{1}^{2}\cos ^{2}\theta _{1})\qquad (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f267960eb67d2217026ee36a173f4c6d254f4472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.423ex; height:3.343ex;" alt="{\displaystyle r_{1}^{2}=e^{2}(h^{2}+2hr_{1}\cos \theta _{1}+r_{1}^{2}\cos ^{2}\theta _{1})\qquad (D)}"></span></dd></dl> <p>Kombinasjon av ligningene <i>A</i>-<i>D</i> gir relasjonen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}^{2}=e^{2}(r_{2}\cos \theta _{2}-h)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}^{2}=e^{2}(r_{2}\cos \theta _{2}-h)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da3862131e9b09de8ff73853e09f0b250ccc692b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.515ex; height:3.343ex;" alt="{\displaystyle r_{2}^{2}=e^{2}(r_{2}\cos \theta _{2}-h)^{2}}"></span></dd></dl> <p>Avstanden mellom de to punktene <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> og <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> er <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>, og størrelsen i parantesen på høyre side i ligningen over er avstanden fra linjen <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> til punktet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. Ligningen viser altå at disse to avstandene er proporsjonale, med samme proporsjonalitetskonstant <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> som det valgte brennpunktet <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> og styrelinjen <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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span>. Paret <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> og <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 L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"></span> er altså et alternativt sett av brennpunkt og styrelinje for hyperbelgrenen. Tilsvarende vil begge to settene også være brennpunkt og styrelinjer for en hyperbelgren som ligger symmetrisk om sentrum i hyperbelen, det vil si symmetrisk om midtpunktet mellom de to brennpunktene. </p><p>Avstanden fra sentrum og et brennpunkt er gitt ved lengden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666526fb52720dea7d26a0f8252ff7f4b928f1cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.415ex; height:2.843ex;" alt="{\displaystyle s/2}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Effekt_av_eksentrisiteten">Effekt av eksentrisiteten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=4" title="Rediger avsnitt: Effekt av eksentrisiteten" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=4" title="Rediger kildekoden til seksjonen Effekt av eksentrisiteten"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbel_eksentrisitet.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hyperbel_eksentrisitet.png/250px-Hyperbel_eksentrisitet.png" decoding="async" width="250" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hyperbel_eksentrisitet.png/375px-Hyperbel_eksentrisitet.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Hyperbel_eksentrisitet.png/500px-Hyperbel_eksentrisitet.png 2x" data-file-width="787" data-file-height="778" /></a><figcaption>Hyperbler med felles brennpunkt, men varierende eksentrisitet</figcaption></figure> <p>Når eksentrisiteten øker mot uendelig vil toppunktet i hyperbelgrenen nærme seg styrelinjen. For en fast avstand mellom brennpunktet og styrelinjen vil latus rectum gå mot uendelig, det vil si at hyperbelen vier seg mer og mer ut. </p><p>Når eksentrisiteten går mot 1, så vil toppunktet nærme seg brennpunktet. Latus rectum avtar mot grenseverdien <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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, slik at hyperbelen klapper mer og mer sammen. </p> <div class="mw-heading mw-heading3"><h3 id="Sammenheng_mellom_geometriske_definisjoner">Sammenheng mellom geometriske definisjoner</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=5" title="Rediger avsnitt: Sammenheng mellom geometriske definisjoner" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=5" title="Rediger kildekoden til seksjonen Sammenheng mellom geometriske definisjoner"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polarformen kan brukes til å vise at de to geometriske definisjonene for hyperbelen er ekvivalente. Gitt et vilkårlig punkt <i>P</i> på hyperbelen, og la avstanden fra dette punktet til de to styrelinjene være henholdsvis <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 l_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b25eeca673386d676f79dce674fe93040693eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.748ex; height:2.509ex;" alt="{\displaystyle l_{1}}"></span> og <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 l_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84708bbc21c20c9834e0e57746dbbc437414c350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.748ex; height:2.509ex;" alt="{\displaystyle l_{2}}"></span>. Tilsvarende la <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> og <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> være avstanden fra punktet til de to brennpunktene. Fra definisjonen med brennpunkt og styrelinje følger det at </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 {\begin{alignedat}{2}r_{1}=el_{1}\\r_{2}=el_{2}\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>e</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>e</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}r_{1}=el_{1}\\r_{2}=el_{2}\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c2ff71b4117fca42356801392ae04854ac48cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.784ex; height:5.843ex;" alt="{\displaystyle {\begin{alignedat}{2}r_{1}=el_{1}\\r_{2}=el_{2}\end{alignedat}}}"></span></dd></dl> <p>Siden <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 |l_{1}-l_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |l_{1}-l_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bca2a94491865e53ac3acd91bc0152ecfc7736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.629ex; height:2.843ex;" alt="{\displaystyle |l_{1}-l_{2}|}"></span> er lengden mellom styrelinjene følger det direkte at differensen av avstandene fra brennpunktet er konstant: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |r_{1}-r_{2}|=e|l_{1}-l_{2}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |r_{1}-r_{2}|=e|l_{1}-l_{2}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57695657869cc9cf5ed4c7b3a923039736a83b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.151ex; height:2.843ex;" alt="{\displaystyle |r_{1}-r_{2}|=e|l_{1}-l_{2}|}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Standardformer_i_kartesiske_koordinater">Standardformer i kartesiske koordinater</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=6" title="Rediger avsnitt: Standardformer i kartesiske koordinater" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=6" title="Rediger kildekoden til seksjonen Standardformer i kartesiske koordinater"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Det eksisterer flere standardformer for hyperbelen, også kalt kanoniske former. En kanonisk form er ligning i kartesiske koordinatene <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}"></span> som framstiller hyperbelen på en enklest mulig måte. Et alternativ for en kanonisk form framkommer når <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-aksen defineres langs hovedaksen, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-aksen defineres parallelt med styrelinjen og <a href="/wiki/Origo" title="Origo">origo</a> velges i sentrum av hyperbelen:<sup id="cite_ref-LAW1_2-0" class="reference"><a href="#cite_note-LAW1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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/a24e3784b3cc27be20faa8b06c0c64e08dcabf7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.372ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}"></span></dd></dl> <p>Ligningen har to parametre, lengdene <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> og <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> av de to <i>halvaksene</i>. For hyperbelen er det ikke et krav at <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\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5d3957d5f94566507526017e4ebb67c02efe81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\geq b}"></span>. Kurven har reelle verdier bare når <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|\geq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≥</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x|\geq a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09d4303fc39317bea390c6b9326db912e95ccba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.952ex; height:2.843ex;" alt="{\displaystyle |x|\geq a}"></span>. </p><p>En korde gjennom sentrum, mellom to punkt på hver sin hyperbelgren, kalles en <i>diameter</i> i hyperbelen. Diameteren mellom de to toppunktene kalles den <i>reelle</i> aksen og har lengden <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>.<sup id="cite_ref-GULD1_3-0" class="reference"><a href="#cite_note-GULD1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Den <i>imaginære</i> aksen står normalt på denne og har lengden <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 2b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da45af0250645a54cab2ef45483c4399e4a40df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.16ex; height:2.176ex;" alt="{\displaystyle 2b}"></span>. Dette er også lengden av en korde mellom asymptotene, normalt på hovedaksen og gjennom toppunktet på en hyperbelgren. Asymptotene har ligningene </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 ax\pm by=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>±</mo> <mi>b</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax\pm by=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69e457e4c41813544bf779f30e2e93ecb93c37d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.814ex; height:2.509ex;" alt="{\displaystyle ax\pm by=0}"></span></dd></dl> <p>Brennpunktene ligger i en avstand <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 ae}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ae}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b8a64aef214b5d91b0a1b5d3d6478a259ac4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.313ex; height:1.676ex;" alt="{\displaystyle ae}"></span> og styrelinjene i avstanden <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/e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1368d6c022671653e7f59cd81ee12772c5fee7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.476ex; height:2.843ex;" alt="{\displaystyle a/e}"></span> fra origo. Eksentrisiteten er gitt fra ligningskoeffisientene ved </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 {\frac {a^{2}+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> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {\frac {a^{2}+b^{2}}{a^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994a893513bb0cf9a6ff1fa299290075c2785d80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.518ex; height:7.509ex;" alt="{\displaystyle e={\sqrt {\frac {a^{2}+b^{2}}{a^{2}}}}}"></span>.</dd></dl> <p>En alternativ kanonisk form framkommer ved å legge origo i det ene brennpunktet:<sup id="cite_ref-LAW1_2-1" class="reference"><a href="#cite_note-LAW1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(x-{\frac {a}{e}})^{2} \over a^{2}}-{y^{2} \over b^{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>e</mi> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <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 {(x-{\frac {a}{e}})^{2} \over a^{2}}-{y^{2} \over b^{2}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e85686cf3ca12780e2b445b2a43bbec601d26b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.728ex; height:6.343ex;" alt="{\displaystyle {(x-{\frac {a}{e}})^{2} \over a^{2}}-{y^{2} \over b^{2}}=1}"></span></dd></dl> <p>En hyperbel der <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=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span> kalles <i>likesidet</i>, <i>ekvilateral</i> eller <i>rektangulær</i>, og for en slik hyperbel brukes også en tredje kanonisk form:<sup id="cite_ref-LAW1_2-2" class="reference"><a href="#cite_note-LAW1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd17f3c891b94f98a5b7d310ceb4d7725f9038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.849ex; height:3.009ex;" alt="{\displaystyle xy=k^{2}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Parametrisk_form">Parametrisk form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=7" title="Rediger avsnitt: Parametrisk form" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=7" title="Rediger kildekoden til seksjonen Parametrisk form"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En hyperbel på standardformen med origo i sentrum kan skrives som en <a href="/wiki/Parameterfremstilling" title="Parameterfremstilling">parameterframstilling</a> på formen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)=\pm a\cosh t\qquad y(t)=b\sinh t\qquad t\in (-\infty ,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mi>a</mi> <mi>cosh</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="2em" /> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="2em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=\pm a\cosh t\qquad y(t)=b\sinh t\qquad t\in (-\infty ,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740cb69229be4248167b288abddfb7a4d7bcfb51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.064ex; height:2.843ex;" alt="{\displaystyle x(t)=\pm a\cosh t\qquad y(t)=b\sinh t\qquad t\in (-\infty ,\infty )}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Konjugerte_hyperbler">Konjugerte hyperbler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=8" title="Rediger avsnitt: Konjugerte hyperbler" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=8" title="Rediger kildekoden til seksjonen Konjugerte hyperbler"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Hyperbler_konjugerte.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Hyperbler_konjugerte.png/250px-Hyperbler_konjugerte.png" decoding="async" width="250" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Hyperbler_konjugerte.png/375px-Hyperbler_konjugerte.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Hyperbler_konjugerte.png/500px-Hyperbler_konjugerte.png 2x" data-file-width="969" data-file-height="783" /></a><figcaption>Konjugerte hyperbler med felles asymptoter</figcaption></figure> <p>To hyperbler er <i>konjugerte</i> dersom hovedaksene til hver av hyperblene står normalt på hverandre, samt at den reelle aksen i den ene er lik den imaginære aksen i den andre og omvendt.<sup id="cite_ref-GULD1_3-1" class="reference"><a href="#cite_note-GULD1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> To konjugerte hyperbler har identiske asymptoter. På standardformen med origo i sentrum kan ligningene for de to konjugerte hyperblene skrives </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 {\begin{alignedat}{2}{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}&=1\\{\frac {x^{2}}{a^{2}}}-{\frac {b^{2}}{a^{2}}}&=-1\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <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> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <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>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> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}&=1\\{\frac {x^{2}}{a^{2}}}-{\frac {b^{2}}{a^{2}}}&=-1\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a2405a3455a7f57c77544f7c368e351ba40610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:16.001ex; height:12.176ex;" alt="{\displaystyle {\begin{alignedat}{2}{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}&=1\\{\frac {x^{2}}{a^{2}}}-{\frac {b^{2}}{a^{2}}}&=-1\end{alignedat}}}"></span></dd></dl> <p>Dersom de to hyperblene har eksentrisitet henholdsvis <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81caf3d4bcb929315801cbabc83543829484ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}"></span> og <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}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4045b5c7cee9bd0681153bbb077489b13269355e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{2}}"></span>, så vil <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 ae_{1}=be_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ae_{1}=be_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dcccfcfceeab42ccb18bdc2031635f2892cb11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.601ex; height:2.509ex;" alt="{\displaystyle ae_{1}=be_{2}}"></span>. De fire brennpunktene til de to hyperblene ligger på en sirkel med radius <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 ae_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ae_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/857b3d5090dd95b5517ec270b6e5bf3fac5a0c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.009ex;" alt="{\displaystyle ae_{1}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Generell_kvadratisk_form">Generell kvadratisk form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=9" title="Rediger avsnitt: Generell kvadratisk form" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=9" title="Rediger kildekoden til seksjonen Generell kvadratisk form"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En generell kvadratisk form </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 f(x,y)=Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>D</mi> <mi>x</mi> <mo>+</mo> <mi>E</mi> <mi>y</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9c4dc31f0a97d3e68fdd7823adfe08e5e0d8df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.451ex; height:3.176ex;" alt="{\displaystyle f(x,y)=Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0}"></span></dd></dl> <p>vil framstillen en hyperbel dersom <i>diskriminanten</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <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> er positiv:<sup id="cite_ref-THOMAS2_4-0" class="reference"><a href="#cite_note-THOMAS2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=B^{2}-4AC>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>A</mi> <mi>C</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=B^{2}-4AC>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97a67e10fd228282b71b966e3890fe712a3fa3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.906ex; height:2.843ex;" alt="{\displaystyle d=B^{2}-4AC>0}"></span></dd></dl> <p>Ligningen kan overføres til standardformen ved hjelp av en koordinattransformasjon: en translasjon og en rotasjon. </p> <div class="mw-heading mw-heading2"><h2 id="Degenerert_hyperbel">Degenerert hyperbel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=10" title="Rediger avsnitt: Degenerert hyperbel" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=10" title="Rediger kildekoden til seksjonen Degenerert hyperbel"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En hyperbel med diskriminant lik null vil degenerere til to kryssende rette linjer dersom determinanten til matriseformen av ligningen er lik null.<sup id="cite_ref-LAW1_2-3" class="reference"><a href="#cite_note-LAW1-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Matriseformen er </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 {\mathsf {x}}{\mathsf {R}}{\mathsf {x}}^{\operatorname {T} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">x</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {x}}{\mathsf {R}}{\mathsf {x}}^{\operatorname {T} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1ba1bf44f874b2d2afab780a6f8cbb16ab21bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.325ex; height:2.676ex;" alt="{\displaystyle {\mathsf {x}}{\mathsf {R}}{\mathsf {x}}^{\operatorname {T} }=0}"></span></dd></dl> <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 {\begin{alignedat}{2}{\mathsf {x}}&=(x,y,1)\\[3pt]{\mathsf {R}}&=\left({\begin{matrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{matrix}}\right)\\[3pt]\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left" rowspacing="0.6em 0.6em" columnspacing="0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">x</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>C</mi> </mtd> <mtd> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{2}{\mathsf {x}}&=(x,y,1)\\[3pt]{\mathsf {R}}&=\left({\begin{matrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{matrix}}\right)\\[3pt]\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d55366296d3d5f70666d08b3e670e26ef2915c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:27.414ex; height:13.509ex;" alt="{\displaystyle {\begin{alignedat}{2}{\mathsf {x}}&=(x,y,1)\\[3pt]{\mathsf {R}}&=\left({\begin{matrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{matrix}}\right)\\[3pt]\end{alignedat}}}"></span></dd></dl> <p>Determinanten <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 \Delta _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02857c65940cb2e79cccb25930bc3a36b3726fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.99ex; height:2.509ex;" alt="{\displaystyle \Delta _{3}}"></span> til matrisen <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 {\mathsf {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4721d2fe15d5584fa7b1a384b079f38ac0d80a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.502ex; height:2.176ex;" alt="{\displaystyle {\mathsf {R}}}"></span> er gitt ved </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 \Delta _{3}=\det {\mathsf {R}}={\begin{vmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">R</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>C</mi> </mtd> <mtd> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta _{3}=\det {\mathsf {R}}={\begin{vmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{vmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a0831ed297bc69853899113c3aba8849ec8d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:33.594ex; height:9.843ex;" alt="{\displaystyle \Delta _{3}=\det {\mathsf {R}}={\begin{vmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{vmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Egenskaper">Egenskaper</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=11" title="Rediger avsnitt: Egenskaper" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=11" title="Rediger kildekoden til seksjonen Egenskaper"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Symmetri">Symmetri</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=12" title="Rediger avsnitt: Symmetri" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=12" title="Rediger kildekoden til seksjonen Symmetri"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hyperbelen er symmetrisk om både den reelle og den imaginære aksen, og dermed også om sentrum. </p> <div class="mw-heading mw-heading3"><h3 id="Tangentlinjer">Tangentlinjer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=13" title="Rediger avsnitt: Tangentlinjer" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=13" title="Rediger kildekoden til seksjonen Tangentlinjer"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For en hyperbel på standardformen med origo i sentrum er ligningen for tangenten i punktet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"></span> gitt ved </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {xx_{0}}{a^{2}}}-{\frac {yy_{0}}{b^{2}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>y</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {xx_{0}}{a^{2}}}-{\frac {yy_{0}}{b^{2}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28176de2bab25b55dcadd7da76f4601f7492b600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.836ex; height:5.176ex;" alt="{\displaystyle {\frac {xx_{0}}{a^{2}}}-{\frac {yy_{0}}{b^{2}}}=0}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Konjungerte_diametre">Konjungerte diametre</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=14" title="Rediger avsnitt: Konjungerte diametre" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=14" title="Rediger kildekoden til seksjonen Konjungerte diametre"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En diameter i hyperbelen er en korde som går gjennom sentrum, mellom to punkt på hyperbelen To diametre i hyperbelen er konjugerte dersom enhver korde parallell med den ene diameteren blir delt i to like deler av den andre.<sup id="cite_ref-SPAIN_5-0" class="reference"><a href="#cite_note-SPAIN-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Anvendelser">Anvendelser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=15" title="Rediger avsnitt: Anvendelser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=15" title="Rediger kildekoden til seksjonen Anvendelser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For en <a href="/wiki/Ideell_gass" title="Ideell gass">ideell gass</a> med konstant temperatur vil variasjon i trykk og volum beskrive en hyperbel. </p> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=16" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=16" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-THOMAS1-1"><b><a href="#cite_ref-THOMAS1_1-0">^</a></b> <span class="reference-text"><a href="#THOMAS">: G. Thomas, R. Finney; <i>Calculus and Analytic Geometry</i></a> s.432 </span> </li> <li id="cite_note-LAW1-2"><b>^</b> <a href="#cite_ref-LAW1_2-0"><sup>a</sup></a> <a href="#cite_ref-LAW1_2-1"><sup>b</sup></a> <a href="#cite_ref-LAW1_2-2"><sup>c</sup></a> <a href="#cite_ref-LAW1_2-3"><sup>d</sup></a> <span class="reference-text"><a href="#LAW">: J.D. Lawrence; <i>A Catalog of Special Plane Curves</i></a> s.61ff </span> </li> <li id="cite_note-GULD1-3"><b>^</b> <a href="#cite_ref-GULD1_3-0"><sup>a</sup></a> <a href="#cite_ref-GULD1_3-1"><sup>b</sup></a> <span class="reference-text"><a href="#GULD">C.M. Guldberg; <i>Analytisk geometri</i></a> s.60ff</span> </li> <li id="cite_note-THOMAS2-4"><b><a href="#cite_ref-THOMAS2_4-0">^</a></b> <span class="reference-text"><a href="#THOMAS">: G. Thomas, R. Finney; <i>Calculus and Analytic Geometry</i></a> s.430 </span> </li> <li id="cite_note-SPAIN-5"><b><a href="#cite_ref-SPAIN_5-0">^</a></b> <span class="reference-text"> <cite class="citation book">Barry Spain (1957). <a rel="nofollow" class="external text" href="http://catalog.hathitrust.org/Record/000660610"><i>Analytical Conics</i></a>. New York: Pergamon Press.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AHyperbel&rft.au=Barry+Spain&rft.btitle=Analytical+Conics&rft.date=1957&rft.genre=book&rft.place=New+York&rft.pub=Pergamon+Press&rft_id=http%3A%2F%2Fcatalog.hathitrust.org%2FRecord%2F000660610&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span> s.39ff</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Litteratur">Litteratur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbel&veaction=edit&section=17" title="Rediger avsnitt: Litteratur" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbel&action=edit&section=17" title="Rediger kildekoden til seksjonen Litteratur"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="LAW" class="citation book">J.Dennis Lawrence (1972). <i>A Catalog of Special Plane Curves</i>. Mineola, New York: Dover Publications. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/978-0-486-60288-2" title="Spesial:Bokkilder/978-0-486-60288-2">978-0-486-60288-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AHyperbel&rft.au=J.Dennis+Lawrence&rft.btitle=A+Catalog+of+Special+Plane+Curves&rft.date=1972&rft.genre=book&rft.isbn=978-0-486-60288-2&rft.place=Mineola%2C+New+York&rft.pub=Dover+Publications&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="THOMAS" class="citation book">George B. Thomas, Ross L. Finney (1995). <i>Calculus and Analytic Geometry</i> (9th edition utg.). Reading, USA: Addison-Wesley. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/0-201-53174-7" title="Spesial:Bokkilder/0-201-53174-7">0-201-53174-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AHyperbel&rft.au=George+B.+Thomas%2C+Ross+L.+Finney&rft.btitle=Calculus+and+Analytic+Geometry&rft.date=1995&rft.edition=9th+edition&rft.genre=book&rft.isbn=0-201-53174-7&rft.place=Reading%2C+USA&rft.pub=Addison-Wesley&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">CS1-vedlikehold: Ekstra tekst (<a href="/wiki/Kategori:CS1-vedlikehold:_Ekstra_tekst" title="Kategori:CS1-vedlikehold: Ekstra tekst">link</a>)</span></li></ul> <ul><li><cite id="GULD" class="citation book">C.M. Guldberg (1941). <a rel="nofollow" class="external text" href="http://urn.nb.no/URN:NBN:no-nb_digibok_2012042608042"><i>Analytisk geometri</i></a>. Oslo: Steensballe. s. 60-69.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AHyperbel&rft.au=C.M.+Guldberg&rft.btitle=Analytisk+geometri&rft.date=1941&rft.genre=book&rft.pages=60-69&rft.place=Oslo&rft.pub=Steensballe&rft_id=http%3A%2F%2Furn.nb.no%2FURN%3ANBN%3Ano-nb_digibok_2012042608042&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li></ul> <ul><li><cite id="refAdams" class="citation book">Adams, Robert A. & Essex, Christopher (2013). <i>Calculus 2</i>. Harlow: Pearson. s. 463-468. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Spesial:Bokkilder/978-1-78365-399-7" title="Spesial:Bokkilder/978-1-78365-399-7">978-1-78365-399-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fno.wikipedia.org%3AHyperbel&rft.au=Adams%2C+Robert+A.+%26+Essex%2C+Christopher&rft.btitle=Calculus+2&rft.date=2013&rft.genre=book&rft.isbn=978-1-78365-399-7&rft.pages=463-468&rft.place=Harlow&rft.pub=Pearson&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span><span class="citation-comment" style="display:none; color:#33aa33; margin-left:0.3em">CS1-vedlikehold: Flere navn: forfatterliste (<a href="/wiki/Kategori:CS1-vedlikehold:_Flere_navn:_forfatterliste" title="Kategori:CS1-vedlikehold: Flere navn: forfatterliste">link</a>)</span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23230704">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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