Hyperbole (mathématiques)

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mw-ui-icon-wikimedia-expand"></span> <span>Afficher / masquer la sous-section Définitions géométriques</span> </button> <ul id="toc-Définitions_géométriques-sublist" class="vector-toc-list"> <li id="toc-Intersection_d'un_cône_et_d'un_plan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intersection_d'un_cône_et_d'un_plan"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Intersection d'un cône et d'un plan</span> </div> </a> <ul id="toc-Intersection_d'un_cône_et_d'un_plan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Définition_par_foyer_et_directrice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_par_foyer_et_directrice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Définition par foyer et directrice</span> </div> </a> <ul id="toc-Définition_par_foyer_et_directrice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Définition_bifocale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_bifocale"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Définition bifocale</span> </div> </a> <ul id="toc-Définition_bifocale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_d'un_cercle_par_une_homographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Image_d'un_cercle_par_une_homographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Image d'un cercle par une homographie</span> </div> </a> <ul id="toc-Image_d'un_cercle_par_une_homographie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relations_entre_les_grandeurs_caractéristiques_d'une_hyperbole" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relations_entre_les_grandeurs_caractéristiques_d'une_hyperbole"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Relations entre les grandeurs caractéristiques d'une hyperbole</span> </div> </a> <ul id="toc-Relations_entre_les_grandeurs_caractéristiques_d'une_hyperbole-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Équations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Équations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Équations</span> </div> </a> <button aria-controls="toc-Équations-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>Afficher / masquer la sous-section Équations</span> </button> <ul id="toc-Équations-sublist" class="vector-toc-list"> <li id="toc-Équation_dans_un_repère_normé_porté_par_les_asymptotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Équation_dans_un_repère_normé_porté_par_les_asymptotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Équation dans un repère normé porté par les asymptotes</span> </div> </a> <ul id="toc-Équation_dans_un_repère_normé_porté_par_les_asymptotes-sublist" class="vector-toc-list"> <li id="toc-Cas_particulier_de_la_fonction_inverse" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cas_particulier_de_la_fonction_inverse"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Cas particulier de la fonction inverse</span> </div> </a> <ul id="toc-Cas_particulier_de_la_fonction_inverse-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cas_général" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cas_général"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Cas général</span> </div> </a> <ul id="toc-Cas_général-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Équations_dans_des_repères_où_l'axe_focal_est_l'axe_principal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Équations_dans_des_repères_où_l'axe_focal_est_l'axe_principal"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Équations dans des repères où l'axe focal est l'axe principal</span> </div> </a> <ul id="toc-Équations_dans_des_repères_où_l'axe_focal_est_l'axe_principal-sublist" class="vector-toc-list"> <li id="toc-Si_le_centre_du_repère_est_le_centre_de_l'hyperbole" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Si_le_centre_du_repère_est_le_centre_de_l'hyperbole"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Si le centre du repère est le centre de l'hyperbole</span> </div> </a> <ul id="toc-Si_le_centre_du_repère_est_le_centre_de_l'hyperbole-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Si_le_centre_du_repère_est_le_foyer_de_l'hyperbole" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Si_le_centre_du_repère_est_le_foyer_de_l'hyperbole"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Si le centre du repère est le foyer de l'hyperbole</span> </div> </a> <ul id="toc-Si_le_centre_du_repère_est_le_foyer_de_l'hyperbole-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Si_le_centre_du_repère_est_le_sommet_de_l'hyperbole" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Si_le_centre_du_repère_est_le_sommet_de_l'hyperbole"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Si le centre du repère est le sommet de l'hyperbole</span> </div> </a> <ul id="toc-Si_le_centre_du_repère_est_le_sommet_de_l'hyperbole-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Équation_générale_de_conique" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Équation_générale_de_conique"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Équation générale de conique</span> </div> </a> <ul id="toc-Équation_générale_de_conique-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Équation_matricielle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Équation_matricielle"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Équation matricielle</span> </div> </a> <ul id="toc-Équation_matricielle-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriétés" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Propriétés"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Propriétés</span> </div> </a> <button aria-controls="toc-Propriétés-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>Afficher / masquer la sous-section Propriétés</span> </button> <ul id="toc-Propriétés-sublist" class="vector-toc-list"> <li id="toc-Intérieur_et_extérieur" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intérieur_et_extérieur"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Intérieur et extérieur</span> </div> </a> <ul id="toc-Intérieur_et_extérieur-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sécantes_et_sommets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sécantes_et_sommets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Sécantes et sommets</span> </div> </a> <ul id="toc-Sécantes_et_sommets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sécantes_et_asymptotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sécantes_et_asymptotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Sécantes et asymptotes</span> </div> </a> <ul id="toc-Sécantes_et_asymptotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperbole_équilatère" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hyperbole_équilatère"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Hyperbole équilatère</span> </div> </a> <ul id="toc-Hyperbole_équilatère-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tangentes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tangentes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Tangentes</span> </div> </a> <ul id="toc-Tangentes-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">4.6</span> <span>Cercles</span> </div> </a> <ul id="toc-Cercles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Longueur_et_aire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Longueur_et_aire"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Longueur et aire</span> </div> </a> <ul id="toc-Longueur_et_aire-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Histoire" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Histoire"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Histoire</span> </div> </a> <ul id="toc-Histoire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes et références</span> </div> </a> <button aria-controls="toc-Notes_et_références-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>Afficher / masquer la sous-section Notes et références</span> </button> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Références" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Références"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Références</span> </div> </a> <ul id="toc-Références-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Annexes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Annexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Annexes</span> </div> </a> <button aria-controls="toc-Annexes-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>Afficher / masquer la sous-section Annexes</span> </button> <ul id="toc-Annexes-sublist" class="vector-toc-list"> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liens_externes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liens_externes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Liens externes</span> </div> </a> <ul id="toc-Liens_externes-sublist" class="vector-toc-list"> </ul> </li> </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="Sommaire" 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="Basculer la table des matières" > <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">Basculer la table des matières</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">Hyperbole (mathématiques)</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="Aller à un article dans une autre langue. Disponible en 74 langues." > <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 langues</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 – arabe" lang="ar" hreflang="ar" data-title="قطع زائد" data-language-autonym="العربية" data-language-local-name="arabe" 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 – asturien" lang="ast" hreflang="ast" data-title="Hipérbola" data-language-autonym="Asturianu" data-language-local-name="asturien" 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) – azerbaïdjanais" lang="az" hreflang="az" data-title="Hiperbola (riyaziyyat)" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaïdjanais" 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) – bachkir" lang="ba" hreflang="ba" data-title="Гипербола (математика)" data-language-autonym="Башҡортса" data-language-local-name="bachkir" 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) – biélorusse" lang="be" hreflang="be" data-title="Гіпербала (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="biélorusse" 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 – bulgare" lang="bg" hreflang="bg" data-title="Хипербола" data-language-autonym="Български" data-language-local-name="bulgare" 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 – bengali" lang="bn" hreflang="bn" data-title="অধিবৃত্ত" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Hiperbola" title="Hiperbola – bosniaque" lang="bs" hreflang="bs" data-title="Hiperbola" data-language-autonym="Bosanski" data-language-local-name="bosniaque" 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 – sorani" lang="ckb" hreflang="ckb" data-title="بڕگەی زیاد" data-language-autonym="کوردی" data-language-local-name="sorani" 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 – tchèque" lang="cs" hreflang="cs" data-title="Hyperbola" data-language-autonym="Čeština" data-language-local-name="tchèque" 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) – tchouvache" lang="cv" hreflang="cv" data-title="Гипербола (математика)" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" 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 – gallois" lang="cy" hreflang="cy" data-title="Hyperbola" data-language-autonym="Cymraeg" data-language-local-name="gallois" 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 – danois" lang="da" hreflang="da" data-title="Hyperbel" data-language-autonym="Dansk" data-language-local-name="danois" 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) – allemand" lang="de" hreflang="de" data-title="Hyperbel (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="allemand" 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) – grec" lang="el" hreflang="el" data-title="Υπερβολή (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="grec" 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 – anglais" lang="en" hreflang="en" data-title="Hyperbola" data-language-autonym="English" data-language-local-name="anglais" 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 – espéranto" lang="eo" hreflang="eo" data-title="Hiperbolo" data-language-autonym="Esperanto" data-language-local-name="espéranto" 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 – espagnol" lang="es" hreflang="es" data-title="Hipérbola" data-language-autonym="Español" data-language-local-name="espagnol" 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 – estonien" lang="et" hreflang="et" data-title="Hüperbool" data-language-autonym="Eesti" data-language-local-name="estonien" 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 – basque" lang="eu" hreflang="eu" data-title="Hiperbola" data-language-autonym="Euskara" data-language-local-name="basque" 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 – persan" lang="fa" hreflang="fa" data-title="هذلولی" data-language-autonym="فارسی" data-language-local-name="persan" 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 – finnois" lang="fi" hreflang="fi" data-title="Hyperbeli" data-language-autonym="Suomi" data-language-local-name="finnois" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Hipearb%C3%B3il" title="Hipearbóil – irlandais" lang="ga" hreflang="ga" data-title="Hipearbóil" data-language-autonym="Gaeilge" data-language-local-name="irlandais" 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) – galicien" lang="gl" hreflang="gl" data-title="Hipérbole (xeometría)" data-language-autonym="Galego" data-language-local-name="galicien" 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 – hébreu" lang="he" hreflang="he" data-title="היפרבולה" data-language-autonym="עברית" data-language-local-name="hébreu" 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 – hindi" lang="hi" hreflang="hi" data-title="अति परवलय" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Hiperbola_(krivulja)" title="Hiperbola (krivulja) – croate" lang="hr" hreflang="hr" data-title="Hiperbola (krivulja)" data-language-autonym="Hrvatski" data-language-local-name="croate" 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) – hongrois" lang="hu" hreflang="hu" data-title="Hiperbola (matematika)" data-language-autonym="Magyar" data-language-local-name="hongrois" 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 – arménien" lang="hy" hreflang="hy" data-title="Հիպերբոլ" data-language-autonym="Հայերեն" data-language-local-name="arménien" 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 – indonésien" lang="id" hreflang="id" data-title="Hiperbola" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" 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 – islandais" lang="is" hreflang="is" data-title="Breiðbogi" data-language-autonym="Íslenska" data-language-local-name="islandais" 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) – italien" lang="it" hreflang="it" data-title="Iperbole (geometria)" data-language-autonym="Italiano" data-language-local-name="italien" 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 – japonais" lang="ja" hreflang="ja" data-title="双曲線" data-language-autonym="日本語" data-language-local-name="japonais" 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 – géorgien" lang="ka" hreflang="ka" data-title="ჰიპერბოლა" data-language-autonym="ქართული" data-language-local-name="géorgien" 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 – khmer" lang="km" hreflang="km" data-title="អ៊ីពែបូល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B3%A1%EC%84%A0" title="쌍곡선 – coréen" lang="ko" hreflang="ko" data-title="쌍곡선" data-language-autonym="한국어" data-language-local-name="coréen" 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 – kirghize" lang="ky" hreflang="ky" data-title="Гипербола математикада" data-language-autonym="Кыргызча" data-language-local-name="kirghize" 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) – lituanien" lang="lt" hreflang="lt" data-title="Hiperbolė (matematika)" data-language-autonym="Lietuvių" data-language-local-name="lituanien" 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 – letton" lang="lv" hreflang="lv" data-title="Hiperbola" data-language-autonym="Latviešu" data-language-local-name="letton" 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 – macédonien" lang="mk" hreflang="mk" data-title="Хипербола" data-language-autonym="Македонски" data-language-local-name="macédonien" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A7%E0%B4%BF%E0%B4%B5%E0%B4%B2%E0%B4%AF%E0%B4%82" title="അധിവലയം – malayalam" lang="ml" hreflang="ml" data-title="അധിവലയം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hyperbool_(meetkunde)" title="Hyperbool (meetkunde) – néerlandais" lang="nl" hreflang="nl" data-title="Hyperbool (meetkunde)" data-language-autonym="Nederlands" data-language-local-name="néerlandais" 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 – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Hyperbel" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien 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 – norvégien bokmål" lang="nb" hreflang="nb" data-title="Hyperbel" data-language-autonym="Norsk bokmål" data-language-local-name="norvégien bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Iperb%C3%B2la_(matematicas)" title="Iperbòla (matematicas) – occitan" lang="oc" hreflang="oc" data-title="Iperbòla (matematicas)" data-language-autonym="Occitan" data-language-local-name="occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Hiperbola_(matematyka)" title="Hiperbola (matematyka) – polonais" lang="pl" hreflang="pl" data-title="Hiperbola (matematyka)" data-language-autonym="Polski" data-language-local-name="polonais" 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 – piémontais" lang="pms" hreflang="pms" data-title="Ipérbol" data-language-autonym="Piemontèis" data-language-local-name="piémontais" 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 – portugais" lang="pt" hreflang="pt" data-title="Hipérbole" data-language-autonym="Português" data-language-local-name="portugais" 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 – roumain" lang="ro" hreflang="ro" data-title="Hiperbolă" data-language-autonym="Română" data-language-local-name="roumain" 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) – russe" lang="ru" hreflang="ru" data-title="Гипербола (математика)" data-language-autonym="Русский" data-language-local-name="russe" 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) – sicilien" lang="scn" hreflang="scn" data-title="Ipèrbuli (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="sicilien" 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 – écossais" lang="sco" hreflang="sco" data-title="Hyperbola" data-language-autonym="Scots" data-language-local-name="écossais" 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 – serbo-croate" lang="sh" hreflang="sh" data-title="Hiperbola" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-croate" 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) – slovaque" lang="sk" hreflang="sk" data-title="Hyperbola (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovaque" 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 – slovène" lang="sl" hreflang="sl" data-title="Hiperbola" data-language-autonym="Slovenščina" data-language-local-name="slovène" 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) – albanais" lang="sq" hreflang="sq" data-title="Hiperbola (matematikë)" data-language-autonym="Shqip" data-language-local-name="albanais" 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 – serbe" lang="sr" hreflang="sr" data-title="Хипербола" data-language-autonym="Српски / srpski" data-language-local-name="serbe" 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 – suédois" lang="sv" hreflang="sv" data-title="Hyperbel" data-language-autonym="Svenska" data-language-local-name="suédois" 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 – tamoul" lang="ta" hreflang="ta" data-title="அதிபரவளைவு" data-language-autonym="தமிழ்" data-language-local-name="tamoul" 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 – thaï" lang="th" hreflang="th" data-title="ไฮเพอร์โบลา" data-language-autonym="ไทย" data-language-local-name="thaï" 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) – ukrainien" lang="uk" hreflang="uk" data-title="Гіпербола (математика)" data-language-autonym="Українська" data-language-local-name="ukrainien" 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 – ouzbek" lang="uz" hreflang="uz" data-title="Giperbola" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ouzbek" 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 – vietnamien" lang="vi" hreflang="vi" data-title="Hyperbol" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" 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class="vector-pinnable-header-label">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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 Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><div class="bandeau-container metadata homonymie hatnote"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Aide:Homonymie" title="Aide:Homonymie"><img alt="Page d’aide sur l’homonymie" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/20px-Logo_disambig.svg.png" decoding="async" width="20" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/30px-Logo_disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Logo_disambig.svg/40px-Logo_disambig.svg.png 2x" data-file-width="512" data-file-height="375" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Hyperbole" class="mw-disambig" title="Hyperbole">Hyperbole</a>. </p> </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Conique_hyperbole.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e1/Conique_hyperbole.png" decoding="async" width="219" height="317" class="mw-file-element" data-file-width="219" data-file-height="317" /></a><figcaption>Hyperbole obtenue comme intersection d'un cône et d'un plan parallèle à l'axe du cône.<br />Si l'on incline légèrement le plan, l'intersection sera encore une hyperbole tant que l'angle d'inclinaison reste inférieur à l'angle que fait une génératrice avec l'axe du cône.</figcaption></figure> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, une <b>hyperbole</b> est une <a href="/wiki/Courbe_plane" title="Courbe plane">courbe plane</a> obtenue comme la <a href="/wiki/Courbe_d%27intersection" title="Courbe d'intersection">double intersection</a> d'un double <a href="/wiki/C%C3%B4ne_(g%C3%A9om%C3%A9trie)" title="Cône (géométrie)">cône</a> de révolution avec un plan. Elle peut également être définie comme <a href="/wiki/Conique" title="Conique">conique</a> d'<a href="/wiki/Excentricit%C3%A9_(math%C3%A9matiques)" title="Excentricité (mathématiques)">excentricité</a> supérieure à <a href="/wiki/1_(nombre)" title="1 (nombre)">1</a>, ou comme ensemble des points dont la différence des distances à deux points fixes est constante. </p><p>Le nom d'« hyperbole » (application <i>par excès</i>) lui est donné par <a href="/wiki/Apollonios_de_Perga" title="Apollonios de Perga">Apollonios de Perga</a>, remarquant, dans sa construction, que l'<a href="/wiki/Aire_(g%C3%A9om%C3%A9trie)" title="Aire (géométrie)">aire</a> du <a href="/wiki/Carr%C3%A9" title="Carré">carré</a> construit sur l'ordonnée excède l'aire d'un <a href="/wiki/Rectangle" title="Rectangle">rectangle</a> de hauteur fixe construit sur l'abscisse (voir section <a href="#Histoire">Histoire</a>). </p><p>Une hyperbole est constituée de deux branches disjointes symétriques l'une de l'autre et possédant deux <a href="/wiki/Asymptote" title="Asymptote">asymptotes</a> communes. </p><p>On peut rencontrer l'hyperbole dans de nombreuses circonstances : </p> <ul><li>lors de la représentation graphique de la <a href="/wiki/Fonction_inverse" title="Fonction inverse">fonction inverse</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\to {\frac {1}{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\to {\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0028f4320e64ccc86dc622cdc002d576579de23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.11ex; height:5.176ex;" alt="{\displaystyle x\to {\frac {1}{x}}}"></span> et de celle de toutes les fonctions qui lui sont associées : <span class="mwe-math-element"><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>,</li></ul> <ul><li>dans l'<a href="/wiki/Ombre" title="Ombre">ombre</a> créée par le pourtour (ou un <a href="/wiki/Abat-jour" title="Abat-jour">abat-jour</a> circulaire) d'une source de <a href="/wiki/Lumi%C3%A8re" title="Lumière">lumière</a> sur un mur</li></ul> <ul><li>dans la trajectoire de certains corps dans l'espace</li> <li>dans les <a href="/wiki/Interf%C3%A9rence" title="Interférence">interférences</a> produites par deux sources d'<a href="/wiki/Onde" title="Onde">ondulations</a> de même <a href="/wiki/Fr%C3%A9quence" title="Fréquence">fréquence</a></li> <li>dans la courbe suivie, pendant une journée, par l'extrémité de l'ombre du <a href="/wiki/Gnomon" title="Gnomon">gnomon</a> d'un <a href="/wiki/Cadran_solaire" title="Cadran solaire">cadran solaire</a> de style polaire.</li></ul> <p>L'hyperbole intervient dans d'autres objets mathématiques comme les <a href="/wiki/Hyperbolo%C3%AFde" title="Hyperboloïde">hyperboloïdes</a>, le <a href="/wiki/Parabolo%C3%AFde_hyperbolique" class="mw-redirect" title="Paraboloïde hyperbolique">paraboloïde hyperbolique</a>, les <a href="/wiki/Fonction_hyperbolique" title="Fonction hyperbolique">fonctions hyperboliques</a> (<a href="/wiki/Sinus_hyperbolique" title="Sinus hyperbolique">sinh</a>, <a href="/wiki/Cosinus_hyperbolique" title="Cosinus hyperbolique">cosh</a>, <a href="/wiki/Tangente_hyperbolique" title="Tangente hyperbolique">tanh</a>). Sa quadrature, c'est-à-dire le calcul de l'aire comprise entre une portion d'hyperbole et son axe principal, est à l'origine de la création de la <a href="/wiki/Logarithme" title="Logarithme">fonction logarithme</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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'hyperbole dessinée par l'ombre créée par une lampe.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définitions_géométriques"><span id="D.C3.A9finitions_g.C3.A9om.C3.A9triques"></span>Définitions géométriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=1" title="Modifier la section : Définitions géométriques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=1" title="Modifier le code source de la section : Définitions géométriques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intersection_d'un_cône_et_d'un_plan"><span id="Intersection_d.27un_c.C3.B4ne_et_d.27un_plan"></span>Intersection d'un cône et d'un plan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=2" title="Modifier la section : Intersection d'un cône et d'un plan" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=2" title="Modifier le code source de la section : Intersection d'un cône et d'un plan"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On considère un cône de révolution engendré par la rotation d'une droite (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle OA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db864cf94e2655d6a7b56c7479f63933e97afb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.516ex; height:2.176ex;" alt="{\displaystyle OA}"></span>) autour d'un axe (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ox}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ox}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d69bd718df9e6a0005953a22f0af2b9794ee70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.103ex; height:2.176ex;" alt="{\displaystyle Ox}"></span>) et on appelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> l'angle géométrique entre ces deux droites. On prend d'autre part un plan dont la normale fait avec l'axe (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ox}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ox}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d69bd718df9e6a0005953a22f0af2b9794ee70f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.103ex; height:2.176ex;" alt="{\displaystyle Ox}"></span>) un angle supérieur à <span class="mwe-math-element"><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 {\pi }{2}}-\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}-\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75cb7a02e622463cac6660609619ad5d37d9edf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.099ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}-\theta }"></span>. Si le plan ne passe pas par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>, il coupe le cône suivant une hyperbole. Si le plan passe par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>, il coupe le cône selon deux droites sécantes en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>. </p><p>Lorsqu'une lampe munie d'un abat-jour est placée non loin d'un mur vertical, la courbe qui délimite, sur le mur, la zone éclairée et la zone ombragée est un arc d'hyperbole. En effet, la lumière est diffusée selon un <a href="/wiki/C%C3%B4ne_(g%C3%A9om%C3%A9trie)" title="Cône (géométrie)">cône</a> — les rayons lumineux partent du centre de l'ampoule et s'appuient sur le cercle de l'ouverture de la lampe — coupé par un plan parallèle à l'axe du cône — le mur. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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>Principe des <a href="/wiki/Arc_diurne" title="Arc diurne">lignes de déclinaison</a> d'un cadran solaire.</figcaption></figure> <p>C'est également ce principe qui explique l'existence d'hyperboles sur <a href="https://commons.wikimedia.org/wiki/Category:Hyperbolas_on_sundials" class="extiw" title="commons:Category:Hyperbolas on sundials">certains cadrans solaires</a>. Au cours d'une journée, les rayons du soleil passant par la pointe du gnomon dessinent une portion de cône dont l'axe, parallèle à l'axe de <a href="/wiki/Rotation_de_la_Terre" title="Rotation de la Terre">rotation de la terre</a>, passe par la pointe du gnomon. L'ombre de cette pointe, sur le plan du cadran solaire dessine alors une portion d'hyperbole<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>note 1<span class="cite_crochet">]</span></a></sup>, appelée <a href="/wiki/Arc_diurne" title="Arc diurne">arc diurne</a> ou ligne de déclinaison, intersection du cône et d'un plan. Au cours de l'année, l'angle du cône varie. Aux équinoxes, il est de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c326d317eddef3ad3e6625e018a708e290a039f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.343ex;" alt="{\displaystyle 90^{\circ }}"></span>, le <i>cône</i> est un plan et l'ombre dessine une droite. Aux solstices, l'angle est de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 66^{\circ }\ 34'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>66</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mtext> </mtext> <msup> <mn>34</mn> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 66^{\circ }\ 34'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666fd65699bbe30c7ad23640730dd045b60efb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.969ex; height:2.509ex;" alt="{\displaystyle 66^{\circ }\ 34'}"></span> et l'ombre dessine une hyperbole. </p><p>La construction de l'hyperbole comme section d'un cône et d'un plan peut être réalisée à l'aide d'un <a href="/wiki/Compas_parfait" title="Compas parfait">compas parfait</a>. </p> <div class="clear" style="clear:both;"></div> <div class="mw-heading mw-heading3"><h3 id="Définition_par_foyer_et_directrice"><span id="D.C3.A9finition_par_foyer_et_directrice"></span>Définition par foyer et directrice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=3" title="Modifier la section : Définition par foyer et directrice" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=3" title="Modifier le code source de la section : Définition par foyer et directrice"><span>modifier le code</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/Fichier:HyperboleFoyerDirectrice.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/260px-HyperboleFoyerDirectrice.svg.png" decoding="async" width="260" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/390px-HyperboleFoyerDirectrice.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/HyperboleFoyerDirectrice.svg/520px-HyperboleFoyerDirectrice.svg.png 2x" data-file-width="472" data-file-height="479" /></a><figcaption>Hyperbole d'excentricité 3/2, avec son foyer F, sa directrice (D), ses asymptotes et son cercle principal.<br /> La distance MF est toujours égale à une fois et demi la distance MH.</figcaption></figure> <p>Soient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span> une <a href="/wiki/Droite_(math%C3%A9matiques)" title="Droite (mathématiques)">droite</a> et <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> un <a href="/wiki/Point_(g%C3%A9om%C3%A9trie)" title="Point (géométrie)">point</a> n'appartenant pas à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span>, et soit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> le plan contenant la droite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span> et le point <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. On appelle hyperbole de <i>(droite) <a href="/wiki/Directrice_(math%C3%A9matiques)" title="Directrice (mathématiques)">directrice</a></i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span> et de <i>foyer</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> l'ensemble des points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> du plan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> vérifiant </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>où <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(M,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(M,F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34681cf7072430781d0f37a05e18af971ed46c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.242ex; height:2.843ex;" alt="{\displaystyle d(M,F)}"></span> mesure la <a href="/wiki/Distance_(math%C3%A9matiques)" title="Distance (mathématiques)">distance</a> du point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> au point <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(M,(D))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(M,(D))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff92e8243d90a3a7cac67a3128678341ec1d28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.235ex; height:2.843ex;" alt="{\displaystyle d(M,(D))}"></span> mesure la distance du point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> à la droite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span>. </p><p>La constante <span class="mwe-math-element"><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> est appelée « excentricité » de l'hyperbole. Elle est caractéristique de la forme de l'hyperbole : si l'on transforme l'hyperbole par une similitude, son excentricité reste inchangée. Elle est donc indépendante du choix arbitraire de repère orthonormé pour ce plan ; elle détermine tous les autres rapports de distances (et toutes les différences angulaires) mesurés sur l'hyperbole. Plus <span class="mwe-math-element"><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> est grand, plus l'hyperbole s'évase, les deux branches se rapprochant de la directrice. Plus <span class="mwe-math-element"><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> s'approche de 1 plus l'hyperbole s'arrondit, les deux branches s'éloignant l'une de l'autre, celle située dans le même demi-plan que le foyer se rapprochant d'une <a href="/wiki/Parabole" title="Parabole">parabole</a>. </p><p>Notons <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> le <a href="/wiki/Projection_orthogonale" title="Projection orthogonale">projeté orthogonal</a> de <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> sur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (KF)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (KF)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dab17af64da6332b2c0eae6005edf2a701270989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle (KF)}"></span> est alors clairement un axe de symétrie de l'hyperbole appelé « axe focal ». </p><p>L'axe focal coupe l'hyperbole en deux points appelés les « sommets » <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span> de l'hyperbole. </p><p>La médiatrice du segment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [SS']}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>S</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [SS']}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ff9b239faecd4f0c8431a10de6553fef9e70ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.999ex; height:3.009ex;" alt="{\displaystyle [SS']}"></span> est également un axe de symétrie de l'hyperbole appelé <i>axe non focal</i>. Le point d'intersection des deux axes, noté <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span>, est alors le <i>centre</i> de symétrie de l'hyperbole. </p><p>Le cercle de diamètre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [SS']}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>S</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [SS']}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ff9b239faecd4f0c8431a10de6553fef9e70ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.999ex; height:3.009ex;" alt="{\displaystyle [SS']}"></span> est appelé « <a href="/wiki/Cercle_principal" title="Cercle principal">cercle principal</a> » de l'hyperbole. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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>Trace du cône <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc76abe9b2a2eb230cee61a42ad476f4bcb92f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.262ex; height:2.843ex;" alt="{\displaystyle (\Gamma )}"></span>, du plan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> , de la sphère <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8116fdd6059c390bf7a7c8a2ee22a31459a3a3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle (\Sigma )}"></span> et du plan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d38298775a33ba3414ffceb15f148e358c7a7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.316ex; height:3.009ex;" alt="{\displaystyle (P')}"></span> dans le plan passant par l'axe du cône et perpendiculaire à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span>. La section du plan et du cône est une hyperbole de foyer <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> de directrice passant par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, de sommet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> et d'excentricité <span class="mwe-math-element"><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 {SF}{SK}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <mi>F</mi> </mrow> <mrow> <mi>S</mi> <mi>K</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {SF}{SK}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211ad74d314e825d8f2196526d18f14b2c9cd397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.401ex; height:5.509ex;" alt="{\displaystyle {\frac {SF}{SK}}}"></span>.</figcaption></figure> <p>La symétrie orthogonale par rapport à l'axe non focal envoie le foyer <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> et la directrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span> en <span class="mwe-math-element"><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"> <msup> <mi>F</mi> <mo>′</mo> </msup> </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/9cb2068c9fcb933aa21d1cfd42103514f5276e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b7487fbaca75c0a5483b60a761c12ff3d9020e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:3.009ex;" alt="{\displaystyle (D')}"></span>. Par symétrie, l'hyperbole est également l'hyperbole de foyer <span class="mwe-math-element"><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"> <msup> <mi>F</mi> <mo>′</mo> </msup> </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/9cb2068c9fcb933aa21d1cfd42103514f5276e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}"></span>, de directrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01b7487fbaca75c0a5483b60a761c12ff3d9020e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:3.009ex;" alt="{\displaystyle (D')}"></span> et d'excentricité <span class="mwe-math-element"><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>. </p><p>Une telle hyperbole possède en outre deux asymptotes passant par <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> et par les points d'intersection du cercle principal et des directrices. Ces points sont également les projetés orthogonaux des foyers sur les asymptotes<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. Lorsque les deux asymptotes sont perpendiculaires, on dit que l'hyperbole est « équilatère ». </p><p>En relation avec la définition précédente, l'hyperbole obtenue comme section de cône et de plan peut être définie par foyer et directrice. On considère une sphère <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8116fdd6059c390bf7a7c8a2ee22a31459a3a3a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle (\Sigma )}"></span> inscrite dans le cône <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\Gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\Gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc76abe9b2a2eb230cee61a42ad476f4bcb92f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.262ex; height:2.843ex;" alt="{\displaystyle (\Gamma )}"></span> et touchant le plan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> en F (<a href="/wiki/Th%C3%A9or%C3%A8me_de_Dandelin" title="Théorème de Dandelin">sphère de Dandelin</a>) et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d38298775a33ba3414ffceb15f148e358c7a7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.316ex; height:3.009ex;" alt="{\displaystyle (P')}"></span> le plan contenant le cercle de tangence de la sphère et du cône. L'hyperbole est de foyer F et de directrice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (D)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (D)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbcb66c83c0cf2a2d0e26d456072eef521e6823" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.734ex; height:2.843ex;" alt="{\displaystyle (D)}"></span> droite d'intersection des deux plans <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>P</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d38298775a33ba3414ffceb15f148e358c7a7b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.316ex; height:3.009ex;" alt="{\displaystyle (P')}"></span>. Dans le plan perpendiculaire à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> et passant par l'axe du cône, se trouvent le point <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, le sommet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> et le point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>. L'excentricité est donnée par le rapport <span class="mwe-math-element"><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 {SF}{SK}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>S</mi> <mi>F</mi> </mrow> <mrow> <mi>S</mi> <mi>K</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {SF}{SK}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211ad74d314e825d8f2196526d18f14b2c9cd397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.401ex; height:5.509ex;" alt="{\displaystyle {\frac {SF}{SK}}}"></span>. Elle ne dépend que de l'inclinaison du plan par rapport à l'axe du cône. Si l'on appelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fef12048a146889ba79830c19cd96794b8ac8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.025ex; height:2.843ex;" alt="{\displaystyle (d)}"></span> la trace de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> dans le plan perpendiculaire à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37919dc239cc39e100a8a628e9d8fd45c6cc0284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle (P)}"></span> passant par l'axe du cône, si l'on note <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> l'angle entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2fef12048a146889ba79830c19cd96794b8ac8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.025ex; height:2.843ex;" alt="{\displaystyle (d)}"></span> et l'axe du cône et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> l'angle du cône, l'excentricité est de <span class="mwe-math-element"><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 {\cos \alpha }{\cos \theta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\cos \alpha }{\cos \theta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25f8befbbe379447cef34bcbf14f1e7e71337a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.822ex; height:4.843ex;" alt="{\displaystyle {\frac {\cos \alpha }{\cos \theta }}}"></span>. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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>Propriété de divergence d'une lentille hyperbolique.</figcaption></figure> <p>Cette relation entre foyer, directrice et excentricité dans une hyperbole est exploitée dans la construction des <a href="/wiki/Lentille_optique" title="Lentille optique">lentilles</a> divergentes : si l'<a href="/wiki/Indice_de_r%C3%A9fraction" title="Indice de réfraction">indice</a> de la lentille par rapport au milieu est de <span class="mwe-math-element"><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>, et si la surface concave de la lentille est une hyperbole de foyer <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> et d'excentricité <span class="mwe-math-element"><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>, le faisceau de rayons parallèles traversant la lentille, se disperse comme si les rayons venaient du foyer <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span><sup id="cite_ref-mathcurve_3-0" class="reference"><a href="#cite_note-mathcurve-3"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p> <div class="clear" style="clear:both;"></div> <hr /> <p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Définition_bifocale"><span id="D.C3.A9finition_bifocale"></span>Définition bifocale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=4" title="Modifier la section : Définition bifocale" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=4" title="Modifier le code source de la section : Définition bifocale"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fichier:Hyperbole_tangente.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hyperbole_tangente.png/290px-Hyperbole_tangente.png" decoding="async" width="290" height="201" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Hyperbole_tangente.png/435px-Hyperbole_tangente.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/eb/Hyperbole_tangente.png 2x" data-file-width="564" data-file-height="390" /></a><figcaption>La tangente en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> est aussi une bissectrice.</figcaption></figure> <p>L'hyperbole est le <a href="/wiki/Lieu_g%C3%A9om%C3%A9trique" title="Lieu géométrique">lieu géométrique</a> des points dont la différence des distances aux deux foyers est constante. </p><p>Géométriquement, cela donne : </p><p>Soient <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> et <span class="mwe-math-element"><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"> <msup> <mi>F</mi> <mo>′</mo> </msup> </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/9cb2068c9fcb933aa21d1cfd42103514f5276e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}"></span> deux points distincts du plan, distants de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8389d669a77876483e5dded26a9112dc4013e23a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.169ex; height:2.176ex;" alt="{\displaystyle 2c}"></span> et soit <span class="mwe-math-element"><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> un réel strictement compris entre 0 et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span>. On appelle hyperbole de foyers <span class="mwe-math-element"><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/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> et <span class="mwe-math-element"><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"> <msup> <mi>F</mi> <mo>′</mo> </msup> </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/9cb2068c9fcb933aa21d1cfd42103514f5276e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.499ex; height:2.509ex;" alt="{\displaystyle F'}"></span> l'ensemble des points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> du plan vérifiant la propriété suivante : </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'<i>axe focal</i> est le nom de la droite portant les deux foyers : c'est l'un des deux axes de symétrie de l'hyperbole, le seul qui la coupe. Pour cette raison, on le nomme aussi <i>axe transverse</i> et ses points communs avec la courbe sont les <i>sommets</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9961844d1f539adee019e432dc18aa2a7ede59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.206ex; height:2.509ex;" alt="{\displaystyle S'}"></span> de l'hyperbole. Le réel <span class="mwe-math-element"><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 définition ci-dessus apparaît comme la moitié de la distance entre les sommets. Les directrices de l'hyperbole passent par les points de contact des tangentes au cercle principal (cercle de diamètre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [SS']}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>S</mi> <msup> <mi>S</mi> <mo>′</mo> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [SS']}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ff9b239faecd4f0c8431a10de6553fef9e70ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.999ex; height:3.009ex;" alt="{\displaystyle [SS']}"></span>) issues des foyers<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. </p><p>En chaque point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> de cette hyperbole, la bissectrice du secteur angulaire <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (FMF')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <mi>M</mi> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (FMF')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f8f148aaa8c4bfd21cd62fc9b6d487de376f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.492ex; height:3.009ex;" alt="{\displaystyle (FMF')}"></span> se trouve être la tangente en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> à la courbe. </p><p>Cette construction de l'hyperbole permet d'expliquer la présence d'hyperbole lors d'<a href="/wiki/Interf%C3%A9rence" title="Interférence">interférences</a> entre deux sources de même fréquence. Au point M le déphasage de l'onde par rapport à la source S<sub>1</sub> est proportionnel à la distance MS<sub>1</sub>. Le déphasage entre l'onde venant de S<sub>1</sub> et l'onde venant de S<sub>2</sub> est donc proportionnel à la différence des distances. Les deux ondes s'annulent quand leur déphasage est égale à <span class="texhtml">(2<i>k</i>+1)π</span>. Les deux ondes s'amplifient quand leur déphasage est de <span class="texhtml">2<i>k</i>π</span>. Les points où l'onde résultante a une amplitude nulle et les points où l'onde résultante a une amplitude maximale dessinent donc un faisceau d'hyperboles de foyers S<sub>1</sub> et S<sub>2</sub>. </p><p>Il existe des mécanismes avec corde et poulie exploitant cette propriété des hyperboles pour effectuer leur tracé. c'est le cas du dispositif conçu par <a href="/wiki/Ibn_Sahl" title="Ibn Sahl">Ibn Sahl</a> au <abbr class="abbr" title="10ᵉ siècle"><span class="romain">X</span><sup style="font-size:72%">e</sup></abbr> siècle<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup>. </p> <table width="100%" align="center" border="0" cellpadding="4" cellspacing="4"> <tbody><tr style="vertical-align:top"> <td width="50%"> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/IbnSahlHyperboleAnim.ogv/220px--IbnSahlHyperboleAnim.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="266" data-durationhint="47" data-mwtitle="IbnSahlHyperboleAnim.ogv" data-mwprovider="wikimediacommons" resource="/wiki/Fichier: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>Construction mécanique d'une hyperbole selon la méthode d'<a href="/wiki/Ibn_Sahl" title="Ibn Sahl">Ibn Sahl</a>. La règle pivote autour du foyer F<sub>2</sub>. Le crayon M, sur la règle s'appuie sur une corde F<sub>1</sub>MD de longueur fixe et dessine une portion d'hyperbole de foyers F<sub>1</sub> et F<sub>2</sub>.</figcaption></figure> </td> <td width="50%"> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Nodalandantinodallines.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Nodalandantinodallines.png/220px-Nodalandantinodallines.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Nodalandantinodallines.png/330px-Nodalandantinodallines.png 1.5x, //upload.wikimedia.org/wikipedia/commons/1/1b/Nodalandantinodallines.png 2x" data-file-width="410" data-file-height="308" /></a><figcaption>Interférences produites par deux sources synchrones avec les lignes nodales et anti-nodales.</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Image_d'un_cercle_par_une_homographie"><span id="Image_d.27un_cercle_par_une_homographie"></span>Image d'un cercle par une homographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=5" title="Modifier la section : Image d'un cercle par une homographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=5" title="Modifier le code source de la section : Image d'un cercle par une homographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Comme toute conique, l'hyperbole peut être considérée comme l'image d'un cercle par une <a href="/wiki/Application_projective" title="Application projective">transformation projective</a>. Plus précisément, si (H) est une hyperbole et (C) un cercle, il existe une transformation projective qui transforme (C) en (H). Par exemple, l'hyperbole est l'image de son cercle principal par une transformation projective dont l'expression analytique, dans un repère orthonormé porté par ses axes de symétrie, est </p><p><span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>où <span class="texhtml mvar" style="font-style:italic;">a</span> est le rayon du cercle principal et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm {\frac {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm {\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeffb4fb450b5fcb30799edfac3d709522d78891" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.874ex; height:5.343ex;" alt="{\displaystyle \pm {\frac {b}{a}}}"></span> les pentes de ses asymptotes. </p><p>Cette propriété permet de transférer à l'hyperbole des propriétés concernant des règles d'incidence de droites dans un cercle, comme le <a href="/wiki/Th%C3%A9or%C3%A8me_de_Pascal" title="Théorème de Pascal">théorème de Pascal</a>. </p><p>Parmi les transformations projectives, il en est de particulières comme les <a href="/wiki/Homologie_(transformation_g%C3%A9om%C3%A9trique)" title="Homologie (transformation géométrique)">homologies</a> harmoniques. Ce sont des homologies <a href="/wiki/Involution_(math%C3%A9matiques)" title="Involution (mathématiques)">involutives</a> de centre I d'axe (d) et de rapport -1. Dans celles-ci, un point M, son image M', le centre I et le point m d'intersection de (IM) avec (d) sont en <a href="/wiki/Division_harmonique" title="Division harmonique">division harmonique</a>. </p><p>Un cercle (C) a pour image une hyperbole par une homologie harmonique de centre I et d'axe (d) si et seulement si l'image (d') de l'axe (d) par l'homothétie de centre I et de rapport 1/2 coupe le cercle en deux points<sup id="cite_ref-Guillerault_6-0" class="reference"><a href="#cite_note-Guillerault-6"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup>. Réciproquement, une hyperbole étant donnée, il est possible de trouver des cercles et des homologies harmoniques associées échangeant hyperbole et cercle. Exemple : </p> <ul><li>une hyperbole de foyer F, de directrice (d) et d'excentricité e est l'image du cercle de centre F et de rayon <i>eh</i> (où <i>h</i> est la distance entre le foyer et la directrice) par l'homologie harmonique de centre F et d'axe (d<sub>1</sub>) image de (d) dans l'homothétie de centre F et de rapport 2<sup id="cite_ref-Guillerault_6-1" class="reference"><a href="#cite_note-Guillerault-6"><span class="cite_crochet">[</span>5<span class="cite_crochet">]</span></a></sup> ;</li> <li>une hyperbole équilatère est l'image de son cercle principal par une homologie harmonique de centre un des sommets et d'axe (d) passant par l'autre sommet et perpendiculaire à l'axe principal de l'hyperbole. Quand l'hyperbole n'est pas équilatère, elle reste l'image d'un cercle par une homologie harmonique de centre un des sommets, mais le cercle n'est plus le cercle principal et l'axe ne passe plus par l'autre sommet. Ce sont des homologies de ce type qui permettent de démontrer le <a href="/wiki/Point_de_Fr%C3%A9gier" title="Point de Frégier">théorème de Frégier</a>.</li></ul> <table width="100%" align="center" border="0" cellpadding="4" cellspacing="4"> <tbody><tr style="vertical-align:top"> <td width="50%"> <figure class="mw-default-size mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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'hyperbole et le cercle sont homologues dans l'homologie harmonique de centre F et 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/Fichier: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'hyperbole équilatère et son cercle principal sont homologues dans l'homologie harmonique de centre S<sub>2</sub> et d'axe (d). Pour construire le point M', on mène par m une tangente au cercle qui touche le cercle en M. La droite (S<sub>2</sub>M) rencontre la perpendiculaire à l'axe focal passant par m en M'.</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Relations_entre_les_grandeurs_caractéristiques_d'une_hyperbole"><span id="Relations_entre_les_grandeurs_caract.C3.A9ristiques_d.27une_hyperbole"></span>Relations entre les grandeurs caractéristiques d'une hyperbole</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=6" title="Modifier la section : Relations entre les grandeurs caractéristiques d'une hyperbole" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=6" title="Modifier le code source de la section : Relations entre les grandeurs caractéristiques d'une hyperbole"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:HyperboleParametre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/260px-HyperboleParametre.svg.png" decoding="async" width="260" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/390px-HyperboleParametre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/HyperboleParametre.svg/520px-HyperboleParametre.svg.png 2x" data-file-width="173" data-file-height="125" /></a><figcaption>Les différentes grandeurs d'une hyperbole.</figcaption></figure> <p>Les grandeurs (géométriques ou numériques) d’une hyperbole sont : </p> <ul><li>la distance entre le centre de l'hyperbole et un de ses sommets généralement notée <span class="texhtml mvar" style="font-style:italic;">a</span> ;</li> <li>la pente (en valeur absolue) que font les asymptotes avec l'axe focal, généralement notée <span class="texhtml mvar" style="font-style:italic;">b/a</span> ;</li> <li>la distance séparant le centre de l'hyperbole et un des foyers, généralement notée <span class="texhtml mvar" style="font-style:italic;">c</span>;</li> <li>la distance séparant un foyer F de sa directrice (<i>d</i>) associée, généralement notée <span class="texhtml mvar" style="font-style:italic;">h</span> ;</li> <li>la distance séparant le centre de l’hyperbole et une de ses deux directrices, généralement notée <span class="texhtml mvar" style="font-style:italic;">f</span> ;</li> <li>l'excentricité de l’hyperbole (strictement supérieure à 1), généralement notée <span class="texhtml mvar" style="font-style:italic;">e</span> ;</li> <li>le « paramètre » de l’hyperbole, généralement noté <span class="texhtml mvar" style="font-style:italic;">p</span>, représentant le demi <i>latus rectum</i> (corde parallèle à la directrice et passant par le foyer).</li></ul> <p>Des relations existent entre ces grandeurs : </p> <ul><li>si l’hyperbole est définie par son excentricité <span class="mwe-math-element"><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> et la distance <span class="mwe-math-element"><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 le foyer F et la directrice (d), alors : <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>si l’hyperbole est donnée par la distance entre le centre et un sommet <span class="mwe-math-element"><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> et la pente des asymptotes <span class="mwe-math-element"><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 {b}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {b}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30304b82a801ef33eaf4c0c0306aa6966e83d2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.066ex; height:5.343ex;" alt="{\displaystyle {\frac {b}{a}}}"></span>, alors : <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>lorsque l’on connait la distance entre le centre et le sommet <span class="mwe-math-element"><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> et l’excentricité <span class="mwe-math-element"><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>enfin, dans la définition bifocale de l'hyperbole où sont connues la longueur 2<span class="texhtml mvar" style="font-style:italic;">a</span> et la distance 2<span class="texhtml mvar" style="font-style:italic;">c</span> entre les foyers : <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="Équations"><span id=".C3.89quations"></span>Équations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=7" title="Modifier la section : Équations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=7" title="Modifier le code source de la section : Équations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Équation_dans_un_repère_normé_porté_par_les_asymptotes"><span id=".C3.89quation_dans_un_rep.C3.A8re_norm.C3.A9_port.C3.A9_par_les_asymptotes"></span>Équation dans un repère normé porté par les asymptotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=8" title="Modifier la section : Équation dans un repère normé porté par les asymptotes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=8" title="Modifier le code source de la section : Équation dans un repère normé porté par les asymptotes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Cas_particulier_de_la_fonction_inverse">Cas particulier de la fonction inverse</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=9" title="Modifier la section : Cas particulier de la fonction inverse" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=9" title="Modifier le code source de la section : Cas particulier de la fonction inverse"><span>modifier le code</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/Fichier: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>Représentation graphique de la fonction inverse. Hyperbole avec ses foyers et ses directrices.</figcaption></figure> <p>L'hyperbole dont l'expression mathématique est la plus simple est la représentation graphique de la <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">fonction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <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> définie par <span class="mwe-math-element"><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>, voir <a href="/wiki/Fonction_inverse" title="Fonction inverse">fonction inverse</a>. </p><p>Cette hyperbole est <i>équilatère</i> car ses deux asymptotes sont orthogonales. Son excentricité vaut <span class="mwe-math-element"><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="NavFrame" style="border: thin solid #aaaaaa; margin:1em 2em; padding: 0 1em; font-size:100%; text-align:justify; overflow:hidden;"> <div class="NavHead" style="background-color:transparent; color:inherit; padding:0;">Démonstration</div><div class="NavContent" style="padding-bottom:0.4em"> <p>Montrons que le graphe de la fonction définie par <span class="mwe-math-element"><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)={\frac {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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce585154e4780be88423541d65e57da942e543e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.682ex; height:5.176ex;" alt="{\displaystyle f(x)={\frac {1}{x}}}"></span> est bien une hyperbole. On pose <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {1}{\sqrt {2}}}(y+x)\,,\,v={\frac {1}{\sqrt {2}}}(y-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {1}{\sqrt {2}}}(y+x)\,,\,v={\frac {1}{\sqrt {2}}}(y-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a53066686e985d1984c3344bfac1f70a181ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.601ex; height:6.176ex;" alt="{\displaystyle u={\frac {1}{\sqrt {2}}}(y+x)\,,\,v={\frac {1}{\sqrt {2}}}(y-x)}"></span></span> donc <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {1}{\sqrt {2}}}(u-v)\,,\,y={\frac {1}{\sqrt {2}}}(u+v).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {1}{\sqrt {2}}}(u-v)\,,\,y={\frac {1}{\sqrt {2}}}(u+v).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db3b4724854e6e44cf85c6580d4730cf6bc5b42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.22ex; height:6.176ex;" alt="{\displaystyle x={\frac {1}{\sqrt {2}}}(u-v)\,,\,y={\frac {1}{\sqrt {2}}}(u+v).}"></span></span> Ainsi à partir de <span class="mwe-math-element"><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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7028e7e873eb4ec50f53be53ad478ded8351c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle xy=1}"></span> qui est l'équation de la courbe de la fonction inverse on obtient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{2}-v^{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{2}-v^{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1f039c333c5887c41b1e4443fbb2df4241569ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.667ex; height:2.843ex;" alt="{\displaystyle u^{2}-v^{2}=2}"></span> </p><p>On prend alors l'équation générale d'une conique dans un repère orthonormé dont l'axe principal passe par le foyer et est perpendiculaire à la directrice : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u-c)^{2}+v^{2}=e^{2}(u-f)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>−</mo> <mi>c</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</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>u</mi> <mo>−</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u-c)^{2}+v^{2}=e^{2}(u-f)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83876eb6e5ed300201c581bdc562fd0bb530d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.611ex; height:3.176ex;" alt="{\displaystyle (u-c)^{2}+v^{2}=e^{2}(u-f)^{2}}"></span></span> où <i>f</i> est la distance du centre du repère à la directrice et <i>c</i> la distance du centre du repère au foyer. </p><p>En développant il vient : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e^{2}-1)u^{2}-v^{2}=(c^{2}-e^{2}f^{2})+2u(e^{2}f-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>c</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> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>u</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e^{2}-1)u^{2}-v^{2}=(c^{2}-e^{2}f^{2})+2u(e^{2}f-c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9102b38d49416061caeb62a415096b0a1268adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.083ex; height:3.176ex;" alt="{\displaystyle (e^{2}-1)u^{2}-v^{2}=(c^{2}-e^{2}f^{2})+2u(e^{2}f-c)}"></span> </p><p>ainsi avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\sqrt {2}},\,c=2f,\,f=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mo>=</mo> <mn>2</mn> <mi>f</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {2}},\,c=2f,\,f=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28e41958998276fe86f336b7f1292d990a052c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.208ex; height:3.009ex;" alt="{\displaystyle e={\sqrt {2}},\,c=2f,\,f=1}"></span> on retrouve bien l'équation recherchée. </p> </div><div class="clear" style="clear:both;"></div> </div> <div class="mw-heading mw-heading4"><h4 id="Cas_général"><span id="Cas_g.C3.A9n.C3.A9ral"></span>Cas général</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=10" title="Modifier la section : Cas général" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=10" title="Modifier le code source de la section : Cas général"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans le repère <span class="mwe-math-element"><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>, où O est le centre de l'hyperbole et <span class="mwe-math-element"><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> et <span class="mwe-math-element"><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> les vecteurs unitaires directeurs des asymptotes, l'hyperbole a pour équation<sup id="cite_ref-mathcurve_3-1" class="reference"><a href="#cite_note-mathcurve-3"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>Réciproquement, si deux droites de vecteurs directeurs <span class="mwe-math-element"><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> et <span class="mwe-math-element"><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 coupent en O et si une courbe, dans le repère <span class="mwe-math-element"><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 pour équation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a68fcbf4e4a86e5465ac0a30d2092b8fab34ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.795ex; height:2.509ex;" alt="{\displaystyle xy=k}"></span> où <span class="texhtml mvar" style="font-style:italic;">k</span> est un réel non nul, alors cette courbe est une hyperbole<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite_crochet">[</span>6<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Équations_dans_des_repères_où_l'axe_focal_est_l'axe_principal"><span id=".C3.89quations_dans_des_rep.C3.A8res_o.C3.B9_l.27axe_focal_est_l.27axe_principal"></span>Équations dans des repères où l'axe focal est l'axe principal</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=11" title="Modifier la section : Équations dans des repères où l'axe focal est l'axe principal" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=11" title="Modifier le code source de la section : Équations dans des repères où l'axe focal est l'axe principal"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Si_le_centre_du_repère_est_le_centre_de_l'hyperbole"><span id="Si_le_centre_du_rep.C3.A8re_est_le_centre_de_l.27hyperbole"></span>Si le centre du repère est le centre de l'hyperbole</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=12" title="Modifier la section : Si le centre du repère est le centre de l'hyperbole" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=12" title="Modifier le code source de la section : Si le centre du repère est le centre de l'hyperbole"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans un repère dont les axes sont de symétrie pour l'hyperbole, l'axe transverse pour axe des abscisses, l'équation cartésienne se met sous la forme <span style="display: block; margin-left:1.6em;"> <span class="mwe-math-element"><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></span> donnant alors les représentations paramétriques <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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> et <span class="mwe-math-element"><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></span> pour chacune des branches. </p><p>Un autre paramétrage possible est : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>Son équation polaire est : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p> <div class="mw-heading mw-heading4"><h4 id="Si_le_centre_du_repère_est_le_foyer_de_l'hyperbole"><span id="Si_le_centre_du_rep.C3.A8re_est_le_foyer_de_l.27hyperbole"></span>Si le centre du repère est le foyer de l'hyperbole</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=13" title="Modifier la section : Si le centre du repère est le foyer de l'hyperbole" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=13" title="Modifier le code source de la section : Si le centre du repère est le foyer de l'hyperbole"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans un repère orthonormé <span class="mwe-math-element"><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> dans lequel <span class="mwe-math-element"><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'hyperbole a pour équation cartésienne : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>Son équation polaire, dans ce même repère est : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>où <span class="texhtml mvar" style="font-style:italic;">p = eh</span> est le paramètre de l'hyperbole. </p> <div class="mw-heading mw-heading4"><h4 id="Si_le_centre_du_repère_est_le_sommet_de_l'hyperbole"><span id="Si_le_centre_du_rep.C3.A8re_est_le_sommet_de_l.27hyperbole"></span>Si le centre du repère est le sommet de l'hyperbole</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=14" title="Modifier la section : Si le centre du repère est le sommet de l'hyperbole" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=14" title="Modifier le code source de la section : Si le centre du repère est le sommet de l'hyperbole"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans un repère orthonormé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S,{\vec {i}},{\vec {j}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>S</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 (S,{\vec {i}},{\vec {j}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5f07643f999e722f4cf4d354b743095061ba1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.564ex; height:3.343ex;" alt="{\displaystyle (S,{\vec {i}},{\vec {j}})}"></span>, l'hyperbole a pour équation cartésienne : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=2px+(e^{2}-1)x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mi>p</mi> <mi>x</mi> <mo>+</mo> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=2px+(e^{2}-1)x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f86b5544ff4e7a344c78988418cf598c08861f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.149ex; height:3.176ex;" alt="{\displaystyle y^{2}=2px+(e^{2}-1)x^{2}}"></span>,</span> </p> <div class="mw-heading mw-heading3"><h3 id="Équation_générale_de_conique"><span id=".C3.89quation_g.C3.A9n.C3.A9rale_de_conique"></span>Équation générale de conique</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=15" title="Modifier la section : Équation générale de conique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=15" title="Modifier le code source de la section : Équation générale de conique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De manière générale, comme toute conique, une hyperbole a une équation cartésienne de la forme <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> avec <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> Pour qu'une telle équation soit celle d'une hyperbole, il est nécessaire<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite_crochet">[</span>7<span class="cite_crochet">]</span></a></sup> que <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> Dans ce cas, la conique a pour centre le point C dont les coordonnées (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) vérifient le système<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite_crochet">[</span>8<span class="cite_crochet">]</span></a></sup> : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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 =0\\\beta x+\gamma y+\epsilon =0.\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> <mi>δ</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> <mi>x</mi> <mo>+</mo> <mi>γ</mi> <mi>y</mi> <mo>+</mo> <mi>ϵ</mi> <mo>=</mo> <mn>0.</mn> </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 =0\\\beta x+\gamma y+\epsilon =0.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91b053a23977fcecfeeda671c04f302c9c48c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.107ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\alpha x+\beta y+\delta =0\\\beta x+\gamma y+\epsilon =0.\end{cases}}}"></span></span> On reconnait les dérivées partielles : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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}{\frac {\partial f}{\partial x}}=0\\{\frac {\partial f}{\partial y}}=0\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\frac {\partial f}{\partial x}}=0\\{\frac {\partial f}{\partial y}}=0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/508bcde7af0eb8027304c9f8487648bcdfcf05db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:9.787ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}{\frac {\partial f}{\partial x}}=0\\{\frac {\partial f}{\partial y}}=0\end{cases}}}"></span></span> </p><p>Un changement de repère, en prenant pour centre le point C, conduit à l'équation suivante : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> qui sera l'équation d'une hyperbole si et seulement si <i>f</i>(<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) est non nul. </p><p>Les directions des asymptotes sont les solutions de l'équation homogène<sup id="cite_ref-LFA_10-0" class="reference"><a href="#cite_note-LFA-10"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup> <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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}=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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha X^{2}+2\beta XY+\gamma Y^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04905e7453b24880f25f73f2ac7e1459d3506ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.945ex; height:3.176ex;" alt="{\displaystyle \alpha X^{2}+2\beta XY+\gamma Y^{2}=0}"></span></span> </p><p>Les asymptotes font un angle φ tel que<sup id="cite_ref-LFA_10-1" class="reference"><a href="#cite_note-LFA-10"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>: <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos ^{2}\varphi ={\frac {(\alpha +\gamma )^{2}}{(\alpha -\gamma )^{2}+4\beta ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>γ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mi>γ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos ^{2}\varphi ={\frac {(\alpha +\gamma )^{2}}{(\alpha -\gamma )^{2}+4\beta ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aca876809bcacc42f373fc7309de48f2c226c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.855ex; height:6.676ex;" alt="{\displaystyle \cos ^{2}\varphi ={\frac {(\alpha +\gamma )^{2}}{(\alpha -\gamma )^{2}+4\beta ^{2}}}}"></span></span> </p><p>L'hyperbole est équilatère si et seulement si <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =-\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77cf6f4e0adeb5d1a9f9591e37c4fe321d04af01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.657ex; height:2.509ex;" alt="{\displaystyle \alpha =-\gamma }"></span>. </p><p>Si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> est non nul, les pentes des asymptotes sont les racines de l'<i>équation aux pentes</i><sup id="cite_ref-LFA_10-2" class="reference"><a href="#cite_note-LFA-10"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</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 \gamma m^{2}+2\beta m+\alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>β</mi> <mi>m</mi> <mo>+</mo> <mi>α</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma m^{2}+2\beta m+\alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da53b148238fc23e08773cb1094120bb3483bf43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.321ex; height:3.176ex;" alt="{\displaystyle \gamma m^{2}+2\beta m+\alpha =0}"></span>, et si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> est nul, les directions ont pour équations : <i>X</i> = 0 et <i>αX</i> + 2<i>βY</i>=0<sup id="cite_ref-LFA_10-3" class="reference"><a href="#cite_note-LFA-10"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>. </p><p>Connaissant ses asymptotes (d1) : <i>ax + by + c</i> = 0 et (d2) : <i>a'x + b'y + c' </i>= 0, et un point M(<i>u</i>,<i>v</i>) de l'hyperbole , <span class="need_ref" title="Une source est souhaitée pour ce passage." style="cursor:help;">son équation est</span><sup class="need_ref_tag" style="padding-left:2px;"><a href="/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" title="Aide:Référence nécessaire">[<abbr class="abbr" title="référence">réf.</abbr> souhaitée]</a></sup> : </p><p><span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ax+by+c)(a'x+b'y+c')=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>x</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>y</mi> <mo>+</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ax+by+c)(a'x+b'y+c')=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518dd61df2923c05e0c682612d902cc09eeae38b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.783ex; height:3.009ex;" alt="{\displaystyle (ax+by+c)(a'x+b'y+c')=k}"></span> avec <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=(au+bv+c)(a'u+b'v+c')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>u</mi> <mo>+</mo> <mi>b</mi> <mi>v</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mi>u</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mi>v</mi> <mo>+</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=(au+bv+c)(a'u+b'v+c')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480be7af4ff5d4863f5f78e2487a9585f5e85c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.727ex; height:3.009ex;" alt="{\displaystyle k=(au+bv+c)(a'u+b'v+c')}"></span></span> </p><p>Ce résultat provient du fait que les directions des asymptotes déterminent à une constante multiplicative près l'équation homogène<sup id="cite_ref-LFA_10-4" class="reference"><a href="#cite_note-LFA-10"><span class="cite_crochet">[</span>9<span class="cite_crochet">]</span></a></sup>, que leur point d'intersection détermine le centre de l'hyperbole et que la constante est fixée par le fait que la courbe passe par M. </p> <div class="mw-heading mw-heading3"><h3 id="Équation_matricielle"><span id=".C3.89quation_matricielle"></span>Équation matricielle</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=16" title="Modifier la section : Équation matricielle" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=16" title="Modifier le code source de la section : Équation matricielle"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'équation précédente peut s'écrire sous forme matricielle : </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>où </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> est la transposée de <b>x</b> ;</li> <li><b>A</b> est une matrice 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> est la transposée de <b>b</b> ;</li></ul> <p>avec toujours les mêmes contraintes. </p> <div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=17" title="Modifier la section : Propriétés" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=17" title="Modifier le code source de la section : Propriétés"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Intérieur_et_extérieur"><span id="Int.C3.A9rieur_et_ext.C3.A9rieur"></span>Intérieur et extérieur</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=18" title="Modifier la section : Intérieur et extérieur" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=18" title="Modifier le code source de la section : Intérieur et extérieur"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'hyperbole partage le plan en 3 zones ou composantes connexes. Comme elle est la section d'un cône et d'un plan, on appelle intérieur de l'hyperbole les portions de plan situées à l'intérieur du cône, ce sont les zones qui contiennent les foyers, et extérieur de l'hyperbole la dernière zone, celle qui contient le centre de l'hyperbole. </p> <div class="mw-heading mw-heading3"><h3 id="Sécantes_et_sommets"><span id="S.C3.A9cantes_et_sommets"></span>Sécantes et sommets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=19" title="Modifier la section : Sécantes et sommets" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=19" title="Modifier le code source de la section : Sécantes et sommets"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:HyperboleDiametre.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/260px-HyperboleDiametre.svg.png" decoding="async" width="260" height="144" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/390px-HyperboleDiametre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/HyperboleDiametre.svg/520px-HyperboleDiametre.svg.png 2x" data-file-width="311" data-file-height="172" /></a><figcaption>Diamètres conjugués dans une hyperbole.</figcaption></figure> <p>Soit M un point de l'hyperbole de sommets S et S' et de centre O. Si par un point N de l'hyperbole, on mène des parallèles à (SM) et (S'M), elles rencontrent l'hyperbole en deux points P et P' symétriques par rapport à O<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite_crochet">[</span>10<span class="cite_crochet">]</span></a></sup>. Par conséquent, dans un faisceau de droites parallèles (d<sub>i</sub>) rencontrant l'hyperbole, les milieux des cordes que déterminent ces droites sur l'hyperbole, sont alignés avec le centre O de l'hyperbole. De plus, si M est le point de l'hyperbole tel que (SM) soit parallèle à (d<sub>i</sub>) alors la droite des milieux est parallèle à (S'M)<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite_crochet">[</span>11<span class="cite_crochet">]</span></a></sup>. Une telle droite passant par le centre de l'hyperbole est appelée diamètre de l'hyperbole<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite_crochet">[</span>12<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Sécantes_et_asymptotes"><span id="S.C3.A9cantes_et_asymptotes"></span>Sécantes et asymptotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=20" title="Modifier la section : Sécantes et asymptotes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=20" title="Modifier le code source de la section : Sécantes et asymptotes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si une droite coupe l'hyperbole en M et M', alors elle coupe les asymptotes en P et P' et les segments [MM'] et [PP'] ont même milieu. </p><p>Soit (d) une droite non parallèle aux asymptotes. Si par un point M de l'hyperbole, on trace une parallèle à (d), elle rencontre les asymptotes en P et P' et le produit <span class="mwe-math-element"><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> est indépendant du point M<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite_crochet">[</span>13<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading3"><h3 id="Hyperbole_équilatère"><span id="Hyperbole_.C3.A9quilat.C3.A8re"></span>Hyperbole équilatère</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=21" title="Modifier la section : Hyperbole équilatère" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=21" title="Modifier le code source de la section : Hyperbole équilatère"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Hyperbeln-gs-3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Hyperbeln-gs-3.svg/220px-Hyperbeln-gs-3.svg.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Hyperbeln-gs-3.svg/330px-Hyperbeln-gs-3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Hyperbeln-gs-3.svg/440px-Hyperbeln-gs-3.svg.png 2x" data-file-width="261" data-file-height="262" /></a><figcaption>Trois hyperboles équilatères d'équation <span class="texhtml"><i>y</i> = <i>A</i>/<i>x</i></span>, où les axes de coordonnées sont les asymptotes (<i>A</i> = 1 en rouge; <i>A</i> = 4 en magenta ; <i>A</i> = 9 en bleu</figcaption></figure> <p>Une hyperbole est dite <b>équilatère</b> lorsque ses deux <a href="/wiki/Asymptote" title="Asymptote">asymptotes</a> sont perpendiculaires. </p> <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid #aaa; text-align: justify;"> <p><strong class="theoreme-nom">Théorème de Brianchon-Poncelet<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite_crochet">[</span>14<span class="cite_crochet">]</span></a></sup></strong><span class="theoreme-tiret"> — </span>Lorsqu'un <a href="/wiki/Triangle" title="Triangle">triangle</a> est inscrit dans une hyperbole équilatère, son <a href="/wiki/Orthocentre" class="mw-redirect" title="Orthocentre">orthocentre</a> est aussi sur l'hyperbole. </p> </div> <p>Ainsi, les <a href="/wiki/Coniques_circonscrites_et_inscrites_%C3%A0_un_triangle#Hyperboles_circonscrites_équilatères" title="Coniques circonscrites et inscrites à un triangle">hyperboles de Kiepert, de Feuerbach et de Jerabek d'un triangle</a> sont équilatères. </p><p>Ce deuxième résultat a été attribué à <a href="/wiki/Karl_Feuerbach" class="mw-redirect" title="Karl Feuerbach">Karl Feuerbach</a> par <a href="/wiki/Julian_Coolidge" title="Julian Coolidge">Julian Coolidge</a><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite_crochet">[</span>15<span class="cite_crochet">]</span></a></sup>, mais il n'apparait dans aucun ouvrage avant ceux de Brianchon et Poncelet. </p> <div class="theoreme" style="margin: 1em 2em; padding: 0.5em 1em 0.4em; border: 1px solid #aaa; text-align: justify;"> <p><strong class="theoreme-nom">Théorème conique de Feuerbach</strong><span class="theoreme-tiret"> — </span>Le lieu des centres des hyperboles équilatères circonscrites à un triangle est le <a href="/wiki/Cercle_d%27Euler" title="Cercle d'Euler">cercle d'Euler</a> du triangle. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Tangentes">Tangentes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=22" title="Modifier la section : Tangentes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=22" title="Modifier le code source de la section : Tangentes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si l'hyperbole a pour équation <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> la tangente au point M de coordonnées (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>) a pour équation<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite_crochet">[</span>16<span class="cite_crochet">]</span></a></sup> : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> </p><p>Comme une tangente est une sécante particulière, la propriété des sécantes et des sommets offre un moyen de tracer une tangente en un point M distinct des sommets : la droite passant par un des sommets S de l'hyperbole et parallèle à (OM) rencontre l'hyperbole en N, la tangente est alors parallèle à (S'N)<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite_crochet">[</span>17<span class="cite_crochet">]</span></a></sup>. </p><p>Si M est un point de l'hyperbole distinct des sommets, la tangente en M est également la bissectrice intérieure de l'angle FMF' où F et F' sont les foyers de l'hyperbole et le produit des distances des foyers à la tangente est toujours égal à b<sup>2</sup><sup id="cite_ref-Tauvel398_19-0" class="reference"><a href="#cite_note-Tauvel398-19"><span class="cite_crochet">[</span>18<span class="cite_crochet">]</span></a></sup>. Cette propriété a quelques applications pratiques. Elle fournit un moyen simple de construire la tangente en un point comme bissectrice intérieure de (F'MF). Dans un miroir de forme hyperbolique, les rayons issus d'un foyer sont réfléchis comme s'il provenaient de l'autre foyer. Bergery préconise donc, pour éviter les déperditions de chaleur, de construire le fond d'un cheminée selon un cylindre hyperbolique<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite_crochet">[</span>19<span class="cite_crochet">]</span></a></sup>. </p><p>La tangente en un point M coupe les asymptotes en deux points P et P' symétriques par rapport à M. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:HyperboleTangentes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/330px-HyperboleTangentes.svg.png" decoding="async" width="330" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/495px-HyperboleTangentes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/HyperboleTangentes.svg/660px-HyperboleTangentes.svg.png 2x" data-file-width="389" data-file-height="239" /></a><figcaption>Construction des deux tangentes à l'hyperbole issues de M à l'aide du cercle directeur.</figcaption></figure> <p>Par le point O ou un point M intérieur à l'hyperbole, il ne passe aucune tangente, par un point situé sur l'hyperbole ou sur une asymptote (distinct de O), il en passe une seule et par un point situé à l'extérieur de l'hyperbole, non situé sur les asymptotes, il passe toujours deux tangentes<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite_crochet">[</span>20<span class="cite_crochet">]</span></a></sup>. Pour construire les deux tangentes issues de M, il suffit de tracer le cercle de centre M passant par un foyer F et le cercle de centre F' et de rayon 2a. Les cercles se rencontrent en N et N', les médiatrices de [FN] et [FN'] sont les tangentes recherchées<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite_crochet">[</span>21<span class="cite_crochet">]</span></a></sup>. Pour trouver le point de contact, il suffit de construire une parallèle à la tangente coupant l'hyperbole en deux points, la droite passant par O et par le milieu de la corde ainsi construite rencontre la tangente en son point de contact. Ou bien, on prend le symétrique d'un des foyers par rapport à la tangente, la droite joignant ce symétrique à l'autre foyer rencontre la tangente à son point de contact<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite_crochet">[</span>22<span class="cite_crochet">]</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=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=23" title="Modifier la section : Cercles" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=23" title="Modifier le code source de la section : Cercles"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Cercle principal</b> : c'est le cercle de diamètre [SS']. Foyer et directrice sont en relation <a href="/wiki/P%C3%B4le_et_polaire" title="Pôle et polaire">pôle/polaire</a> par rapport au cercle principal. Le cercle principal, privé de ses intersections avec les asymptotes, est également la <a href="/wiki/Podaire" title="Podaire">podaire</a> de l'hyperbole par rapport à l'un des foyers, c'est-à-dire le lieu des projetés orthogonaux de ce foyer sur les tangentes<sup id="cite_ref-Tauvel398_19-1" class="reference"><a href="#cite_note-Tauvel398-19"><span class="cite_crochet">[</span>18<span class="cite_crochet">]</span></a></sup>, ce qui fait de l'hyperbole, l'antipodaire de son cercle principal par rapport à un de ses foyers. </p><p><b>Cercle directeur</b> : c'est un cercle passant par un foyer et de rayon égal à <span class="texhtml">2<i>a</i></span>. Selon la définition bifocale de l'hyperbole, l'hyperbole est le lieu des centres des cercles passant par F et tangents intérieurement ou extérieurement au cercle directeur de centre F'. L'ensemble des médiatrices des segments [FM], où M parcourt le cercle directeur de centre F' donne l'ensemble des tangentes à l'hyperbole, le cercle directeur est donc l'<a href="/wiki/Podaire" title="Podaire">orthotomique</a> de l'hyperbole par rapport à un foyer, c'est-à-dire l'ensemble des symétriques de F par rapport aux tangentes<sup id="cite_ref-Tauvel398_19-2" class="reference"><a href="#cite_note-Tauvel398-19"><span class="cite_crochet">[</span>18<span class="cite_crochet">]</span></a></sup>. </p><p><b> Cercle orthoptique</b> : si l’excentricité est strictement comprise entre 1 et <span class="racine">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>, il existe des points M par lesquelles passent deux tangentes orthogonales. L'ensemble de ces points M dessine un cercle de centre O et de rayon <span class="racine texhtml">√<span style="border-top:1px solid; padding:0 0.1em;"><i>a</i><sup>2</sup>-<i>b</i><sup>2</sup></span></span>, appelé <i>cercle orthoptique</i> ou <i>cercle de Monge</i> de l'hyperbole<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite_crochet">[</span>23<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite_crochet">[</span>24<span class="cite_crochet">]</span></a></sup>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier:HyperboleCentreCourbure.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/330px-HyperboleCentreCourbure.svg.png" decoding="async" width="330" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/495px-HyperboleCentreCourbure.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/HyperboleCentreCourbure.svg/660px-HyperboleCentreCourbure.svg.png 2x" data-file-width="707" data-file-height="425" /></a><figcaption>Construction du centre de courbure au point M, du cercle osculateur et <a href="/wiki/D%C3%A9velopp%C3%A9e" title="Développée">développée</a> de l'hyperbole.</figcaption></figure> <p><b> Cercles osculateurs</b> : en tout point M de l'hyperbole, il existe un cercle possédant un point de contact triple avec l'hyperbole. C'est le cercle osculateur à l'hyperbole au point M. Son centre, appelé centre de courbure, est situé sur la normale à la courbe (qui est aussi la bissectrice extérieure de l'angle FMF') a une distance de M égale au rayon de courbure. Si l'hyperbole a pour équation : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> le rayon du cercle osculateur au point d'abscisse <span class="texhtml"><i>x</i><sub>0</sub></span> a pour valeur<sup id="cite_ref-⁇?_26-0" class="reference"><a href="#cite_note-⁇?-26"><span class="cite_crochet">[</span>25<span class="cite_crochet">]</span></a></sup> : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> Il est possible de construire géométriquement le centre du cercle osculateur en M. Si (t) et (n) sont respectivement la tangente et la normale à l'hyperbole au point M, on trace le symétrique M' de M par rapport à l'axe principal et le symétrique (t') de (t) par rapport à (MM'), cette droite rencontre la droite (OM') en N. La perpendiculaire à (t') en N rencontre la normale (n) au centre de courbure<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite_crochet">[</span>26<span class="cite_crochet">]</span></a></sup>. </p><p>La <a href="/wiki/D%C3%A9velopp%C3%A9e" title="Développée">développée</a> de l'hyperbole, c'est-à-dire le lieu des centres de courbure est une <a href="/wiki/Courbe_de_Lam%C3%A9" title="Courbe de Lamé">courbe de Lamé</a> d'équation<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite_crochet">[</span>27<span class="cite_crochet">]</span></a></sup> : <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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)^{2/3}-\left({\frac {by}{c^{2}}}\right)^{2/3}=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>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</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>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {ax}{c^{2}}}\right)^{2/3}-\left({\frac {by}{c^{2}}}\right)^{2/3}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91a0d45f3b8209dfd29439ca0018b4631410934f" 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)^{2/3}-\left({\frac {by}{c^{2}}}\right)^{2/3}=1}"></span></span> </p> <div class="mw-heading mw-heading3"><h3 id="Longueur_et_aire">Longueur et aire</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=24" title="Modifier la section : Longueur et aire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=24" title="Modifier le code source de la section : Longueur et aire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si M(<i>t</i><sub>0</sub>) est un point de l'hyperbole d'équation paramétrée, <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> où <i>t</i><sub>0</sub> est compris entre 0 et π/2, la longueur de l'arc SM est <span style="display: block; margin-left:1.6em;"><span class="mwe-math-element"><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></span> dont l'intégration nécessite l'utilisation des <a href="/wiki/Int%C3%A9grale_elliptique" title="Intégrale elliptique">intégrales elliptiques</a><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite_crochet">[</span>28<span class="cite_crochet">]</span></a></sup>. </p><p>L'aire d'une portion de plan délimitée par un arc d'hyperbole est à l'origine de la création de la fonction logarithme et des fonctions hyperboliques. L'aire de la surface délimitée par l'hyperbole d'équation <i>yx</i> = 1, l'axe des abscisses et les droites d'équation <i>x</i> = <i>u</i> et <i>x</i> = <i>v</i> est égale à |ln(<i>v</i>/<i>u</i>)|. </p> <div class="mw-heading mw-heading2"><h2 id="Histoire">Histoire</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=25" title="Modifier la section : Histoire" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=25" title="Modifier le code source de la section : Histoire"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Conique#Histoire" title="Conique">Conique#Histoire</a>.</div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fichier: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>Égalité d'aire, dans une hyperbole, entre le carré mené sur l'ordonnée et le rectangle bleu mené sur l'abscisse. Ce rectangle est <b>plus grand</b> que le rectangle de hauteur SP (paramètre de l'hyperbole) d'où le nom de <i>hyperbole</i> (ajustement par excès) donné à la courbe par <a href="/wiki/Apollonios_de_Perga" title="Apollonios de Perga">Apollonius</a>.</figcaption></figure> <p>L'hyperbole est étudiée, dans le cadre des coniques, durant la période grecque. Ainsi <a href="/wiki/M%C3%A9nechme" title="Ménechme">Ménechme</a> résout un problème de double proportionnelle en utilisant une hyperbole d'équation XY= constante<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite_crochet">[</span>29<span class="cite_crochet">]</span></a></sup>. Mais pour Menechme et ses successeurs, Euclide et Aristée, cette courbe ne possède qu'une composante. Ils l'appellent «conique obtusangle» car ils la définissent comme l'intersection d'un cône obtusangle (cône engendré par la rotation d'un triangle ABC, rectangle en B, autour de AB et tel que l'angle de sommet A soit supérieur à 45 °) avec un plan perpendiculaire à sa génératrice<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite_crochet">[</span>30<span class="cite_crochet">]</span></a></sup>. <a href="/wiki/Apollonios_de_Perga" title="Apollonios de Perga">Apollonius de Perge</a> semble être le premier à envisager les deux composantes de l'hyperbole<sup id="cite_ref-apollonius_32-0" class="reference"><a href="#cite_note-apollonius-32"><span class="cite_crochet">[</span>31<span class="cite_crochet">]</span></a></sup>. C'est également lui qui lui donne le nom d'«hyperbole» (ajustement par excès) ayant remarqué que l'aire du carré dessiné sur l'ordonnée est supérieure à celle du rectangle dessiné sur l'abscisse et dont la hauteur serait fixe<sup id="cite_ref-apollonius_32-1" class="reference"><a href="#cite_note-apollonius-32"><span class="cite_crochet">[</span>31<span class="cite_crochet">]</span></a></sup>. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fichier:Hyperbolic_functions-2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/220px-Hyperbolic_functions-2.svg.png" decoding="async" width="220" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/330px-Hyperbolic_functions-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Hyperbolic_functions-2.svg/440px-Hyperbolic_functions-2.svg.png 2x" data-file-width="500" data-file-height="437" /></a><figcaption>Une demi-droite passant par l'origine intersecte l'hyperbole d'équation <span class="mwe-math-element"><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}\ =\ 1}"> <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> <mtext> </mtext> <mo>−</mo> <mtext> </mtext> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}\ -\ y^{2}\ =\ 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e685a58d3346d2b9550bdef7a3c46d6dd6c987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.023ex; height:3.009ex;" alt="{\displaystyle x^{2}\ -\ y^{2}\ =\ 1}"></span> au point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cosh \,a,\,\sinh \,a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cosh</mi> <mspace width="thinmathspace" /> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>sinh</mi> <mspace width="thinmathspace" /> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cosh \,a,\,\sinh \,a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce456e0dfe2fc4c6140f27ba9e76c94535117b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.79ex; height:2.843ex;" alt="{\displaystyle (\cosh \,a,\,\sinh \,a)}"></span>, où <span class="mwe-math-element"><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> est le double de l'aire algébrique de la surface délimitée par la demi-droite, l'hyperbole et l'axe des <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>.</figcaption></figure> <p>La recherche de l'aire sous l'hyperbole est entreprise, en 1647, par <a href="/wiki/Gr%C3%A9goire_de_Saint-Vincent" title="Grégoire de Saint-Vincent">Grégoire de Saint-Vincent</a>, qui met en évidence sa propriété logarithmique<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite_crochet">[</span>32<span class="cite_crochet">]</span></a></sup>. </p><p>En 1757-1762, <a href="/wiki/Vincenzo_Riccati" title="Vincenzo Riccati">Vincenzo Riccati</a> établit une relation entre l'aire d'un secteur angulaire dans une hyperbole et les coordonnées d'un point et définit les fonctions cosinus hyperbolique et sinus hyperbolique par analogie à la relation existant dans le cercle<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite_crochet">[</span>33<span class="cite_crochet">]</span></a></sup>. </p> <div class="clear" style="clear:both;"></div> <hr /> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=26" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=26" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Notes">Notes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=27" title="Modifier la section : Notes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=27" title="Modifier le code source de la section : Notes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Du moins cela est vrai pour tous les <a href="/wiki/Cadran_horizontal" title="Cadran horizontal">cadrans horizontaux</a> situés entre les <a href="/wiki/Cercle_polaire" title="Cercle polaire">cercles polaires</a> et tous les <a href="/wiki/Cadran_vertical" title="Cadran vertical">cadrans verticaux</a> situés au-delà des <a href="/wiki/Tropique" title="Tropique">tropiques</a>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading3"><h3 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=28" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=28" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 392.</span> </li> <li id="cite_note-mathcurve-3"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-mathcurve_3-0">a</a> et <a href="#cite_ref-mathcurve_3-1">b</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 noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text">Conséquence de <a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 392, prop. 24.1.16.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink noprint"><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 class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Histoire+des+sciences+arabes&rft.atitle=G%C3%A9om%C3%A9trie&rft.pub=Seuil&rft.aulast=Rosenfeld&rft.aufirst=Boris+A.&rft.au=Youshkevitch%2C+Adolf+P.&rft.date=1997&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span>, p. 97.</span> </li> <li id="cite_note-Guillerault-6"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Guillerault_6-0">a</a> et <a href="#cite_ref-Guillerault_6-1">b</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 noprint"><a href="#cite_ref-7">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 392-393.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink noprint"><a href="#cite_ref-8">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 414.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink noprint"><a href="#cite_ref-9">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 412.</span> </li> <li id="cite_note-LFA-10"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-LFA_10-0">a</a> <a href="#cite_ref-LFA_10-1">b</a> <a href="#cite_ref-LFA_10-2">c</a> <a href="#cite_ref-LFA_10-3">d</a> et <a href="#cite_ref-LFA_10-4">e</a></sup> </span><span class="reference-text"> <span class="ouvrage" id="Lelong-FerrandArnaudiès1977"><span class="ouvrage" id="Jacqueline_Lelong-FerrandJean-Marie_Arnaudiès1977"><a href="/wiki/Jacqueline_Lelong-Ferrand" title="Jacqueline Lelong-Ferrand">Jacqueline Lelong-Ferrand</a> et <a href="/wiki/Jean-Marie_Arnaudi%C3%A8s" title="Jean-Marie Arnaudiès">Jean-Marie Arnaudiès</a>, <cite class="italique">Cours de mathématiques : géométrie et Cinématique</cite>, <abbr class="abbr" title="tome">t.</abbr> 3, <a href="/wiki/%C3%89ditions_Bordas" title="Éditions Bordas">Bordas</a>, <time>1977</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cours+de+math%C3%A9matiques&rft.pub=Bordas&rft.stitle=g%C3%A9om%C3%A9trie+et+Cin%C3%A9matique&rft.aulast=Lelong-Ferrand&rft.aufirst=Jacqueline&rft.au=Jean-Marie+Arnaudi%C3%A8s&rft.date=1977&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span>, pp 97 ; 98 ; 154</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink noprint"><a href="#cite_ref-11">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 145 prop. 218.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink noprint"><a href="#cite_ref-12">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 148 prop. 223).</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink noprint"><a href="#cite_ref-13">↑</a> </span><span class="reference-text"><i>Encyclopædia Universalis</i>, 1990, vol. 6, p. 386(b).</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink noprint"><a href="#cite_ref-14">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 401.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink noprint"><a href="#cite_ref-15">↑</a> </span><span class="reference-text"><span class="ouvrage" id="OdaniTakase1999"><span class="ouvrage" id="Kenzi_OdaniShihomi_Takase1999"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Kenzi Odani et Shihomi Takase, « <cite style="font-style:normal" lang="en">On a theorem of Brianchon and Poncelet</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a></span></i>,‎ <time>1999</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.4439&rep=rep1&type=pdf">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=On+a+theorem+of+Brianchon+and+Poncelet&rft.jtitle=The+Mathematical+Gazette&rft.aulast=Odani&rft.aufirst=Kenzi&rft.au=Shihomi+Takase&rft.date=1999&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span>.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink noprint"><a href="#cite_ref-16">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Coolidge1959"><span class="ouvrage" id="J._L._Coolidge1959"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> J. L. Coolidge, <cite class="italique" lang="en">A Treatise on Algebraic Plane Curves</cite>, New York: Dover, <time>1959</time>, 198 <abbr class="abbr" title="pages">p.</abbr><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Algebraic+Plane+Curves&rft.place=New+York%3A+Dover&rft.aulast=Coolidge&rft.aufirst=J.+L.&rft.date=1959&rft.tpages=198&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span>.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink noprint"><a href="#cite_ref-17">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 396.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink noprint"><a href="#cite_ref-18">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 163 prop. 249.</span> </li> <li id="cite_note-Tauvel398-19"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Tauvel398_19-0">a</a> <a href="#cite_ref-Tauvel398_19-1">b</a> et <a href="#cite_ref-Tauvel398_19-2">c</a></sup> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 398.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink noprint"><a href="#cite_ref-20">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 177.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink noprint"><a href="#cite_ref-21">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 397.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink noprint"><a href="#cite_ref-22">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 171 prob. b.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink noprint"><a href="#cite_ref-23">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 172 prob. e.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink noprint"><a href="#cite_ref-24">↑</a> </span><span class="reference-text"><a href="#Tauvel2005">Tauvel 2005</a>, <abbr class="abbr" title="page(s)">p.</abbr> 402.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink noprint"><a href="#cite_ref-25">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Bondil_et_Charles_Boubel2009"><span class="ouvrage" id="Romain_Bondil_et_Charles_Boubel2009">Romain Bondil et Charles Boubel, « <a rel="nofollow" class="external text" href="http://irma.math.unistra.fr/~boubel/orthoptique_bondil_boubel1.pdf"><cite style="font-style:normal;">Courbe orthoptique d’une conique</cite></a> » <abbr class="abbr indicateur-format format-pdf" title="Document au format Portable Document Format (PDF) d'Adobe">[PDF]</abbr>, <time class="nowrap" datetime="2009-04-16" data-sort-value="2009-04-16">16 avril 2009</time></span></span></span> </li> <li id="cite_note-⁇?-26"><span class="mw-cite-backlink noprint"><a href="#cite_ref-⁇?_26-0">↑</a> </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-27"><span class="mw-cite-backlink noprint"><a href="#cite_ref-27">↑</a> </span><span class="reference-text"><a href="#Bergery1843">Bergery 1843</a>, <abbr class="abbr" title="page(s)">p.</abbr> 180.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink noprint"><a href="#cite_ref-28">↑</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-29"><span class="mw-cite-backlink noprint"><a href="#cite_ref-29">↑</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-30"><span class="mw-cite-backlink noprint"><a href="#cite_ref-30">↑</a> </span><span class="reference-text"><a href="#Vitrac">Vitrac</a>, <a rel="nofollow" class="external text" href="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-31"><span class="mw-cite-backlink noprint"><a href="#cite_ref-31">↑</a> </span><span class="reference-text"><a href="#Vitrac">Vitrac</a>, <a rel="nofollow" class="external text" href="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-32"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-apollonius_32-0">a</a> et <a href="#cite_ref-apollonius_32-1">b</a></sup> </span><span class="reference-text"><a href="#Vitrac">Vitrac</a>, <a rel="nofollow" class="external text" href="http://culturemath.ens.fr/histoire%20des%20maths/htm/Vitrac/grec-8.html#apollonius">L'approche d'Apollonius</a>.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink noprint"><a href="#cite_ref-33">↑</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>, <a href="/wiki/Institut_de_recherche_sur_l%27enseignement_des_math%C3%A9matiques" title="Institut de recherche sur l'enseignement des mathématiques">IREM</a> de Lyon, <time>1989</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=La+D%C3%A9monstration+math%C3%A9matique+dans+l%27histoire&rft.atitle=De+la+m%C3%A9thode+dite+d%27exhaustion+%3A+Gr%C3%A9goire+de+Saint-Vincent+%281584+-+1667%29&rft.pub=IREM&rft.aulast=Legoff&rft.aufirst=Jean-Pierre&rft.date=1989&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span>, p. 215.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink noprint"><a href="#cite_ref-34">↑</a> </span><span class="reference-text">Robert E. Bradley, Lawrence A. D'Antonio, C. Edward Sandifer, <i><span class="lang-en" lang="en">Euler at 300: An Appreciation</span></i>, MAA, 2007, <a rel="nofollow" class="external text" href="https://books.google.fr/books?id=tK_KRmTf9nUC&pg=PA99#v=onepage&q&f=false">p. 99</a>.</span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Annexes">Annexes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=29" title="Modifier la section : Annexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=29" title="Modifier le code source de la section : Annexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="commons"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Hyperbolas?uselang=fr">Hyperbole</a>, sur <span class="project">Wikimedia Commons</span></li> </ul> </div> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=30" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=30" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Tauvel2005"><span class="ouvrage" id="Patrice_Tauvel2005">Patrice <span class="nom_auteur">Tauvel</span>, <cite class="italique">Géométrie : agrégation, licence 3e année, master</cite>, Paris, <a href="/wiki/%C3%89ditions_Dunod" title="Éditions Dunod">Dunod</a>, <time>2005</time>, 532 <abbr class="abbr" title="pages">p.</abbr> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/2-10-049413-9" title="Spécial:Ouvrages de référence/2-10-049413-9"><span class="nowrap">2-10-049413-9</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%A9om%C3%A9trie&rft.place=Paris&rft.pub=Dunod&rft.stitle=agr%C3%A9gation%2C+licence+3e+ann%C3%A9e%2C+master&rft.aulast=Tauvel&rft.aufirst=Patrice&rft.date=2005&rft.tpages=532&rft.isbn=2-10-049413-9&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span></li> <li><span class="ouvrage" id="Bergery1843"><span class="ouvrage" id="Claude_Lucien_Bergery1843">Claude Lucien <span class="nom_auteur">Bergery</span>, <cite class="italique">Géométrie des courbes appliquée aux arts</cite>, Thiel, <time>1843</time> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://books.google.fr/books?id=aO42AAAAMAAJ&pg=PR3">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%A9om%C3%A9trie+des+courbes+appliqu%C3%A9e+aux+arts&rft.pub=Thiel&rft.aulast=Bergery&rft.aufirst=Claude+Lucien&rft.date=1843&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AHyperbole+%28math%C3%A9matiques%29"></span></span></span></li> <li>Jean-Denis Eiden, Géométrie analytique classique, Calvage & Mounet, 2009. <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-916352-08-4" title="Spécial:Ouvrages de référence/978-2-916352-08-4"><span class="nowrap">978-2-916352-08-4</span></a>)</small></li> <li>Jean Fresnel, <i>Méthodes modernes en géométrie</i></li> <li>Bruno Ingrao, <i>Coniques affines, euclidiennes et projectives</i>, C&M. <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-2-916352-12-1" title="Spécial:Ouvrages de référence/978-2-916352-12-1"><span class="nowrap">978-2-916352-12-1</span></a>)</small></li> <li><span class="ouvrage" id="Vitrac">Bernard Vitrac, « <a rel="nofollow" class="external text" href="http://culturemath.ens.fr/histoire%20des%20maths/htm/Vitrac/grec-8.html"><cite style="font-style:normal;">Les géomètres de l'antiquité 8- Apollonius de Perge et la tradition des coniques</cite></a> », sur <span class="italique">cultureMATH</span></span> (ressources pour les enseignants de mathématiques, site expert des Écoles normales supérieures françaises et du ministère de l'Éducation nationale française)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=31" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=31" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fonction_hyperbolique" title="Fonction hyperbolique">Fonction hyperbolique</a></li> <li><a href="/wiki/Conique" title="Conique">Coniques</a> <ul><li><a href="/wiki/Ellipse_(math%C3%A9matiques)" title="Ellipse (mathématiques)">Ellipse</a></li> <li><a href="/wiki/Parabole" title="Parabole">Parabole</a></li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&veaction=edit&section=32" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Hyperbole_(math%C3%A9matiques)&action=edit&section=32" title="Modifier le code source de la section : Liens externes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.mathcurve.com/courbes2d/hyperbole/hyperbole.shtml">Hyperbole</a></li> <li><a rel="nofollow" class="external text" href="http://xavier.hubaut.info/coursmath/2de/belges.htm">Les théorèmes belges</a> (coniques et théorème de Dandelin)</li></ul> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Courbes" title="Modèle:Palette Courbes"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Courbes&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%">Exemples de <a href="/wiki/Courbe" title="Courbe">courbes</a></div></th> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Conique" title="Conique">Coniques</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Cercle" title="Cercle">Cercle</a></li> <li><a href="/wiki/Ellipse_(math%C3%A9matiques)" title="Ellipse (mathématiques)">Ellipse</a></li> <li><a class="mw-selflink selflink">Hyperbole</a></li> <li><a href="/wiki/Parabole" title="Parabole">Parabole</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Cisso%C3%AFde" title="Cissoïde">Cissoïdes</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Cisso%C3%AFde_de_Diocl%C3%A8s" title="Cissoïde de Dioclès">Cissoïde de Dioclès</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Courbe_cyclo%C3%AFdale" title="Courbe cycloïdale">Courbes cycloïdales</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Cyclo%C3%AFde" title="Cycloïde">Cycloïde</a></li> <li><a href="/wiki/%C3%89picyclo%C3%AFde" title="Épicycloïde">Épicycloïde</a> <ul><li><a href="/wiki/Cardio%C3%AFde" title="Cardioïde">Cardioïde</a></li> <li><a href="/wiki/N%C3%A9phro%C3%AFde" title="Néphroïde">Néphroïde</a></li></ul></li> <li><a href="/wiki/Hypocyclo%C3%AFde" title="Hypocycloïde">Hypocycloïde</a> <ul><li><a href="/wiki/Astro%C3%AFde" title="Astroïde">Astroïde</a></li> <li><a href="/wiki/Delto%C3%AFde_(courbe)" title="Deltoïde (courbe)">Deltoïde</a></li></ul></li> <li><a href="/wiki/%C3%89pitrocho%C3%AFde" title="Épitrochoïde">Épitrochoïde</a> <ul><li><a href="/wiki/Lima%C3%A7on_de_Pascal" title="Limaçon de Pascal">Limaçon de Pascal</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Spirale" title="Spirale">Spirales</a> (<a href="/wiki/Liste_de_spirales_math%C3%A9matiques" title="Liste de spirales mathématiques">Liste</a>)</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Spirale_d%27Archim%C3%A8de" title="Spirale d'Archimède">Archimède</a></li> <li><a href="/wiki/Spirale_%C3%A0_centres_multiples" title="Spirale à centres multiples">À centre multiple</a></li> <li><a href="/wiki/Spirale_de_Cotes" title="Spirale de Cotes">Cotes</a></li> <li><a href="/wiki/%C3%89pi_(courbe)" title="Épi (courbe)">Épi</a></li> <li><a href="/wiki/Clotho%C3%AFde" title="Clothoïde">Euler</a></li> <li><a href="/wiki/Spirale_de_Fermat" title="Spirale de Fermat">Fermat</a></li> <li><a href="/wiki/Spirale_hyperbolique" title="Spirale hyperbolique">Hyperbolique</a></li> <li><a href="/wiki/Lituus_(courbe)" title="Lituus (courbe)">Lituus</a></li> <li><a href="/wiki/Spirale_logarithmique" title="Spirale logarithmique">Logarithmique</a> <ul><li><a href="/wiki/Spirale_d%27or" title="Spirale d'or">D'or</a></li></ul></li> <li><a href="/wiki/Spirale_de_Poinsot" title="Spirale de Poinsot">Poinsot</a></li> <li><a href="/wiki/Spirale_sinuso%C3%AFdale" title="Spirale sinusoïdale">Sinusoïdale</a></li> <li><a href="/wiki/Spirale_d%27Ulam" title="Spirale d'Ulam">Ulam</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Lemniscate" title="Lemniscate">Lemniscates</a></th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Lemniscate_de_Bernoulli" title="Lemniscate de Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Lemniscate_de_Booth" title="Lemniscate de Booth">Booth</a></li> <li><a href="/wiki/Courbe_du_diable" title="Courbe du diable">Courbe du diable</a></li> <li><a href="/wiki/Lemniscate_de_Gerono" title="Lemniscate de Gerono">Gerono</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style=""><a href="/wiki/Isochrone" title="Isochrone">Isochrones</a></th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Isochrone_de_Leibniz" title="Isochrone de Leibniz">Isochrone de Leibniz</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Autres</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Courbe_brachistochrone" title="Courbe brachistochrone">Brachistochrone</a></li> <li><a href="/wiki/Cha%C3%AEnette" title="Chaînette">Chaînette</a></li> <li><a href="/wiki/Concho%C3%AFde" title="Conchoïde">Conchoïde</a></li> <li><a href="/wiki/Folium_de_Descartes" title="Folium de Descartes">Folium de Descartes</a></li> <li><a href="/wiki/H%C3%A9lice_(g%C3%A9om%C3%A9trie)" title="Hélice (géométrie)">Hélice</a></li> <li><a href="/wiki/Hypotrocho%C3%AFde" title="Hypotrochoïde">Hypotrochoïde</a></li> <li><a href="/wiki/Ovale_de_Cassini" title="Ovale de Cassini">Ovale de Cassini</a></li> <li><a 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Hyperbole (mathématiques) — Wikipédia