Iperbòla (matematicas)

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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">Escondre</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">Començament</div> </a> </li> <li id="toc-Definicions_geometricas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definicions_geometricas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicions geometricas</span> </div> </a> <button aria-controls="toc-Definicions_geometricas-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>Commuta la subsecció Definicions geometricas</span> </button> <ul id="toc-Definicions_geometricas-sublist" class="vector-toc-list"> <li id="toc-Interseccion_d'un_còn_e_d'un_plan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interseccion_d'un_còn_e_d'un_plan"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Interseccion d'un còn e d'un plan</span> </div> </a> <ul id="toc-Interseccion_d'un_còn_e_d'un_plan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definicion_per_fogal_e_directritz" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definicion_per_fogal_e_directritz"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Definicion per fogal e directritz</span> </div> </a> <ul id="toc-Definicion_per_fogal_e_directritz-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definicion_bifocala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definicion_bifocala"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Definicion bifocala</span> </div> </a> <ul id="toc-Definicion_bifocala-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Imatge_d'un_cercle_par_una_omografia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Imatge_d'un_cercle_par_una_omografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Imatge d'un cercle par una omografia</span> </div> </a> <ul id="toc-Imatge_d'un_cercle_par_una_omografia-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relacions_entre_las_grandors_caracteristicas_d'una_iperbòla" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relacions_entre_las_grandors_caracteristicas_d'una_iperbòla"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relacions entre las grandors caracteristicas d'una iperbòla</span> </div> </a> <ul id="toc-Relacions_entre_las_grandors_caracteristicas_d'una_iperbòla-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equacions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Equacions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Equacions</span> </div> </a> <button aria-controls="toc-Equacions-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>Commuta la subsecció Equacions</span> </button> <ul id="toc-Equacions-sublist" class="vector-toc-list"> <li id="toc-Equacion_dins_una_marca_normada_portada_per_las_asimptòtas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacion_dins_una_marca_normada_portada_per_las_asimptòtas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Equacion dins una marca normada portada per las asimptòtas</span> </div> </a> <ul id="toc-Equacion_dins_una_marca_normada_portada_per_las_asimptòtas-sublist" class="vector-toc-list"> <li id="toc-Cas_particular_de_la_foncion_invèrsa" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cas_particular_de_la_foncion_invèrsa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Cas particular de la foncion invèrsa</span> </div> </a> <ul id="toc-Cas_particular_de_la_foncion_invèrsa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cas_general" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cas_general"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Cas general</span> </div> </a> <ul id="toc-Cas_general-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equacions_dins_de_marcas_ont_l'axe_focal_es_l'axe_principal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacions_dins_de_marcas_ont_l'axe_focal_es_l'axe_principal"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Equacions dins de marcas ont l'axe focal es l'axe principal</span> </div> </a> <ul id="toc-Equacions_dins_de_marcas_ont_l'axe_focal_es_l'axe_principal-sublist" class="vector-toc-list"> <li id="toc-Se_le_centre_de_la_marca_es_le_centre_de_l'iperbòla" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Se_le_centre_de_la_marca_es_le_centre_de_l'iperbòla"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Se le centre de la marca es le centre de l'iperbòla</span> </div> </a> <ul id="toc-Se_le_centre_de_la_marca_es_le_centre_de_l'iperbòla-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_le_centre_de_la_marca_es_lo_fogal_de_l'iperbòla" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Se_le_centre_de_la_marca_es_lo_fogal_de_l'iperbòla"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Se le centre de la marca es lo fogal de l'iperbòla</span> </div> </a> <ul id="toc-Se_le_centre_de_la_marca_es_lo_fogal_de_l'iperbòla-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equacion_generala_de_conica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacion_generala_de_conica"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Equacion generala de conica</span> </div> </a> <ul id="toc-Equacion_generala_de_conica-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equacion_matriciala" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equacion_matriciala"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Equacion matriciala</span> </div> </a> <ul id="toc-Equacion_matriciala-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proprietats" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Proprietats"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Proprietats</span> </div> </a> <button aria-controls="toc-Proprietats-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>Commuta la subsecció Proprietats</span> </button> <ul id="toc-Proprietats-sublist" class="vector-toc-list"> <li id="toc-Interior_e_exterior" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interior_e_exterior"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Interior e exterior</span> </div> </a> <ul id="toc-Interior_e_exterior-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secantas_e_soms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Secantas_e_soms"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Secantas e soms</span> </div> </a> <ul id="toc-Secantas_e_soms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secantas_e_asimptòtas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Secantas_e_asimptòtas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Secantas e asimptòtas</span> </div> </a> <ul id="toc-Secantas_e_asimptòtas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangentas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangentas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Tangentas</span> </div> </a> <ul id="toc-Tangentas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cercles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cercles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Cercles</span> </div> </a> <ul id="toc-Cercles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Longor_e_airal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Longor_e_airal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Longor e airal</span> </div> </a> <ul id="toc-Longor_e_airal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Istòria" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Istòria"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Istòria</span> </div> </a> <ul id="toc-Istòria-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Annèxas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Annèxas"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Annèxas</span> </div> </a> <button aria-controls="toc-Annèxas-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>Commuta la subsecció Annèxas</span> </button> <ul id="toc-Annèxas-sublist" class="vector-toc-list"> <li id="toc-Articles_connèxes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connèxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Articles connèxes</span> </div> </a> <ul id="toc-Articles_connèxes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligams_extèrnes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligams_extèrnes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Ligams extèrnes</span> </div> </a> <ul id="toc-Ligams_extèrnes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nòtas_e_referéncias" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nòtas_e_referéncias"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Nòtas e referéncias</span> </div> </a> <ul id="toc-Nòtas_e_referéncias-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="Somari" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Commuta la taula de continguts." > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Commuta la taula de continguts.</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Iperbòla (matematicas)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vés a un article en una altra llengua. Disponible en 74 llengües" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-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 lengas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/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 - arabi" lang="ar" hreflang="ar" data-title="قطع زائد" data-language-autonym="العربية" data-language-local-name="arabi" 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 - asturian" lang="ast" hreflang="ast" data-title="Hipérbola" data-language-autonym="Asturianu" data-language-local-name="asturian" 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) - azerbaijani" lang="az" hreflang="az" data-title="Hiperbola (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%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) - baixkir" lang="ba" hreflang="ba" data-title="Гипербола (математика)" data-language-autonym="Башҡортса" data-language-local-name="baixkir" 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) - belarús" lang="be" hreflang="be" data-title="Гіпербала (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="belarús" 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) - Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Гіпэрбала (геамэтрыя)" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола - bulgar" lang="bg" hreflang="bg" data-title="Хипербола" data-language-autonym="Български" data-language-local-name="bulgar" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A7%E0%A6%BF%E0%A6%AC%E0%A7%83%E0%A6%A4%E0%A7%8D%E0%A6%A4" title="অধিবৃত্ত - bengalin" lang="bn" hreflang="bn" data-title="অধিবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengalin" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Hiperbola" title="Hiperbola - bosniac" lang="bs" hreflang="bs" data-title="Hiperbola" data-language-autonym="Bosanski" data-language-local-name="bosniac" 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 - catalan" lang="ca" hreflang="ca" data-title="Hipèrbola" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</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 - kurd central" lang="ckb" hreflang="ckb" data-title="بڕگەی زیاد" data-language-autonym="کوردی" data-language-local-name="kurd central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hyperbola" title="Hyperbola - chèc" lang="cs" hreflang="cs" data-title="Hyperbola" data-language-autonym="Čeština" data-language-local-name="chèc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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) - chovash" lang="cv" hreflang="cv" data-title="Гипербола (математика)" data-language-autonym="Чӑвашла" data-language-local-name="chovash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Hyperbola" title="Hyperbola - gal·lès" lang="cy" hreflang="cy" data-title="Hyperbola" data-language-autonym="Cymraeg" data-language-local-name="gal·lès" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hyperbel" title="Hyperbel - danés" lang="da" hreflang="da" data-title="Hyperbel" data-language-autonym="Dansk" data-language-local-name="danés" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hyperbel_(Mathematik)" title="Hyperbel (Mathematik) - alemand" lang="de" hreflang="de" data-title="Hyperbel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%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) - grèc" lang="el" hreflang="el" data-title="Υπερβολή (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="grèc" 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 - anglés" lang="en" hreflang="en" data-title="Hyperbola" data-language-autonym="English" data-language-local-name="anglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola" title="Hipérbola - espanhòl" lang="es" hreflang="es" data-title="Hipérbola" data-language-autonym="Español" data-language-local-name="espanhòl" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/H%C3%BCperbool" title="Hüperbool - estonian" lang="et" hreflang="et" data-title="Hüperbool" data-language-autonym="Eesti" data-language-local-name="estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Hiperbola" title="Hiperbola - basc" lang="eu" hreflang="eu" data-title="Hiperbola" data-language-autonym="Euskara" data-language-local-name="basc" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D8%B0%D9%84%D9%88%D9%84%DB%8C" title="هذلولی - perse" lang="fa" hreflang="fa" data-title="هذلولی" data-language-autonym="فارسی" data-language-local-name="perse" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hyperbeli" title="Hyperbeli - finlandés" lang="fi" hreflang="fi" data-title="Hyperbeli" data-language-autonym="Suomi" data-language-local-name="finlandés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Hyperbole_(math%C3%A9matiques)" title="Hyperbole (mathématiques) - francés" lang="fr" hreflang="fr" data-title="Hyperbole (mathématiques)" data-language-autonym="Français" data-language-local-name="francés" 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 - irlandés" lang="ga" hreflang="ga" data-title="Hipearbóil" data-language-autonym="Gaeilge" data-language-local-name="irlandés" 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) - galician" lang="gl" hreflang="gl" data-title="Hipérbole (xeometría)" data-language-autonym="Galego" data-language-local-name="galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%99%D7%A4%D7%A8%D7%91%D7%95%D7%9C%D7%94" title="היפרבולה - ebrèu" lang="he" hreflang="he" data-title="היפרבולה" data-language-autonym="עברית" data-language-local-name="ebrèu" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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 - Indi" lang="hi" hreflang="hi" data-title="अति परवलय" data-language-autonym="हिन्दी" data-language-local-name="Indi" 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) - croat" lang="hr" hreflang="hr" data-title="Hiperbola (krivulja)" data-language-autonym="Hrvatski" data-language-local-name="croat" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hiperbola_(matematika)" title="Hiperbola (matematika) - ongrés" lang="hu" hreflang="hu" data-title="Hiperbola (matematika)" data-language-autonym="Magyar" data-language-local-name="ongrés" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%BA%D5%A5%D6%80%D5%A2%D5%B8%D5%AC" title="Հիպերբոլ - armèni" lang="hy" hreflang="hy" data-title="Հիպերբոլ" data-language-autonym="Հայերեն" data-language-local-name="armèni" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Hiperbola" title="Hiperbola - indonesian" lang="id" hreflang="id" data-title="Hiperbola" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Brei%C3%B0bogi" title="Breiðbogi - islandés" lang="is" hreflang="is" data-title="Breiðbogi" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Iperbole_(geometria)" title="Iperbole (geometria) - italian" lang="it" hreflang="it" data-title="Iperbole (geometria)" data-language-autonym="Italiano" data-language-local-name="italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A" title="双曲線 - japonés" lang="ja" hreflang="ja" data-title="双曲線" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</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 - georgian" lang="ka" hreflang="ka" data-title="ჰიპერბოლა" data-language-autonym="ქართული" data-language-local-name="georgian" class="interlanguage-link-target"><span>ქართული</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 - cambojian" lang="km" hreflang="km" data-title="អ៊ីពែបូល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="cambojian" class="interlanguage-link-target"><span>ភាសាខ្មែរ</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 - corean" lang="ko" hreflang="ko" data-title="쌍곡선" data-language-autonym="한국어" data-language-local-name="corean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%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 - kirguís" lang="ky" hreflang="ky" data-title="Гипербола математикада" data-language-autonym="Кыргызча" data-language-local-name="kirguís" 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-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Hiperbol%C4%97_(matematika)" title="Hiperbolė (matematika) - lituan" lang="lt" hreflang="lt" data-title="Hiperbolė (matematika)" data-language-autonym="Lietuvių" data-language-local-name="lituan" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Hiperbola" title="Hiperbola - leton" lang="lv" hreflang="lv" data-title="Hiperbola" data-language-autonym="Latviešu" data-language-local-name="leton" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола - macedonian" lang="mk" hreflang="mk" data-title="Хипербола" data-language-autonym="Македонски" data-language-local-name="macedonian" 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 - malaiàlam" lang="ml" hreflang="ml" data-title="അധിവലയം" data-language-autonym="മലയാളം" data-language-local-name="malaiàlam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hyperbool_(meetkunde)" title="Hyperbool (meetkunde) - neerlandés" lang="nl" hreflang="nl" data-title="Hyperbool (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Hyperbel" title="Hyperbel - norvegian nynorsk" lang="nn" hreflang="nn" data-title="Hyperbel" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegian nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Hyperbel" title="Hyperbel - norvegian bokmål" lang="nb" hreflang="nb" data-title="Hyperbel" data-language-autonym="Norsk bokmål" data-language-local-name="norvegian bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Hiperbola_(matematyka)" title="Hiperbola (matematyka) - polonés" lang="pl" hreflang="pl" data-title="Hiperbola (matematyka)" data-language-autonym="Polski" data-language-local-name="polonés" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Ip%C3%A9rbol" title="Ipérbol - piemontès" lang="pms" hreflang="pms" data-title="Ipérbol" data-language-autonym="Piemontèis" data-language-local-name="piemontès" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Hip%C3%A9rbole" title="Hipérbole - portugués" lang="pt" hreflang="pt" data-title="Hipérbole" data-language-autonym="Português" data-language-local-name="portugués" 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 - romanés" lang="ro" hreflang="ro" data-title="Hiperbolă" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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) - rus" lang="ru" hreflang="ru" data-title="Гипербола (математика)" data-language-autonym="Русский" data-language-local-name="rus" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Гипербола - ruthène" lang="rue" hreflang="rue" data-title="Гипербола" data-language-autonym="Русиньскый" data-language-local-name="ruthène" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Ip%C3%A8rbuli_(matim%C3%A0tica)" title="Ipèrbuli (matimàtica) - sicilian" lang="scn" hreflang="scn" data-title="Ipèrbuli (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Hyperbola" title="Hyperbola - escossés" lang="sco" hreflang="sco" data-title="Hyperbola" data-language-autonym="Scots" data-language-local-name="escossés" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hiperbola" title="Hiperbola - serbocroat" lang="sh" hreflang="sh" data-title="Hiperbola" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroat" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Hyperbola" title="Hyperbola - Simple English" lang="en-simple" hreflang="en-simple" data-title="Hyperbola" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Hyperbola_(matematika)" title="Hyperbola (matematika) - eslovac" lang="sk" hreflang="sk" data-title="Hyperbola (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovac" 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 - eslovèn" lang="sl" hreflang="sl" data-title="Hiperbola" data-language-autonym="Slovenščina" data-language-local-name="eslovèn" class="interlanguage-link-target"><span>Slovenščina</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) - albanés" lang="sq" hreflang="sq" data-title="Hiperbola (matematikë)" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B0" title="Хипербола - serbi" lang="sr" hreflang="sr" data-title="Хипербола" data-language-autonym="Српски / srpski" data-language-local-name="serbi" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hyperbel" title="Hyperbel - suedés" lang="sv" hreflang="sv" data-title="Hyperbel" data-language-autonym="Svenska" data-language-local-name="suedés" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%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 - tamol" lang="ta" hreflang="ta" data-title="அதிபரவளைவு" data-language-autonym="தமிழ்" data-language-local-name="tamol" 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 - tai" lang="th" hreflang="th" data-title="ไฮเพอร์โบลา" data-language-autonym="ไทย" data-language-local-name="tai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Hiperbol" title="Hiperbol - turc" lang="tr" hreflang="tr" data-title="Hiperbol" data-language-autonym="Türkçe" data-language-local-name="turc" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%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) - ucrainés" lang="uk" hreflang="uk" data-title="Гіпербола (математика)" data-language-autonym="Українська" data-language-local-name="ucrainés" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Giperbola" title="Giperbola - ozbèc" lang="uz" hreflang="uz" data-title="Giperbola" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ozbèc" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Hyperbol" title="Hyperbol - vietnamian" lang="vi" hreflang="vi" data-title="Hyperbol" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamian" class="interlanguage-link-target"><span>Tiếng Việt</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 - xinès wu" lang="wuu" hreflang="wuu" data-title="双曲线" data-language-autonym="吴语" data-language-local-name="xinès wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%BA%BF" title="双曲线 - chinés" lang="zh" hreflang="zh" data-title="双曲线" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9B%99%E6%9B%B2%E7%B7%9A" title="雙曲線 - 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Imprimir / exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libre&bookcmd=book_creator&referer=Iperb%C3%B2la+%28matematicas%29"><span>Crear un libre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Iperb%C3%B2la_%28matematicas%29&action=show-download-screen"><span>Telecargar coma PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&printable=yes" title="Version imprimibla d'aquesta pagina [p]" accesskey="p"><span>Version imprimibla</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dins d'autres projèctes </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparença</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mòu a la barra laterala</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">Escondre</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">Un article de Wikipèdia, l'enciclopèdia liura.</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="oc" dir="ltr"><div class="noprint" style="width: 100%; background-color: #f5f5f5; margin-bottom: 1em; position:relative;"> <table style="background-color: #f5f5f5" cellspacing="5" class="dablink"> <tbody><tr> <td style="width: 25px; vertical-align: top"><span typeof="mw:File"><span title="talha="><img alt="talha=" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/22px-Disambig_grey.svg.png" decoding="async" width="22" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/33px-Disambig_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Disambig_grey.svg/44px-Disambig_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></span></span> </td> <td style="font-style: italic;font-size: 12px;"><i>Pels articles omonims, vejatz <a href="/wiki/Iperb%C3%B2la" title="Iperbòla">Iperbòla</a>.</i> </td></tr></tbody></table></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Cono_-_hip%C3%A9rbola.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Cono_-_hip%C3%A9rbola.svg/227px-Cono_-_hip%C3%A9rbola.svg.png" decoding="async" width="227" height="396" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Cono_-_hip%C3%A9rbola.svg/341px-Cono_-_hip%C3%A9rbola.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Cono_-_hip%C3%A9rbola.svg/455px-Cono_-_hip%C3%A9rbola.svg.png 2x" data-file-width="2240" data-file-height="3900" /></a><figcaption>Iperbòla obtenguda coma interseccion d'un còn e d'un plan</figcaption></figure> <p>En <a href="/wiki/Matematicas" title="Matematicas">matematicas</a>, una <b>iperbòla</b> es une corba plana obtenguda coma la dobla interseccion d'un doble <a href="/wiki/C%C3%B2n" title="Còn">còn</a> de revolucion amb un plan. Pòt tanben se definir coma una <a href="/wiki/Conica" title="Conica">conica</a> d'excentricitat superiora a 1 o coma l'ensems dels punts que la diferéncia de las distàncias a dos punts fixes es constanta. </p><p>Una iperbòla es constituida de dos brancas disjonchas simetricas l'una de l'autra e possedissent doas asimptotas comunas. </p><p>Se pòt encontrar l'iperbòla dins fòrça circonstanças coma quand se representa graficament la foncion invèrsa, <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\to 1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to 1/x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60c49d7ec69a7c86a641374853b0272a6bf57337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.598ex; height:2.843ex;" alt="{\displaystyle x\to 1/x}"></span> e aquela de totas las foncions que li son associadas: <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\to ax+b+{\frac {c}{x+d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>x</mi> <mo>+</mo> <mi>d</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to ax+b+{\frac {c}{x+d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fd7ab994c57aa4b20a4af27b3ea435210e82da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.404ex; height:5.009ex;" alt="{\displaystyle x\to ax+b+{\frac {c}{x+d}}}"></span>, o encara dins l'ombra creada per una font de lutz sus una paret, dins la trajectòria d'unes còs dins l'espaci o dins las interferéncias produchas per doas fonts d'ondulacions de meteissa frequéncia. Es tanben la corba seguida, pendent una jornada, pel tèrme de l'ombre del gnomòn de la mòstra de solelh  d'estil polar. </p><p>L'iperbòla interven dins d'autres objèctes matematics coma los iperboloíds, lo paraboloíd iperbolic, las foncions iperbolicas. Sa quadratura, es a dire lo calcul de l'airal comprés entre una porcion d'iperbòla e son axe principal, es a l'origina de la creation de la <a href="/wiki/Logaritme" title="Logaritme">foncion logaritme</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperbolaeDrawnByHalogenLamp.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HyperbolaeDrawnByHalogenLamp.jpg/220px-HyperbolaeDrawnByHalogenLamp.jpg" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/HyperbolaeDrawnByHalogenLamp.jpg/330px-HyperbolaeDrawnByHalogenLamp.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/HyperbolaeDrawnByHalogenLamp.jpg/440px-HyperbolaeDrawnByHalogenLamp.jpg 2x" data-file-width="1536" data-file-height="2048" /></a><figcaption>Arc d'iperbòla dessenhada per l'ombre creada per una lampa</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicions_geometricas">Definicions geometricas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=1" title="Modificar la seccion : Definicions geometricas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=1" title="Edita el codi de la secció: Definicions geometricas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Interseccion_d'un_còn_e_d'un_plan"><span id="Interseccion_d.27un_c.C3.B2n_e_d.27un_plan"></span>Interseccion d'un còn e d'un plan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=2" title="Modificar la seccion : Interseccion d'un còn e d'un plan" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=2" title="Edita el codi de la secció: Interseccion d'un còn e d'un plan"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se considèra un còn de revolucion engendrada per la rotacion d'una drecha (OA) a l'entorn d'un axe (Ox) e se nomena θ l'angle geometric entre aquelas doas drechas. Se pren d'autre mena un plan que la normala fach amb l'axe (Ox) un angle superieur a π/2-θ. Se lo plan passe pas per O, trenca lo còn seguent una iperbòla. Se lo plan passe per O, trenca lo còn segon doas drechas secantas en O. </p><p>Quand una lampa amb paralum es plaçada près d'una paret verticala, la corba que delimita, sus la paret, la zona enlusida e la zona ombrejada es un arc d'iperbòla. De fach, la lutz es difusada segon un <a href="/wiki/C%C3%B2n" title="Còn">còn</a> — los rais luminoses partisson del centre de l'ampola e s'apièjan sulcercle de l'overtura de la lampa — trencat par un plan parallèl a l'axe del còn — la paret. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Gnomonische_Projektion.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Gnomonische_Projektion.png/220px-Gnomonische_Projektion.png" decoding="async" width="220" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Gnomonische_Projektion.png/330px-Gnomonische_Projektion.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Gnomonische_Projektion.png/440px-Gnomonische_Projektion.png 2x" data-file-width="803" data-file-height="531" /></a><figcaption>Principi de las linhas de declinason d'una mòstra de solelh</figcaption></figure> <p>Es tanben lo meteis principi qu'explica l'existéncia d'iperbòlas sus una mena de mòstra de solelh. Pendent una jornada, los rais solars passant per la punta del gnomòn dessenhan una partida de còn que l'axe, parallèl a l'axe de rotacion de la Tèrra, passa per la punta del gnomòn. L'ombre d'aquela punta, sul plan de la mòstra del solelh dessenha alara una partida d'iperbòla<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, nomenada arc diurn o linha de declinason, intersection del còn e d'un plan. Pendent l'annada, l'angle del còn varia. Als equinòccis, es de 90°, lo <i>còn</i> es un plan e l'ombra dessenha una drecha. Als solsticis, l'angle es de 66° 34' e l'ombra dessenha una iperbòla. </p><p>La construccion de l'iperbòla coma seccion d'un còn e d'un plan se pòt relizar amb un compàs perfièch. </p> <div class="mw-heading mw-heading3"><h3 id="Definicion_per_fogal_e_directritz">Definicion per fogal e directritz</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=3" title="Modificar la seccion : Definicion per fogal e directritz" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=3" title="Edita el codi de la secció: Definicion per fogal e directritz"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleFoyerDirectrice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/220px-HyperboleFoyerDirectrice.svg.png" decoding="async" width="220" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/330px-HyperboleFoyerDirectrice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/440px-HyperboleFoyerDirectrice.svg.png 2x" data-file-width="472" data-file-height="479" /></a><figcaption>Iperbòla d'excentricitat 3/2, amb son fogal F, sa directritz (D), sas asimptotas e son cercle principal.<br /> La distància MF es totjorn egala a un còp e mièg la distància MH</figcaption></figure> <p>Sián (D) una <a href="/wiki/Drecha_(matematicas)" title="Drecha (matematicas)">drecha</a> e F un <a href="/wiki/Punt_(geometria)" title="Punt (geometria)">punt</a> apartenent pas a (D), e siá P lo plan contenent la drecha (D) e lo punt F. Se nomena iperbòla de <i>drecha directritz</i> (D) e de <i>fogal</i> F l'ensems dels punts M del plan P verificant </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 \qquad {\frac {\mathrm {d} (\mathrm {M} ,\mathrm {F} )}{\mathrm {d} (\mathrm {M} ,(\mathrm {D} ))}}=e\qquad e>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo>,</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>e</mi> <mspace width="2em" /> <mi>e</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad {\frac {\mathrm {d} (\mathrm {M} ,\mathrm {F} )}{\mathrm {d} (\mathrm {M} ,(\mathrm {D} ))}}=e\qquad e>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b903ffe5f556f7e6b30f6c0c5b4eac90b3c4eba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.504ex; height:6.509ex;" alt="{\displaystyle \qquad {\frac {\mathrm {d} (\mathrm {M} ,\mathrm {F} )}{\mathrm {d} (\mathrm {M} ,(\mathrm {D} ))}}=e\qquad e>1}"></span></dd></dl> <p>ont d(M, F) mesura la <a href="/w/index.php?title=Dist%C3%A0ncia_(matematica)&action=edit&redlink=1" class="new" title="Distància (matematica) (la pagina existís pas)">distància</a> del punt M al punt F e d(M, (D)) mesura la distància del punt M a la drecha (D). </p><p>La constanta <i>e</i> es nomenda <i>excentricitat</i> de l'iperbòla. Es caracteristica de la forma de l'iperbòla: se se transfòrma l'iperbòla per una similitud, son excentricitat demora incambiada. Es donc independenta de la causida arbitrària de la marca ortonormada per aquel plan; determina totes los autres rapòrts de distàcias (e totas las diferéncias angularas) mesurats sus l'iperbòla. Mai <i>e</i> es grand, mai l'iperbòla s'envasa, las doas brancas s'apròchant de la directritz. Mai <i>e</i> s'apròcha d'1 mai l'iperbòla se redondís, las doas brancas s'alunhant l'una de l'autra, aquela situada dins lo meteis miègplan que lo fogal s'aprochant d'una <a href="/wiki/Parab%C3%B2la" title="Parabòla">parabòla</a>. </p><p>Notam K lo projectat ortogonal d'F sus (D). (KF) es alara clarament un axe de simetria de l'iperbòla nomenada <i>axe focal</i>. </p><p> L'axe focal trenca l'iperbòla en dos punts nomenats los <i>soms</i> S e S' de l'iperbòla.</p><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:ConeHyperbole.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/ConeHyperbole.svg/220px-ConeHyperbole.svg.png" decoding="async" width="220" height="285" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/ConeHyperbole.svg/330px-ConeHyperbole.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/ConeHyperbole.svg/440px-ConeHyperbole.svg.png 2x" data-file-width="730" data-file-height="945" /></a><figcaption>Traça del còn (Γ), del plan (P) , de l'esfèra (Σ) e del plan (P') dins lo plan passant per l'axe del còn e perpendicular a (P). La seccion del plan e del còn es una iperbòla de fogal F de directritz passant per K, de som S e d'excentricitat SF/SK.</figcaption></figure><p>La mediatritz del segment [SS'] es, tanben un axe de simetria de l'iperbòla nomenat <i>axe non focal</i>. Lo punt d'interseccion dels dos axes, notat O, es alara lo <i>centre</i> de simetria de l'iperbòla. </p><p>Lo cercle de diamètre [SS'] es nomenat <i>cercle principal</i> de l'iperbòla. </p><p>La simetria ortogonala al respècte de l'axe non focal envia lo fogal F e la directritz (D) en F' e (D'). Per simetria, l'iperbòla es tanben l'iperbòla de fogal F', de directritz (D') e d'excentricitat <i>e</i>. </p><p>Una tala iperbòla possedís en mai doas asimptòtas passant per O e pels punts d'interseccion del cercle principal e de las directriças. Aqueles punts son tanben los projectats ortogonals dels fogals sus las asimptotas<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. Aquelas doas asimptòtas son perpendicularas, se dich que l'iperbòla es <i>equilatèra</i>. </p><p>En relacion amb la definition precedenta, l'iperbòla obtenguda coma seccion de còn e de plan pòt èsser definida per fogal e directritz. Se considèra una esfèra (Σ) inscricha dins lo còn (Γ) e tocant lo plan (P) en F (esphèra de Dandelin) e (P') lo plan contenent lo cercle de tangéncia de l'esfèra e del còn. L'iperbòla es de fogal F e de directritz (D) drecha d'interseccion dels dos plans (P) e (P'). Dins lo plan perpendicular a (P) e passant per l'axe del còn, se trapan lo punt F, lo som S e lo punt K. L'excentricitat es donada pel rapòrt SF/SK. Depend pas que de l'inclinason del plan al respècte de l'axe del còn. Se se nomena (d) la traça de (P) dins lo plan perpendicular a (P) passant per l'axe del còn, se se nota α l'angle entre (d) e l'axe del còn e θ l'angle du còn, l'excentricitat es de cos(α)/cos(θ). </p><p>Aquela relacion entre fogal, directritz e excentricitat dins una iperbòla es expleitada dins la construccion de las lentilhas divergentas: se l'indici de la lentilha al respècte de la mitat es de <i>e</i>, e se la superfícia concava de la lentilha es una iperbòla de fogal F e d'excentricitat <i>e</i>, lo fais de rais parallèls passant la lentilha, s'espandís coma se los rais venián del fogal F<sup id="cite_ref-mathcurve_3-0" class="reference"><a href="#cite_note-mathcurve-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Definicion_bifocala">Definicion bifocala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=4" title="Modificar la seccion : Definicion bifocala" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=4" title="Edita el codi de la secció: Definicion bifocala"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleLentille.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/HyperboleLentille.svg/220px-HyperboleLentille.svg.png" decoding="async" width="220" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/HyperboleLentille.svg/330px-HyperboleLentille.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/HyperboleLentille.svg/440px-HyperboleLentille.svg.png 2x" data-file-width="324" data-file-height="342" /></a><figcaption>Proprietat de divergéncia d'una lentilha iperbolica</figcaption></figure> <p>L'iperbòla es lo luòc geometric dels punts que la diferéncia de las distància als dos fogals es constanta. </p><p>Geometricament, aquò fa: </p><p>Sián F e F' dos punts distinctes del plan, distants de 2<i>c</i> e siá <i>a</i> un real estrictament comprés entre 0 e <i>c</i>. Se nomena iperbole de fogals F e F' l'ensems dels punts M del plan verificant la proprietat seguenta: </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 \qquad \mid \mathrm {d} (\mathrm {M} ,\mathrm {F} )-\mathrm {d} (\mathrm {M} ,\mathrm {F} ')\mid =2a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="2em" /> <mo stretchy="false">∣</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>∣=</mo> <mn>2</mn> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \qquad \mid \mathrm {d} (\mathrm {M} ,\mathrm {F} )-\mathrm {d} (\mathrm {M} ,\mathrm {F} ')\mid =2a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07c183c683f5e248a295af82c9689d11936ad9ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.171ex; height:3.009ex;" alt="{\displaystyle \qquad \mid \mathrm {d} (\mathrm {M} ,\mathrm {F} )-\mathrm {d} (\mathrm {M} ,\mathrm {F} ')\mid =2a.}"></span></dd></dl> <p>L'axe focal es lo nom de la drecha portant ambedos fogals: es un dels dos axes de simetria de l'iperbòla, lo sol que la trenca. Per aquela rason, se lo nomena atal <i>axe traversièr</i> e sos punts comuns amb la corba son los <i>soms</i> S e S' de l'iperbòla. Lo real <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> de la definition çai dessús aparéis coma la mitat de la distància entre los soms. Las directriças de l'iperbòla passan pel punt del contacte de las tangentas al cercle principal (cercle de diamètre [SS']) eissits dels fogals<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>. </p><p>En cada punt M d'aquela iperbòla, la bisectritz del sector angular (FMF') se trapan èsser a tangenta en M a la corba. </p><p>Aquela construccion de l'iperbòla permet d'explicar la preséncia d'iperbòla pendent d'interferéncias entre doas fonts de meteissa frequéncia. Al punt M lo desfasatge de l'onda al respècte de la font S<sub>1</sub> es proporcional a la distància MS<sub>1</sub>. Lo desfasatge entre l'onda venent de S<sub>1</sub> e l'onda venent de S<sub>2</sub> es donc proporcional a la diferéncia de las distàncias. Ambedoas ondas s'anullan quand lor desfasatge es egal a (2k+1)π. Ambedoas ondas s'amplifican quand lor desfasatge es de 2kπ. Los punts ont l'onda resultanta a una amplitud nulla e los punts ont l'onda resultanta a una amplitud maximala dessenhant donc un fais d'iperbòlas de fogals S<sub>1</sub> e S<sub>2</sub>. </p><p>Existisson de mecanismes amb còrda e carèla expleitant aquela proprietat de las iperbòlas per dessenhar lor traçat. es lo cas del dispositiu realizat per <a href="/w/index.php?title=Ibn_Sahl&action=edit&redlink=1" class="new" title="Ibn Sahl (la pagina existís pas)">Ibn Sahl</a> al <a href="/wiki/S%C3%A8gle_X" title="Sègle X">sègle X</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>. </p> <table style="margin-bottom:10px;" class="cx-highlight" width="100%" align="center" border="0" cellpadding="4" cellspacing="4"> <tbody><tr> <td width="50%"> <figure class="mw-halign-center" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/IbnSahlHyperboleAnim.ogv/207px--IbnSahlHyperboleAnim.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="207" height="251" data-durationhint="47" data-mwtitle="IbnSahlHyperboleAnim.ogv" data-mwprovider="wikimediacommons" resource="/wiki/Fichi%C3%A8r:IbnSahlHyperboleAnim.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/5/5b/IbnSahlHyperboleAnim.ogv" type="video/ogg; codecs="theora"" data-width="380" data-height="460" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5b/IbnSahlHyperboleAnim.ogv/IbnSahlHyperboleAnim.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="118" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5b/IbnSahlHyperboleAnim.ogv/IbnSahlHyperboleAnim.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="198" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5b/IbnSahlHyperboleAnim.ogv/IbnSahlHyperboleAnim.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="298" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/5b/IbnSahlHyperboleAnim.ogv/IbnSahlHyperboleAnim.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="298" data-height="360" /></video></span><figcaption>Construccion mecanica d'una iperbòla segon lo metòde d'Ibn Sahl. La règla pivòta a l'entorn del fogal F<sub>2</sub>. Lo gredon M, sus la règla s'apièja sus una còrda F<sub>1</sub>MD de longor fixe e dessenha una partida d'iperbòla de fogals F<sub>1</sub> e F<sub>2</sub>.</figcaption></figure> </td> <td width="50%"> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:Nodalandantinodallines.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Nodalandantinodallines.png/310px-Nodalandantinodallines.png" decoding="async" width="310" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1b/Nodalandantinodallines.png 1.5x" data-file-width="410" data-file-height="308" /></a><figcaption>Interferéncias produchas per doas fonts sincrònas amb las linhas nodalas e antinodalas</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Imatge_d'un_cercle_par_una_omografia"><span id="Imatge_d.27un_cercle_par_una_omografia"></span>Imatge d'un cercle par una omografia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=5" title="Modificar la seccion : Imatge d'un cercle par una omografia" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=5" title="Edita el codi de la secció: Imatge d'un cercle par una omografia"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Coma tota conica, l'iperbòla se pòt considerar coma l'imatge d'un cercle per una transformacion projectiva. Mai precisament, se (H) es una iperbòla e (C) un cercle, existís una transformacion projectiva que transforma (C) en (H). Per exemple, l'iperbòla es l'imatge de son cercle principal per una transformacion projectiva que l'expression analitica, dins una marca ortonormada portada pels sieus axes de simetria, es </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x'={\frac {a^{2}}{x}}\\y'={\frac {by}{x}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>y</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x'={\frac {a^{2}}{x}}\\y'={\frac {by}{x}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e21240f69b293b893b74caa46764e20d065a8f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:10.574ex; height:7.509ex;" alt="{\displaystyle {\begin{cases}x'={\frac {a^{2}}{x}}\\y'={\frac {by}{x}}.\end{cases}}}"></span> </p><p>ont <i>a</i> es lo rai del cercle principal e ± <i>b/a</i> los penjals de las sieunas asimptòtas. </p><p>Aquela proprietat permet de transferir a l'iperbòla de las proprietats tocant de las règlas d'incidéncia de drechas dins un cercle, coma lo teorèma de Pascal. </p><p>D'entra las transformacions projectivas, n'i a de particularas coma las omologias armonicas. Son d'omologias involutivas de centre I d'axe (d) e de rapòrt -1. Dins aquelas, un punt M, son imatge M', lo centre I e lo punt m d'interseccion de (IM) amb (d) son en division armonica. </p><p>Un cercle (C) a per imatge una iperbòla per una omologia armonica de centre I e d'axe (d) se e pas que se l'imatge (d') de l'axe (d) per l'omotecia de centre I e de rapòrt 1/2 trenca lo cercle en dos punts<sup id="cite_ref-Guillerault_6-0" class="reference"><a href="#cite_note-Guillerault-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>. Reciprocament, una iperbòla essent donada, es possible de trobar de cercles e de las omologias armonicas associadas escambiant iperbòla e cercle. Per exemple: </p> <ul><li>Una iperbòla de fogal F, de directritz (d) e d'excentricitat <i>e</i> es l'imatge del cercle de centre F e de rai <i>eh</i> (ont <i>h</i> es la distància entre lo fogal e la directritz) per l'omologia armonica de centre F e d'axe (d<sub>1</sub>) imatge de (d) dins l'homotecia de centre F e de rapòrt 2<sup id="cite_ref-Guillerault_6-1" class="reference"><a href="#cite_note-Guillerault-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>.</li> <li>Una iperbòla equilatèra es l'imatge de son cercle principal per una omologia armonica de centre un dels soms e d'axe (d) passant per l'autre som e perpendicular a l'axe principal de l'iperbòla. Quand l'iperbòla es pas equilatèra, demora l'imatge d'un cercle per una omologia armonica de centre un dels soms, mas lo cercle es pas mai le cercle principal e l'axe passa pas mai per l'autre som. Son d'omologias d'aquel tipe que permeton de montrar le teorèma de Frégier.</li></ul> <table class="center" border="0" cellpadding="4" cellspacing="4"> <tbody><tr> <td width="50%"> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HomologieFoyerDirectrice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HomologieFoyerDirectrice.svg/220px-HomologieFoyerDirectrice.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HomologieFoyerDirectrice.svg/330px-HomologieFoyerDirectrice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/HomologieFoyerDirectrice.svg/440px-HomologieFoyerDirectrice.svg.png 2x" data-file-width="580" data-file-height="426" /></a><figcaption>L'iperbòla e lo cercle son omològs dins l'omologia armonica de centre F e d'axe (d<sub>1</sub>)</figcaption></figure> </td> <td width="50%"> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HomologieHyperboleEquilatere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/HomologieHyperboleEquilatere.svg/220px-HomologieHyperboleEquilatere.svg.png" decoding="async" width="220" height="161" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/HomologieHyperboleEquilatere.svg/330px-HomologieHyperboleEquilatere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/HomologieHyperboleEquilatere.svg/440px-HomologieHyperboleEquilatere.svg.png 2x" data-file-width="180" data-file-height="132" /></a><figcaption>L'iperbòla équilatèra e son cercle principal son omològs dns l'omologia armonica de centre S<sub>2</sub> e d'axe (d). Per construire lo punt M' se dessenha per m una tangenta al cercle qui tòca lo cercle en M. La drecha (S<sub>2</sub>M) encontra la perpendiculara a l'axe focal passant per m en M'</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Relacions_entre_las_grandors_caracteristicas_d'una_iperbòla"><span id="Relacions_entre_las_grandors_caracteristicas_d.27una_iperb.C3.B2la"></span>Relacions entre las grandors caracteristicas d'una iperbòla</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=6" title="Modificar la seccion : Relacions entre las grandors caracteristicas d'una iperbòla" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=6" title="Edita el codi de la secció: Relacions entre las grandors caracteristicas d'una iperbòla"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleParametre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/220px-HyperboleParametre.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/330px-HyperboleParametre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/440px-HyperboleParametre.svg.png 2x" data-file-width="173" data-file-height="125" /></a><figcaption>Las diferentas grandors d'una iperbòla</figcaption></figure> <p>Las grandors (geometricas o numericas) d’una iperbòla son: </p> <ul><li>la distància entre lo centre de l'iperbòla e un dels sieus soms mai sovent notada <i>a</i>;</li> <li>lo penjal (en valor absoluda) que fan las asimptòtas amb l'axe focal, mai sovent notat <i>b/a</i>;</li> <li>la distància separant lo centre de l'iperbòla e un dels fogals, mai sovent notada <i>c</i>;</li> <li>la distància separant un fogal F de sa directritz (<i>d</i>) associada, mai sovent notada <i>h</i>;</li> <li>la distància separant lo centre de l’iperbòla e una de sas doas directriças, mai sovent notada <i>f</i>;</li> <li>l'excentricitat de l’iperbòla (estrictament superiora a 1), mai sovent notada <i>e</i>;</li> <li>lo « paramètre » de l’iperbòla, mai sovent notat <i>p</i>, representant lo mièg <i>latus rectum</i> (còrda parallèla a la directritz e passant pel fogal).</li></ul> <p>De relacions existisson entre aquelas grandors: </p> <ul><li>se l'iperbòla es definida per son excentricitat <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> e la distància <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> entre lo fogal F e la directritz (d), alara: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a={eh \over e^{2}-1},\,b={eh \over {\sqrt {e^{2}-1}}},\,c={e^{2}h \over e^{2}-1},\,f={h \over e^{2}-1},\,p=eh\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <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> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>e</mi> <mi>h</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>=</mo> <mi>e</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={eh \over e^{2}-1},\,b={eh \over {\sqrt {e^{2}-1}}},\,c={e^{2}h \over e^{2}-1},\,f={h \over e^{2}-1},\,p=eh\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b23c35640a46f2fd5dd7d773086b8097adf421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.546ex; height:7.009ex;" alt="{\displaystyle a={eh \over e^{2}-1},\,b={eh \over {\sqrt {e^{2}-1}}},\,c={e^{2}h \over e^{2}-1},\,f={h \over e^{2}-1},\,p=eh\,;}"></span></dd></dl></li> <li>se l'iperbòla es donada per la distància entre lo centre e un som <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> e lo penjal de las asimptòtas <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/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b/a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0082708c4a962619f1b493a22dc776b5fbab62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.39ex; height:2.843ex;" alt="{\displaystyle b/a}"></span> alara: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c={\sqrt {a^{2}+b^{2}}},\,h={b^{2} \over {\sqrt {a^{2}+b^{2}}}},\,f={a^{2} \over {\sqrt {a^{2}+b^{2}}}},\,e={{\sqrt {a^{2}+b^{2}}} \over a},\,p={b^{2} \over a}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {a^{2}+b^{2}}},\,h={b^{2} \over {\sqrt {a^{2}+b^{2}}}},\,f={a^{2} \over {\sqrt {a^{2}+b^{2}}}},\,e={{\sqrt {a^{2}+b^{2}}} \over a},\,p={b^{2} \over a}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2958a62899964c9626bbd9412264d641ff88f75b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:71.484ex; height:7.509ex;" alt="{\displaystyle c={\sqrt {a^{2}+b^{2}}},\,h={b^{2} \over {\sqrt {a^{2}+b^{2}}}},\,f={a^{2} \over {\sqrt {a^{2}+b^{2}}}},\,e={{\sqrt {a^{2}+b^{2}}} \over a},\,p={b^{2} \over a}\,;}"></span></dd></dl></li> <li>quand se conéis la distància entre lo centre e lo som <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> e l’excentricitat <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> : <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=a{\sqrt {e^{2}-1}},\,c=ae,\,h={a(e^{2}-1) \over e},\,f={a \over e},\,p=a(e^{2}-1)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mi>e</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>e</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>e</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a{\sqrt {e^{2}-1}},\,c=ae,\,h={a(e^{2}-1) \over e},\,f={a \over e},\,p=a(e^{2}-1)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd80f594ed9578819034b8b1bc1438ab3929c209" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:61.271ex; height:5.843ex;" alt="{\displaystyle b=a{\sqrt {e^{2}-1}},\,c=ae,\,h={a(e^{2}-1) \over e},\,f={a \over e},\,p=a(e^{2}-1)\,.}"></span></dd></dl></li> <li>Fin finala, dins la definicion bifocala de l'iperbòla ont son conegudas la longor 2<i>a</i> e la distància 2<i>c</i> entre los fogals: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\sqrt {c^{2}-a^{2}}},\,e={\frac {c}{a}},\,h={c^{2}-a^{2} \over c},\,f={a^{2} \over c},\,p={c^{2}-a^{2} \over a}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>a</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\sqrt {c^{2}-a^{2}}},\,e={\frac {c}{a}},\,h={c^{2}-a^{2} \over c},\,f={a^{2} \over c},\,p={c^{2}-a^{2} \over a}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95f31d01e536c5b8fc4427aa1d913442d60a4f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:58.817ex; height:5.676ex;" alt="{\displaystyle b={\sqrt {c^{2}-a^{2}}},\,e={\frac {c}{a}},\,h={c^{2}-a^{2} \over c},\,f={a^{2} \over c},\,p={c^{2}-a^{2} \over a}\,.}"></span></dd></dl></li></ul> <div class="mw-heading mw-heading2"><h2 id="Equacions">Equacions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=7" title="Modificar la seccion : Equacions" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=7" title="Edita el codi de la secció: Equacions"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Equacion_dins_una_marca_normada_portada_per_las_asimptòtas"><span id="Equacion_dins_una_marca_normada_portada_per_las_asimpt.C3.B2tas"></span>Equacion dins una marca normada portada per las asimptòtas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=8" title="Modificar la seccion : Equacion dins una marca normada portada per las asimptòtas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=8" title="Edita el codi de la secció: Equacion dins una marca normada portada per las asimptòtas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Cas_particular_de_la_foncion_invèrsa"><span id="Cas_particular_de_la_foncion_inv.C3.A8rsa"></span>Cas particular de la foncion invèrsa</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=9" title="Modificar la seccion : Cas particular de la foncion invèrsa" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=9" title="Edita el codi de la secció: Cas particular de la foncion invèrsa"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:FonctionInverseFoyersDirectrices.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/FonctionInverseFoyersDirectrices.svg/220px-FonctionInverseFoyersDirectrices.svg.png" decoding="async" width="220" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/FonctionInverseFoyersDirectrices.svg/330px-FonctionInverseFoyersDirectrices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/FonctionInverseFoyersDirectrices.svg/440px-FonctionInverseFoyersDirectrices.svg.png 2x" data-file-width="205" data-file-height="204" /></a><figcaption>Representacion grafica de la foncion invèrsa. Iperbòla amb sos fogals e sas directriças</figcaption></figure> <p>L'iperbòla que l'expression matematica es mai simple es la representacion grafica de la foncion <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> definida per <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)=1/x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1/x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049b4bda6d9e222e496f2670248d9ecfb75841d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.171ex; height:2.843ex;" alt="{\displaystyle f(x)=1/x}"></span>, vejatz foncion invèrsa. </p><p>Aquela iperbòla es <i>equilatèra</i> que sa doas asimptòtas son ortogonalas. Son excentricitat val <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Cas_general">Cas general</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=10" title="Modificar la seccion : Cas general" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=10" title="Edita el codi de la secció: Cas general"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dins la marca <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (O,{\vec {u}},{\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (O,{\vec {u}},{\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316416ad8cbe7cc49613f820107b70239318fc85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.156ex; height:2.843ex;" alt="{\displaystyle (O,{\vec {u}},{\vec {v}})}"></span>, ont O es lo centre de l'iperbòla e <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 {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> e <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> los vectors unitaris directors de las asimptòtas, l'iperbòla a per equacion<sup id="cite_ref-mathcurve_3-1" class="reference"><a href="#cite_note-mathcurve-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> : <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={\frac {c^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy={\frac {c^{2}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3882300dd901cc59633d2c17416d4dc0d8872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.481ex; height:5.676ex;" alt="{\displaystyle xy={\frac {c^{2}}{4}}}"></span>. </p><p>Reciprocament, se doas drechas de vectors directors <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 {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> e <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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> se trencan en O e se una corba, dins la marca <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (O,{\vec {u}},{\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (O,{\vec {u}},{\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316416ad8cbe7cc49613f820107b70239318fc85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.156ex; height:2.843ex;" alt="{\displaystyle (O,{\vec {u}},{\vec {v}})}"></span>, a per equacion xy= Csta o Csta es un real non nul, alara aqula corba es una iperbòla<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Equacions_dins_de_marcas_ont_l'axe_focal_es_l'axe_principal"><span id="Equacions_dins_de_marcas_ont_l.27axe_focal_es_l.27axe_principal"></span>Equacions dins de marcas ont l'axe focal es l'axe principal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=11" title="Modificar la seccion : Equacions dins de marcas ont l'axe focal es l'axe principal" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=11" title="Edita el codi de la secció: Equacions dins de marcas ont l'axe focal es l'axe principal"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Se_le_centre_de_la_marca_es_le_centre_de_l'iperbòla"><span id="Se_le_centre_de_la_marca_es_le_centre_de_l.27iperb.C3.B2la"></span>Se le centre de la marca es le centre de l'iperbòla</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=12" title="Modificar la seccion : Se le centre de la marca es le centre de l'iperbòla" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=12" title="Edita el codi de la secció: Se le centre de la marca es le centre de l'iperbòla"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dins una marca que los axes son de simetrie per l'iperbòla, l'axe tracersièr per axe de las abscissis, l'equacion cartesiana se met jos la forma <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> </p><p>donant alara las representacions parametricas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto \left(a\,\cosh(t),b\,\sinh(t)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto \left(a\,\cosh(t),b\,\sinh(t)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f50197950b12d659b936aa61408103bd53db40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.922ex; height:2.843ex;" alt="{\displaystyle t\mapsto \left(a\,\cosh(t),b\,\sinh(t)\right)}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\mapsto \left(-a\,\cosh(t),b\,\sinh(t)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto \left(-a\,\cosh(t),b\,\sinh(t)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad80259390c8e12a2794af8a4d8f33eda7a23f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.73ex; height:2.843ex;" alt="{\displaystyle t\mapsto \left(-a\,\cosh(t),b\,\sinh(t)\right)}"></span> per caduna de las brancas. </p><p>Un autre parametratge possible es: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mi>tan</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="1em" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/125fcad170b5e193af844ddaf29ed084048967ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.15ex; height:4.676ex;" alt="{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}}}"></span> </p><p>Son equacion polara es: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {b}{\sqrt {e^{2}\cos ^{2}(\theta )-1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <msqrt> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {b}{\sqrt {e^{2}\cos ^{2}(\theta )-1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c69d5dcae9e1874f9297adb5743b7ef3176ff31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.053ex; height:6.676ex;" alt="{\displaystyle \rho ={\frac {b}{\sqrt {e^{2}\cos ^{2}(\theta )-1}}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Se_le_centre_de_la_marca_es_lo_fogal_de_l'iperbòla"><span id="Se_le_centre_de_la_marca_es_lo_fogal_de_l.27iperb.C3.B2la"></span>Se le centre de la marca es lo fogal de l'iperbòla</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=13" title="Modificar la seccion : Se le centre de la marca es lo fogal de l'iperbòla" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=13" title="Edita el codi de la secció: Se le centre de la marca es lo fogal de l'iperbòla"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dins la marca ortonormada <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,{\vec {i}},{\vec {j}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F,{\vec {i}},{\vec {j}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9c6667879d0c2d290e91562be2b4551d2f4f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.806ex; height:3.343ex;" alt="{\displaystyle (F,{\vec {i}},{\vec {j}})}"></span> ont <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 {\vec {i}}={\frac {1}{FK}}{\overrightarrow {FK}}={\frac {1}{h}}{\overrightarrow {FK}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>F</mi> <mi>K</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>K</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>h</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>F</mi> <mi>K</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {i}}={\frac {1}{FK}}{\overrightarrow {FK}}={\frac {1}{h}}{\overrightarrow {FK}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94bab9a9776b7b930b4b7d4702086f71c2095cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-top: -0.384ex; width:22.103ex; height:5.509ex;" alt="{\displaystyle {\vec {i}}={\frac {1}{FK}}{\overrightarrow {FK}}={\frac {1}{h}}{\overrightarrow {FK}}}"></span>, l'iperbòla a per equacion cartesiana: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}=e^{2}(x-h)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <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 x^{2}+y^{2}=e^{2}(x-h)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a259ac5ea7cc124e99ef466d98edc57b850c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.048ex; height:3.176ex;" alt="{\displaystyle x^{2}+y^{2}=e^{2}(x-h)^{2}}"></span> </p><p>Son equacion polara, dins la metaissa marca es: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {p}{1+e\cos(\theta )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>e</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {p}{1+e\cos(\theta )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c45875ee385dd1a33006e636094c015fe1f6c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.621ex; height:5.676ex;" alt="{\displaystyle \rho ={\frac {p}{1+e\cos(\theta )}}}"></span> </p><p>ont <i>p = eh</i> es lo paramètre de l'iperbòla. </p> <div class="mw-heading mw-heading3"><h3 id="Equacion_generala_de_conica">Equacion generala de conica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=14" title="Modificar la seccion : Equacion generala de conica" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=14" title="Edita el codi de la secció: Equacion generala de conica"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De biais general, coma tota conica, una iperbòla a una equacion cartesiana de la forma <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)=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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.868ex; height:2.843ex;" alt="{\displaystyle f(x,y)=0}"></span> amb <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)=\alpha x^{2}+2\beta xy+\gamma y^{2}+2\delta x+2\epsilon y+\phi .}"> <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>α</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>δ</mi> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>ϵ</mi> <mi>y</mi> <mo>+</mo> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=\alpha x^{2}+2\beta xy+\gamma y^{2}+2\delta x+2\epsilon y+\phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db9fd1bbebb4e7f0a31f744054513b19dc2ec45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.071ex; height:3.176ex;" alt="{\displaystyle f(x,y)=\alpha x^{2}+2\beta xy+\gamma y^{2}+2\delta x+2\epsilon y+\phi .}"></span> Per qu'una tala equacion siá aquela d'una iperbòla, cal<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> que <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 \alpha \gamma -\beta ^{2}<0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>γ</mi> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \gamma -\beta ^{2}<0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69aa72f214fc5d212199cf10c68b5918ab7dd127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.889ex; height:3.176ex;" alt="{\displaystyle \alpha \gamma -\beta ^{2}<0.}"></span> Dins aquel cas, la conica a per centre lo punt C que las coordonadas (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) verifican lo sistèma<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>: <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{cases}\alpha x+\beta y=-\delta \\\beta x+\gamma y=-\epsilon .\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>α</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>δ</mi> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> <mi>x</mi> <mo>+</mo> <mi>γ</mi> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>ϵ</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\alpha x+\beta y=-\delta \\\beta x+\gamma y=-\epsilon .\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9fd045014c3c2de83ed8ea433ea5068daf32510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.912ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\alpha x+\beta y=-\delta \\\beta x+\gamma y=-\epsilon .\end{cases}}}"></span> Un cambiament de marca, prenent per centre lo punt C, mena a l'equacion seguenta: <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 \alpha X^{2}+2\beta XY+\gamma Y^{2}+f(x_{0},y_{0})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mi>X</mi> <mi>Y</mi> <mo>+</mo> <mi>γ</mi> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>f</mi> <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> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha X^{2}+2\beta XY+\gamma Y^{2}+f(x_{0},y_{0})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b234db7eb9ac94734cd1571434a1b4601d6c79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.485ex; height:3.176ex;" alt="{\displaystyle \alpha X^{2}+2\beta XY+\gamma Y^{2}+f(x_{0},y_{0})=0}"></span> que serà l'equacion d'una iperbòla se e pas que se <i>f</i>(<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) es non nul. </p><p>L'iperbòla es equilatèra se e pas que se <i>α = - γ</i> </p> <div class="mw-heading mw-heading3"><h3 id="Equacion_matriciala">Equacion matriciala</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=15" title="Modificar la seccion : Equacion matriciala" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=15" title="Edita el codi de la secció: Equacion matriciala"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'equacion precedenta pòt s'escriure jos forma matriciala: </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 ^{\mathrm {t} }\mathbf {x} \mathbf {A} \mathbf {x} +^{\mathrm {t} }\mathbf {b} \mathbf {x} +f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{\mathrm {t} }\mathbf {x} \mathbf {A} \mathbf {x} +^{\mathrm {t} }\mathbf {b} \mathbf {x} +f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d64c68d6f5c765b4e13b78a10ae4661788b017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.702ex; height:2.843ex;" alt="{\displaystyle ^{\mathrm {t} }\mathbf {x} \mathbf {A} \mathbf {x} +^{\mathrm {t} }\mathbf {b} \mathbf {x} +f=0}"></span></dd></dl> <p>ont </p> <ul><li><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 \mathbf {x} ={\begin{pmatrix}x\\y\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{pmatrix}x\\y\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acee7ca4cea12080d6b1830b4a32e3f3945c615c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.012ex; height:6.176ex;" alt="{\displaystyle \mathbf {x} ={\begin{pmatrix}x\\y\end{pmatrix}}}"></span> ; <sup>t</sup><b>x</b> es la transpausada de <b>x</b>;</li> <li><b>A</b> es una matritz 2×2, <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 \mathbf {A} ={\begin{pmatrix}\alpha &\beta \\\beta &\gamma \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> </mtd> <mtd> <mi>γ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}\alpha &\beta \\\beta &\gamma \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9cb8f929fccd161558a2295d27ceaa4e1053e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.433ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{pmatrix}\alpha &\beta \\\beta &\gamma \end{pmatrix}}}"></span>;</li> <li><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 \mathbf {b} ={\begin{pmatrix}2\delta \\2\epsilon \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mi>δ</mi> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>ϵ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} ={\begin{pmatrix}2\delta \\2\epsilon \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b053bb924f2630c4f9e3c76110437ee98feb69d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.968ex; height:6.176ex;" alt="{\displaystyle \mathbf {b} ={\begin{pmatrix}2\delta \\2\epsilon \end{pmatrix}}}"></span> ; <sup>t</sup><b>b</b> es la transpausada de <b>b</b>;</li></ul> <p>Totjorn amb la meteissa constenchas. </p> <div class="mw-heading mw-heading2"><h2 id="Proprietats">Proprietats</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=16" title="Modificar la seccion : Proprietats" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=16" title="Edita el codi de la secció: Proprietats"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Interior_e_exterior">Interior e exterior</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=17" title="Modificar la seccion : Interior e exterior" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=17" title="Edita el codi de la secció: Interior e exterior"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'iperbòla partetja lo plan en 3 zonas o compausantas connèxas. Coma es la seccion d'un còn e d'un plan, se nomena interior de l'iperbòla las partidas de plan situadas a l'interior del còn, so las zonas que contenon los fogals, e exterior de l'iperbòla la darrièra zona, aquela que conten lo centre de l'iperbòla. </p> <div class="mw-heading mw-heading3"><h3 id="Secantas_e_soms">Secantas e soms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=18" title="Modificar la seccion : Secantas e soms" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=18" title="Edita el codi de la secció: Secantas e soms"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleDiametre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/220px-HyperboleDiametre.svg.png" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/330px-HyperboleDiametre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/440px-HyperboleDiametre.svg.png 2x" data-file-width="311" data-file-height="172" /></a><figcaption>Diamètres conjugats dins una iperbòla</figcaption></figure> <p>Siá M un punt de l'iperbòla de soms S e S' e de centre O. Se per un punt N de l'iperbòla, se mena de parallèlas a (SM) e (S'M), encontran l'iperbòla en dos punts P e P' simetrics al respècte de O<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>. En consequéncia, dins un fais de drechas parallèlas (d<sub>i</sub>) encontrant l'iperbòla, los miègs de las còrdas que determinan aquelas drechas sus l'iperbòla, son alinhadas amb leo centre O de l'iperbòla. De mai, se M es lo punt de l'iperbòlas tal que (SM) siá parallèl a (d<sub>i</sub>) alara la drecha dels miègs es parallèl a (S'M)<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>. Una tala drecha passant pel centre de l'iperbòla es nomenada diamètre de l'iperbòla<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Secantas_e_asimptòtas"><span id="Secantas_e_asimpt.C3.B2tas"></span>Secantas e asimptòtas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=19" title="Modificar la seccion : Secantas e asimptòtas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=19" title="Edita el codi de la secció: Secantas e asimptòtas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se una drecha trenca l'iperbòla en M e M', alara trenca las asimptòtas en P e P' e los segments [MM'] e [PP'] on lo meteis mièg. </p><p>Siá (d) una drecha non parallèla a las asimptòtas. Se per un punt M de l'iperbòla, se dessenha una parallèla a (d), encontra las asimptotas en P e P' e lo produch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {MP}}\times {\overline {MP'}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <mi>P</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>M</mi> <msup> <mi>P</mi> <mo>′</mo> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {MP}}\times {\overline {MP'}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c1e839827ac6df233677ee674570f1cf0ae605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.371ex; height:3.176ex;" alt="{\displaystyle {\overline {MP}}\times {\overline {MP'}}}"></span> es independent del punt M<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Tangentas">Tangentas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=20" title="Modificar la seccion : Tangentas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=20" title="Edita el codi de la secció: Tangentas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se l'iperbòla a per equacion <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> <mo>,</mo> </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/2b6d87e3798ff72cac48672d3a0aec41442b7e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.019ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,}"></span> la tangenta al punt M de coordonadas (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) a per equacion<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>: <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_{0}}{a^{2}}}x-{\frac {y_{0}}{b^{2}}}y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x_{0}}{a^{2}}}x-{\frac {y_{0}}{b^{2}}}y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f66bfe0eede7d074109bd54be300ce6ae67b2d0e" 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 {x_{0}}{a^{2}}}x-{\frac {y_{0}}{b^{2}}}y=1}"></span> </p><p>Coma una tangenta es una secanta particulara, la proprietat de las secantas e dels soms ofrisson un mejan de dessenhar una tangenta en un punt M distincte dels soms: la drecha passant per un dels soms S de l'iperbòla e parallèla a (OM) encontra l'iperbòla en N, la tangenta es alara parallèla a (S'N)<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>. </p><p>Se M es un punt de l'iperbòla distinct dels soms, la tangenta en M es tanben la bissectritz interiora de l'angle FMF' ont F e F' sont los fogals de l'iperbòla e lo produch de las distàncias dels fogals a la tangenta es totjorn egal a b²<sup id="cite_ref-Tauvel398_16-0" class="reference"><a href="#cite_note-Tauvel398-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>. Aquela proprietat a unas aplicacions practicas. Dona un mejans simple de construire la tangenta en un punt coma bissectritz interiora de (F'MF). Dins un miralh de forma iperbolica, los rais eissits d'un fogal son rebatuts coma se vinian d'un autre fogal. Bergery preconisa donc, per evitar las pèrdas de calor, de basir lo fond d'un chiminèa segon un cilindre iperbolic<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>. </p><p>La tangenta en un punt M trenca las asimptòtas en dos punts P e P' simetrics al respècte de M. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleTangentes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/220px-HyperboleTangentes.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/330px-HyperboleTangentes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/440px-HyperboleTangentes.svg.png 2x" data-file-width="389" data-file-height="239" /></a><figcaption>Dessenhs de doas tangentas a l'iperbòla eissidas de M mejans lo cercle director</figcaption></figure> <p>Pel punt O o un punt M interior a l'iperbòla, passa pas cap de tangenta, per un punt situat sus l'iperbòla sus una asimptòta (distinct de O), ne passa pas qu'una e per un punt situat a l'exterior de l'iperbòla, non situada sus las asimptòtas, passa totjorn doas tangentas<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>. Per bastir las doas tangentas eissidas de M, sufís de dessenhar lo cercle de centre M passant per un fogal F e lo cercle de centre F' e de rai 2a. Los cercles s'encontran en N e N', las mediatriças de [FN] e [FN'] son las tangentas cercadas<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>. Per trobar lo punt de contacte, sufís de bastir una parallèla a la tangenta trencant l'iperbòla en dos punts, la drecha passant per O e la mitat de la còrdatanben bastida encontra la tangenta en son punt de contacte. O alara, se pren lo simetric d'un dels fogals al respècte de la tangenta, la drecha jonhent aquel simetric a l'autre fogal encontra la tangenta a son punt de contacte<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Cercles">Cercles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=21" title="Modificar la seccion : Cercles" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=21" title="Edita el codi de la secció: Cercles"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Cercle principal</b>: es lo cercle de diamètre [SS']. Fogal e directritz son en relacion pòl/polar al respècte del cercle principal. Lo cercle principal es tanben la podària de l'iperbòla al respècte de l'un dels fogals (se s'exclusís las interseccions del cercle amb las asimptòtas), es a dire lo luòc dels projectats ortoganals d'aquel ce fogals sus las tangentas<sup id="cite_ref-Tauvel398_16-1" class="reference"><a href="#cite_note-Tauvel398-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>, çò que fa de l'iperbòla, l'antipodària de son cercle principal al respècte d'un dels sieus fogals. </p><p><b>Cercle director</b>: es un cercle passant per fogal e de rai egal a 2a. Segon la definicion bifocala de l'iperbòla, l'iperbòla es lo luòc dels centres dels cercles passant per F e tangents interiorament o exteriorament al cercle director de centre F'. L'ensems de las médiatriças dels segments [FM], onr M parcors lo cercle director de centre F' dona l'ensems de las tangentas a l'iperbòla, lo cercle director es donc l'ortotomic de l'iperbòla al respècte d'un fogal, es a dire l'ensems dels simetrics de F al respècte de las tangentas<sup id="cite_ref-Tauvel398_16-2" class="reference"><a href="#cite_note-Tauvel398-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>. </p><p><b> Cercle ortoptic</b>: se l’excentricitat es estrictament compresa entre 1 e <span style="font-size:larger; letter-spacing:0px;">√</span><span style="border-top:1px solid black; padding:1px 0 0 3px;">2</span><span class="racine" style="position:relative"></span>, existís de punts M per ont passan dos tangentas ortogonalas. L'ensems d'aqueles punts M dessenha un cercle de centre O e de rai <span style="font-size:larger; letter-spacing:0px;">√</span><span style="border-top:1px solid black; padding:1px 0 0 3px;">a²-b²</span><span class="racine texhtml" style="position:relative"></span>, nomenat cercle ortoptic de l'iperbòla<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleCentreCourbure.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/220px-HyperboleCentreCourbure.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/330px-HyperboleCentreCourbure.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/440px-HyperboleCentreCourbure.svg.png 2x" data-file-width="707" data-file-height="425" /></a><figcaption>Construccion del centre de corbadura al punt M, du cercle osculator e desvelopada de l'iperbòla</figcaption></figure> <p><b> Cercles osculators</b>: en tot punt M de l'iperbòla, existís un cercle possedant un punt de contacte triple avec l'iperbòla. Es lo cercle osculator a l'iperbòla al punt M. Son centre, nomenat centre de corbadura, es situat sus la normala a la corba (qu'es tanben la bissectritz exteriora de l'angle FMF') a una distància de M egala al rai de corbadura. Se l'iperbòla a per equacion: <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> <mo>,</mo> </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/2b6d87e3798ff72cac48672d3a0aec41442b7e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.019ex; height:6.009ex;" alt="{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,}"></span> lo rai del cercle osculator al punt d'abscissi x<sub>0</sub> a per valor<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>: <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={\frac {\left(e^{2}x_{0}^{2}-a^{2}\right)^{3/2}}{ab}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow> <mi>a</mi> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\frac {\left(e^{2}x_{0}^{2}-a^{2}\right)^{3/2}}{ab}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3fdb382cbda3d0b89d945366debd8f7180c07c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.104ex; height:6.843ex;" alt="{\displaystyle r={\frac {\left(e^{2}x_{0}^{2}-a^{2}\right)^{3/2}}{ab}}.}"></span> Es possible de dessenhar geometricament lo centre del cercle osculator en M. Se (t) e (n) son respectivament la tangenta e la normala a l'iperbòla al punt M, se traça lo simetric M' de M al respècte de l'axe principal e lo simetric (t') de (t) al respècte de (MM'), aquela drecha encontra la drecha (OM') en N. La perpendiculara a (t') en N encontra la normala (n) al centre de corbadura<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>. </p><p>La desvelopada de l'iperbòla, es dire lo luòc dels centres de corbadura es una corba de Lamé d'equacion<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {ax}{c^{2}}}\right)^{3/2}-\left({\frac {by}{c^{2}}}\right)^{3/2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>x</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mi>y</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {ax}{c^{2}}}\right)^{3/2}-\left({\frac {by}{c^{2}}}\right)^{3/2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966819cfc15c5493a18646b47287d948aed4ed82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.725ex; height:6.676ex;" alt="{\displaystyle \left({\frac {ax}{c^{2}}}\right)^{3/2}-\left({\frac {by}{c^{2}}}\right)^{3/2}=1}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Longor_e_airal">Longor e airal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=22" title="Modificar la seccion : Longor e airal" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=22" title="Edita el codi de la secció: Longor e airal"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se M(<i>t</i><sub>0</sub>) es un punt de l'iperbòla d'equacion parametrada, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>b</mi> <mi>tan</mi> <mo>⁡</mo> <mi>t</mi> <mspace width="1em" /> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5115f5dc66c5dfe4519a20261157f62f047bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.796ex; height:4.676ex;" alt="{\displaystyle y=b\tan t\quad x={\frac {a}{\cos t}},}"></span> ont <i>t</i><sub>0</sub> es comprés entre 0 e π/2, la longor de l'arc SM es <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 \int _{0}^{t_{0}}{\frac {\sqrt {b^{2}+a^{2}\sin ^{2}t}}{\cos ^{2}t}}\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>t</mi> </msqrt> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t_{0}}{\frac {\sqrt {b^{2}+a^{2}\sin ^{2}t}}{\cos ^{2}t}}\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0ed74dde698a51e51c51bc08a4a4dd9bff1c67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.501ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{t_{0}}{\frac {\sqrt {b^{2}+a^{2}\sin ^{2}t}}{\cos ^{2}t}}\mathrm {d} t}"></span> que l'integracion necessita l'utilizacion de las integralas ellipticas<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup>. </p><p>L'airal d'una partida de plan terminada per un arc d'iperbòl es a l'origina de la creacion de la foncion logaritme e de las foncions iperbolicas. L'airal de la superfícia terminada per l'iperbòla d'equacion <i>yx</i>=1, l'axe de las abscissis e las drechas d'equacion <i>x=u</i> et <i>x=v</i> es egala a |ln(<i>v/u</i>)|. </p> <div class="mw-heading mw-heading2"><h2 id="Istòria"><span id="Ist.C3.B2ria"></span>Istòria</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=23" title="Modificar la seccion : Istòria" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=23" title="Edita el codi de la secció: Istòria"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichi%C3%A8r:HyperboleApollonios.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/HyperboleApollonios.svg/220px-HyperboleApollonios.svg.png" decoding="async" width="220" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/HyperboleApollonios.svg/330px-HyperboleApollonios.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/HyperboleApollonios.svg/440px-HyperboleApollonios.svg.png 2x" data-file-width="255" data-file-height="217" /></a><figcaption>Egalitat d'airal, dins una iperbòla, entre lo carrat menat sus l'ordonada e lo rectangle blau menat sus l'abscissi. Aquel rectangle es <b>mai grand</b> que lo rectangle de nautor SP (paramètre de l'iperbòla) d'ont lo nom d'<i>iperbòla</i> (ajustament par excès) donèt a la corba per Apollonios</figcaption></figure> <p>L'iperbòla es estudiada, dins l'encastre de las conicas, pendent lo periòde grèc. Alara Menechme resolguèt un problèma de dobla proporcionala utilisant una iperbòla d'equacion XY= constanta<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>. Mas per Menechme e sos successors, Euclides e Aristèu, aquela corba possèda qu'una compausanta. La nomenan «conica obtusangla» que la definisson coma l'interseccion d'un còn obtusangla (còn engendrat per la rotacion d'un triangle ABC, rectangle en B, a l'entorn de AB e de tal biais que l'angle de som A siá superior a 45 °) amb un plan perpendicular a sa generatritz<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>. Apollonius de Perge sembla èsser lo primièr d'envisatjar las dos compausantas de l'iperbòla<sup id="cite_ref-apollonius_28-0" class="reference"><a href="#cite_note-apollonius-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>. Es tanben el qui li dona lo nom d'«iperbòla» (ajustament per excès) avent remarcat que l'airal du carrat dessenhat sus l'ordonada es superiora a aquela del rectangle dessenhat sus l'abscissi e que la nautor seriá fixa<sup id="cite_ref-apollonius_28-1" class="reference"><a href="#cite_note-apollonius-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>. </p><p>La recerca de l'airal jos l'iperbòla es entrepresa, en 1647, per Grégoire de Saint-Vincent, que mòstra sa proprietat logaritmica<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p>En 1757-1762, Vincenzo Riccati establís una relacion entre l'airal d'un sector angular dins una iperbòla e las coordonadas d'un punt e definís las foncions cosinus iperbolic e sinus iperbolic per analogia a la relacion existissent dins lo cercle<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=24" title="Modificar la seccion : Bibliografia" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=24" title="Edita el codi de la secció: Bibliografia"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation book" style="font-style:normal" id="CITEREF"> {{{títol}}}. <span style="font-size:90%; white-space:nowrap;"><a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/2-10-049413-9" title="Especial:Obratge de referéncia/2-10-049413-9">ISBN 2-10-049413-9</a></span>.</span><span style="display: none;"> </span></li> <li><span class="citation book" style="font-style:normal" id="CITEREF"> {{{títol}}}.</span><span style="display: none;"> </span></li> <li>Jean-Denis Eiden, Géométrie analytique classique, Calvage & Mounet, 2009, <a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/9782916352084" class="internal mw-magiclink-isbn">ISBN 978-2-916352-08-4</a></li> <li>Méthodes modernes en géométrie de Jean Fresnel</li> <li>Bruno Ingrao, Coniques affines, euclidiennes et projectives, C&M, <a href="/wiki/Especial:Obratge_de_refer%C3%A9ncia/9782916352121" class="internal mw-magiclink-isbn">ISBN 978-2-916352-12-1</a></li> <li class="mw-empty-elt"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Annèxas"><span id="Ann.C3.A8xas"></span>Annèxas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=25" title="Modificar la seccion : Annèxas" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=25" title="Edita el codi de la secció: Annèxas"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connèxes"><span id="Articles_conn.C3.A8xes"></span>Articles connèxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=26" title="Modificar la seccion : Articles connèxes" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=26" title="Edita el codi de la secció: Articles connèxes"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Foncion_iperbolica&action=edit&redlink=1" class="new" title="Foncion iperbolica (la pagina existís pas)">Foncion iperbolica</a></li> <li><a href="/wiki/Conica" title="Conica">Conicas</a> <ul><li><a href="/wiki/Ellipsa" title="Ellipsa">Ellipsa</a></li> <li><a href="/wiki/Parab%C3%B2la" title="Parabòla">Parabòla</a></li></ul></li> <li>Conica degenerada<a href="https://en.wikipedia.org/wiki/Degenerate_conic" class="extiw" title="en:Degenerate conic"><span class="indicateur-langue" title="Équivalent de l’article « Conique dégénérée » dans une autre langue"></span></a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ligams_extèrnes"><span id="Ligams_ext.C3.A8rnes"></span>Ligams extèrnes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=27" title="Modificar la seccion : Ligams extèrnes" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=27" title="Edita el codi de la secció: Ligams extèrnes"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En francés">(<abbr title="">fr</abbr>)</span> <span title="Nombre escrit en chifras romanas" style="font-variant:small-caps;">ferréol</span>, Robert; <span title="Nombre escrit en chifras romanas" style="font-variant:small-caps;">mandonnet</span>, Jacques; <span title="Nombre escrit en chifras romanas" style="font-variant:small-caps;">esculier</span>, Alain, <a rel="nofollow" class="external text" href="http://www.mathcurve.com/courbes2d/hyperbole/hyperbole.shtml"><i>Hyperbole</i></a>.</li> <li><span style="font-size:0.95em; font-weight:bold; color:var(--color-subtle, #54595d);" title="En francés">(<abbr title="">fr</abbr>)</span> <span title="Nombre escrit en chifras romanas" style="font-variant:small-caps;">hubaut</span>, Xavier, professor emerit de l'Universitat Liura Brussèlas, <a rel="nofollow" class="external text" href="http://xavier.hubaut.info/coursmath/2de/belges.htm"><i>Les théorèmes belges</i></a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Nòtas_e_referéncias"><span id="N.C3.B2tas_e_refer.C3.A9ncias"></span>Nòtas e referéncias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&veaction=edit&section=28" title="Modificar la seccion : Nòtas e referéncias" class="mw-editsection-visualeditor"><span>modificar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Iperb%C3%B2la_(matematicas)&action=edit&section=28" title="Edita el codi de la secció: Nòtas e referéncias"><span>Modificar lo còdi</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist references-column-count references-column-count-4" style="column-count: 4; -moz-column-count: 4; -webkit-column-count: 4; list-style-type:decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Du moins cela est vrai pour tous les cadrans horizontaux situés entre les <a href="/wiki/Cercle_polar" title="Cercle polar">cercles polaires</a> et tous les cadrans verticaux situés au delà des <a href="/wiki/Tropic" title="Tropic">tropiques</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 392</span> </li> <li id="cite_note-mathcurve-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-mathcurve_3-0">3,0</a></sup> et <sup><a href="#cite_ref-mathcurve_3-1">3,1</a></sup></span> <span class="reference-text">Robert Ferréol, <a rel="nofollow" class="external text" href="http://www.mathcurve.com/courbes2d/hyperbole/hyperbole.shtml">hyperbole</a>, sur <a rel="nofollow" class="external text" href="http://www.mathcurve.com/">L'Encyclopédie des formes mathématiques remarquables</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Conséquence de <a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 392, prop. 24.1.16</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="ouvrage" id="RosenfeldYoushkevitch1997"><span class="ouvrage" id="Boris_A._RosenfeldAdolf_P._Youshkevitch1997">Boris A. <span class="nom_auteur">Rosenfeld</span> et Adolf P. <span class="nom_auteur">Youshkevitch</span>, <cite style="font-style:normal">« Géométrie »</cite>, dans <cite class="italique">Histoire des sciences arabes</cite>, <abbr class="abbr" title="tome">t.</abbr> 2, Seuil,‎ <time>1997</time></span></span><span class="ouvrage" id="RosenfeldYoushkevitch1997"></span>, p. 97</span> </li> <li id="cite_note-Guillerault-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Guillerault_6-0">6,0</a></sup> et <sup><a href="#cite_ref-Guillerault_6-1">6,1</a></sup></span> <span class="reference-text">Extraits de la conférence de Michel Guillerault lors de l’université d'été 93, <a rel="nofollow" class="external text" href="http://www-cabri.imag.fr/abracadabri/Coniques/Guillerault/HomologieH.html">3 - Conique comme transformée de cercle par homologie harmonique</a>, sur le site de Cabri-Geomètre.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 392-393</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 414</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 412</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><a href="#cite_ref-10">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 145 prop.218</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 148 prop. 223)</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text">Encyclopedia universalis, 1990, vol.6, p.386(b)</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><a href="#cite_ref-13">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 401</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><a href="#cite_ref-14">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 396</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><a href="#cite_ref-15">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 163 prop. 249</span> </li> <li id="cite_note-Tauvel398-16"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Tauvel398_16-0">16,0</a></sup> <sup><a href="#cite_ref-Tauvel398_16-1">16,1</a></sup> et <sup><a href="#cite_ref-Tauvel398_16-2">16,2</a></sup></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 398</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><a href="#cite_ref-17">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 177</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><a href="#cite_ref-18">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 397.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><a href="#cite_ref-19">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 171 prob.b</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><a href="#cite_ref-20">↑</a></span> <span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 172 prob. e</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><a href="#cite_ref-21">↑</a></span> <span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 402</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><a href="#cite_ref-22">↑</a></span> <span class="reference-text">d'après Robert Ferréol, <a rel="nofollow" class="external text" href="http://www.mathcurve.com/courbes2d/hyperbole/hyperbole.shtml">Hyperbole</a>, sur <a rel="nofollow" class="external text" href="http://www.mathcurve.com/">L'Encyclopédie des formes mathématiques remarquables</a></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><a href="#cite_ref-23">↑</a></span> <span class="reference-text">d'après <a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="pagina(s)">p.</abbr> 180</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><a href="#cite_ref-24">↑</a></span> <span class="reference-text">Serge Mehl, <a rel="nofollow" class="external text" href="http://serge.mehl.free.fr/anx/devHyperb.html">développée de l'hyperbole</a> sur le site <a rel="nofollow" class="external text" href="http://serge.mehl.free.fr/">ChronoMath</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><a href="#cite_ref-25">↑</a></span> <span class="reference-text">Serge Mehl, <i>Fonction et intégrale elliptique - <a rel="nofollow" class="external text" href="http://serge.mehl.free.fr/anx/int_elli.html#hyperb">Arc d'hyperbole</a></i>, sur le site <a rel="nofollow" class="external text" href="http://serge.mehl.free.fr/">ChronoMath</a></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><a href="#cite_ref-26">↑</a></span> <span class="reference-text"><a href="https://fr.wikipedia.org/wiki/Hyperbole_(math%C3%A9matiques)#Vitrac" class="extiw" title="fr:Hyperbole (mathématiques)">Vitrac</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706124923/http://culturemath.ens.fr/histoire%20des%20maths/htm/Vitrac/grec-8.html#inventeur">Ménechme, l'inventeur des sections coniques ?</a></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><a href="#cite_ref-27">↑</a></span> <span class="reference-text"><a href="https://fr.wikipedia.org/wiki/Hyperbole_(math%C3%A9matiques)#Vitrac" class="extiw" title="fr:Hyperbole (mathématiques)">Vitrac</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706124923/http://culturemath.ens.fr/histoire%20des%20maths/htm/Vitrac/grec-8.html#aristee">La génération des coniques selon Aristée</a></span> </li> <li id="cite_note-apollonius-28"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-apollonius_28-0">28,0</a></sup> et <sup><a href="#cite_ref-apollonius_28-1">28,1</a></sup></span> <span class="reference-text"><a href="https://fr.wikipedia.org/wiki/Hyperbole_(math%C3%A9matiques)#Vitrac" class="extiw" title="fr:Hyperbole (mathématiques)">Vitrac</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170706124923/http://culturemath.ens.fr/histoire%20des%20maths/htm/Vitrac/grec-8.html#apollonius">L'approche d'Apollonius</a> <span class="error mw-ext-cite-error" lang="oc" dir="ltr">Error de citacion : Etiqueta <code><ref></code> no vàlida; el nom «apollonius» està definit diverses vegades amb contingut diferent.</span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><a href="#cite_ref-29">↑</a></span> <span class="reference-text"><span class="ouvrage" id="Legoff1989"><span class="ouvrage" id="Jean-Pierre_Legoff1989">Jean-Pierre <span class="nom_auteur">Legoff</span>, <cite style="font-style:normal">« De la méthode dite d'exhaustion : Grégoire de Saint-Vincent (1584 - 1667) »</cite>, dans <cite class="italique">La Démonstration mathématique dans l'histoire</cite>, IREM de Lyon,‎ <time>1989</time></span></span><span class="ouvrage" id="Legoff1989"></span>, p. 215</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><a href="#cite_ref-30">↑</a></span> <span class="reference-text">Robert E. Bradley, Lawrence A. D'Antonio, C. Edward Sandifer, <i>Euler at 300: An Appreciation</i>, MAA, 2007, <a rel="nofollow" class="external text" href="http://books.google.fr/books?id=tK_KRmTf9nUC&pg=PA99#v=onepage&q&f=false">p. 99</a></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Recuperada de « <a dir="ltr" href="https://oc.wikipedia.org/w/index.php?title=Iperbòla_(matematicas)&oldid=2416333">https://oc.wikipedia.org/w/index.php?title=Iperbòla_(matematicas)&oldid=2416333</a> »</div></div> <div id="catlinks" class="catlinks catlinks-allhidden" data-mw="interface"><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias amagadas : <ul><li><a href="/wiki/Categoria:Paginas_amb_d%27errors_de_refer%C3%A9ncia" title="Categoria:Paginas amb d'errors de referéncia">Paginas amb d'errors de referéncia</a></li><li><a href="/wiki/Categoria:Paginas_utilizant_de_ligams_magics_ISBN" title="Categoria:Paginas utilizant de ligams magics ISBN">Paginas utilizant de ligams magics ISBN</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Aquela pagina es estada modificada pel darrièr còp lo 3 genièr de 2024 a 14.55.</li> <li id="footer-info-copyright"><span style="white-space: normal"><a href="/wiki/Wikip%C3%A8dia:Drech_d%27autor" title="Wikipèdia:Drech d'autor">Drech d'autor</a> : Los tèxtes son disponibles jos <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.fr">licéncia Creative Commons paternitat pertatge a l’identic</a> ; d’autras condicions se pòdon aplicar. 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Iperbòla (matematicas) — Wikipèdia