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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Zaloguj się</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Strony dla anonimowych edytorów <a href="/wiki/Pomoc:Pierwsze_kroki" aria-label="Dowiedz się więcej na temat edytowania"><span>dowiedz się więcej</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Specjalna:M%C3%B3j_wk%C5%82ad" title="Lista edycji wykonanych z tego adresu IP [y]" accesskey="y"><span>Edycje</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Specjalna:Moja_dyskusja" title="Dyskusja użytkownika dla tego adresu IP [n]" accesskey="n"><span>Dyskusja</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Witryna"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Spis treści" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Spis treści</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ukryj</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Początek</div> </a> </li> <li id="toc-Sfera_w_euklidesowej_przestrzeni_trójwymiarowej" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sfera_w_euklidesowej_przestrzeni_trójwymiarowej"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Sfera w euklidesowej przestrzeni trójwymiarowej</span> </div> </a> <ul id="toc-Sfera_w_euklidesowej_przestrzeni_trójwymiarowej-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Związane_pojęcia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Związane_pojęcia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Związane pojęcia</span> </div> </a> <ul id="toc-Związane_pojęcia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sfera_w_n-wymiarowej_przestrzeni_euklidesowej" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sfera_w_n-wymiarowej_przestrzeni_euklidesowej"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Sfera w n-wymiarowej przestrzeni euklidesowej</span> </div> </a> <ul id="toc-Sfera_w_n-wymiarowej_przestrzeni_euklidesowej-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uogólnienia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uogólnienia"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Uogólnienia</span> </div> </a> <ul id="toc-Uogólnienia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linki_zewnętrzne" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linki_zewnętrzne"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Linki zewnętrzne</span> </div> </a> <ul id="toc-Linki_zewnętrzne-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Spis treści" 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="Przełącz stan spisu treści" > <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">Przełącz stan spisu treści</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">Sfera</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="Przejdź do artykułu w innym języku. Treść dostępna w 106 językach" > <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-106" 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">106 języków</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/Sfeer" title="Sfeer – afrikaans" lang="af" hreflang="af" data-title="Sfeer" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%89%E1%88%8D" title="ሉል – amharski" lang="am" hreflang="am" data-title="ሉል" data-language-autonym="አማርኛ" data-language-local-name="amharski" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D8%B1%D8%A9" title="كرة – arabski" lang="ar" hreflang="ar" data-title="كرة" data-language-autonym="العربية" data-language-local-name="arabski" 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/Esfera" title="Esfera – asturyjski" lang="ast" hreflang="ast" data-title="Esfera" data-language-autonym="Asturianu" data-language-local-name="asturyjski" 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/Sfera" title="Sfera – azerbejdżański" lang="az" hreflang="az" data-title="Sfera" data-language-autonym="Azərbaycanca" data-language-local-name="azerbejdżański" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%DA%A9%D9%88%D8%B1%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="کوره (هندسه) – South Azerbaijani" lang="azb" hreflang="azb" data-title="کوره (هندسه)" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8B%E0%A6%B2%E0%A6%95" title="গোলক – bengalski" lang="bn" hreflang="bn" data-title="গোলক" data-language-autonym="বাংলা" data-language-local-name="bengalski" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ki%C3%BB-b%C4%ABn" title="Kiû-bīn – minnański" lang="nan" hreflang="nan" data-title="Kiû-bīn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="minnański" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – baszkirski" lang="ba" hreflang="ba" data-title="Сфера" data-language-autonym="Башҡортса" data-language-local-name="baszkirski" 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%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – białoruski" lang="be" hreflang="be" data-title="Сфера" data-language-autonym="Беларуская" data-language-local-name="białoruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%BE" title="गोला – Bhojpuri" lang="bh" hreflang="bh" data-title="गोला" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – bułgarski" lang="bg" hreflang="bg" data-title="Сфера" data-language-autonym="Български" data-language-local-name="bułgarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sfera" title="Sfera – bośniacki" lang="bs" hreflang="bs" data-title="Sfera" data-language-autonym="Bosanski" data-language-local-name="bośniacki" 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/Esfera" title="Esfera – kataloński" lang="ca" hreflang="ca" data-title="Esfera" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – czuwaski" lang="cv" hreflang="cv" data-title="Сфера" data-language-autonym="Чӑвашла" data-language-local-name="czuwaski" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sf%C3%A9ra_(matematika)" title="Sféra (matematika) – czeski" lang="cs" hreflang="cs" data-title="Sféra (matematika)" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Mburungwa" title="Mburungwa – shona" lang="sn" hreflang="sn" data-title="Mburungwa" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Sff%C3%AAr" title="Sffêr – walijski" lang="cy" hreflang="cy" data-title="Sffêr" data-language-autonym="Cymraeg" data-language-local-name="walijski" 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/Kugle" title="Kugle – duński" lang="da" hreflang="da" data-title="Kugle" data-language-autonym="Dansk" data-language-local-name="duński" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%83%D9%88%D8%B1%D8%A9" title="كورة – marokański arabski" lang="ary" hreflang="ary" data-title="كورة" data-language-autonym="الدارجة" data-language-local-name="marokański arabski" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kugel" title="Kugel – niemiecki" lang="de" hreflang="de" data-title="Kugel" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Sf%C3%A4%C3%A4r" title="Sfäär – estoński" lang="et" hreflang="et" data-title="Sfäär" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%86%CE%B1%CE%AF%CF%81%CE%B1" title="Σφαίρα – grecki" lang="el" hreflang="el" data-title="Σφαίρα" data-language-autonym="Ελληνικά" data-language-local-name="grecki" 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/Sphere" title="Sphere – angielski" lang="en" hreflang="en" data-title="Sphere" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Esfera" title="Esfera – hiszpański" lang="es" hreflang="es" data-title="Esfera" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Sfero" title="Sfero – esperanto" lang="eo" hreflang="eo" data-title="Sfero" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Esfera" title="Esfera – baskijski" lang="eu" hreflang="eu" data-title="Esfera" data-language-autonym="Euskara" data-language-local-name="baskijski" 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/%DA%A9%D8%B1%D9%87_(%D9%87%D9%86%D8%AF%D8%B3%D9%87)" title="کره (هندسه) – perski" lang="fa" hreflang="fa" data-title="کره (هندسه)" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sph%C3%A8re" title="Sphère – francuski" lang="fr" hreflang="fr" data-title="Sphère" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sf%C3%A9ar" title="Sféar – irlandzki" lang="ga" hreflang="ga" data-title="Sféar" data-language-autonym="Gaeilge" data-language-local-name="irlandzki" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cruinne" title="Cruinne – szkocki gaelicki" lang="gd" hreflang="gd" data-title="Cruinne" data-language-autonym="Gàidhlig" data-language-local-name="szkocki gaelicki" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Esfera" title="Esfera – galicyjski" lang="gl" hreflang="gl" data-title="Esfera" data-language-autonym="Galego" data-language-local-name="galicyjski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – gan" lang="gan" hreflang="gan" data-title="球面" data-language-autonym="贛語" data-language-local-name="gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%97%E0%AB%8B%E0%AA%B3%E0%AB%8B" title="ગોળો – gudżarati" lang="gu" hreflang="gu" data-title="ગોળો" data-language-autonym="ગુજરાતી" data-language-local-name="gudżarati" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%AC_(%EA%B8%B0%ED%95%98%ED%95%99)" title="구 (기하학) – koreański" lang="ko" hreflang="ko" data-title="구 (기하학)" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A5%8B%E0%A4%B2%E0%A4%BE" title="गोला – hindi" lang="hi" hreflang="hi" data-title="गोला" 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/Sfera" title="Sfera – chorwacki" lang="hr" hreflang="hr" data-title="Sfera" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Sfero" title="Sfero – ido" lang="io" hreflang="io" data-title="Sfero" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bola_(geometri)" title="Bola (geometri) – indonezyjski" lang="id" hreflang="id" data-title="Bola (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezyjski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Sphera" title="Sphera – interlingua" lang="ia" hreflang="ia" data-title="Sphera" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/K%C3%BAla" title="Kúla – islandzki" lang="is" hreflang="is" data-title="Kúla" data-language-autonym="Íslenska" data-language-local-name="islandzki" 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/Sfera" title="Sfera – włoski" lang="it" hreflang="it" data-title="Sfera" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%A4%D7%99%D7%A8%D7%94_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="ספירה (גאומטריה) – hebrajski" lang="he" hreflang="he" data-title="ספירה (גאומטריה)" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8B%E0%B2%B3" title="ಗೋಳ – kannada" lang="kn" hreflang="kn" data-title="ಗೋಳ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%A4%E1%83%94%E1%83%A0%E1%83%9D" title="სფერო – gruziński" lang="ka" hreflang="ka" data-title="სფერო" data-language-autonym="ქართული" data-language-local-name="gruziński" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – kazachski" lang="kk" hreflang="kk" data-title="Сфера" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Tufe" title="Tufe – suahili" lang="sw" hreflang="sw" data-title="Tufe" data-language-autonym="Kiswahili" data-language-local-name="suahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Esf%C3%A8" title="Esfè – kreolski haitański" lang="ht" hreflang="ht" data-title="Esfè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="kreolski haitański" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Sf%C3%A8r" title="Sfèr – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Sfèr" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Sphaera" title="Sphaera – łaciński" lang="la" hreflang="la" data-title="Sphaera" data-language-autonym="Latina" data-language-local-name="łaciński" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sf%C4%93ra" title="Sfēra – łotewski" lang="lv" hreflang="lv" data-title="Sfēra" data-language-autonym="Latviešu" data-language-local-name="łotewski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sfera" title="Sfera – litewski" lang="lt" hreflang="lt" data-title="Sfera" data-language-autonym="Lietuvių" data-language-local-name="litewski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6mb" title="Gömb – węgierski" lang="hu" hreflang="hu" data-title="Gömb" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – macedoński" lang="mk" hreflang="mk" data-title="Сфера" data-language-autonym="Македонски" data-language-local-name="macedoński" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Bola_(je%C3%B4metria)" title="Bola (jeômetria) – malgaski" lang="mg" hreflang="mg" data-title="Bola (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgaski" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8B%E0%B4%B3%E0%B4%82" title="ഗോളം – malajalam" lang="ml" hreflang="ml" data-title="ഗോളം" data-language-autonym="മലയാളം" data-language-local-name="malajalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sfera" title="Sfera – malajski" lang="ms" hreflang="ms" data-title="Sfera" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mdf mw-list-item"><a href="https://mdf.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0%D1%81%D1%8C" title="Сферась – moksza" lang="mdf" hreflang="mdf" data-title="Сферась" data-language-autonym="Мокшень" data-language-local-name="moksza" class="interlanguage-link-target"><span>Мокшень</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D3%A9%D0%BC%D0%B1%D3%A9%D0%BB%D3%A9%D0%B3" title="Бөмбөлөг – mongolski" lang="mn" hreflang="mn" data-title="Бөмбөлөг" data-language-autonym="Монгол" data-language-local-name="mongolski" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%85%E1%80%80%E1%80%BA%E1%80%9C%E1%80%AF%E1%80%B6%E1%80%B8" title="စက်လုံး – birmański" lang="my" hreflang="my" data-title="စက်လုံး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmański" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Vuravura_(Geometry)" title="Vuravura (Geometry) – fidżijski" lang="fj" hreflang="fj" data-title="Vuravura (Geometry)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fidżijski" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sfeer_(wiskunde)" title="Sfeer (wiskunde) – niderlandzki" lang="nl" hreflang="nl" data-title="Sfeer (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – japoński" lang="ja" hreflang="ja" data-title="球面" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kuugel" title="Kuugel – północnofryzyjski" lang="frr" hreflang="frr" data-title="Kuugel" data-language-autonym="Nordfriisk" data-language-local-name="północnofryzyjski" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kule" title="Kule – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Kule" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kule" title="Kule – norweski (nynorsk)" lang="nn" hreflang="nn" data-title="Kule" data-language-autonym="Norsk nynorsk" data-language-local-name="norweski (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Esf%C3%A8ra" title="Esfèra – oksytański" lang="oc" hreflang="oc" data-title="Esfèra" data-language-autonym="Occitan" data-language-local-name="oksytański" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Duqunqula" title="Duqunqula – oromo" lang="om" hreflang="om" data-title="Duqunqula" data-language-autonym="Oromoo" data-language-local-name="oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sfera" title="Sfera – uzbecki" lang="uz" hreflang="uz" data-title="Sfera" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbecki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%8B%E0%A8%B2%E0%A8%BC%E0%A8%BE" title="ਗੋਲ਼ਾ – pendżabski" lang="pa" hreflang="pa" data-title="ਗੋਲ਼ਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="pendżabski" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Sfier" title="Sfier – jamajski" lang="jam" hreflang="jam" data-title="Sfier" data-language-autonym="Patois" data-language-local-name="jamajski" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sfera" title="Sfera – piemoncki" lang="pms" hreflang="pms" data-title="Sfera" data-language-autonym="Piemontèis" data-language-local-name="piemoncki" 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/Esfera" title="Esfera – portugalski" lang="pt" hreflang="pt" data-title="Esfera" data-language-autonym="Português" data-language-local-name="portugalski" 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/Sfer%C4%83" title="Sferă – rumuński" lang="ro" hreflang="ro" data-title="Sferă" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Lunq%27u" title="Lunq'u – keczua" lang="qu" hreflang="qu" data-title="Lunq'u" data-language-autonym="Runa Simi" data-language-local-name="keczua" class="interlanguage-link-target"><span>Runa Simi</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – rosyjski" lang="ru" hreflang="ru" data-title="Сфера" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – jakucki" lang="sah" hreflang="sah" data-title="Сфера" data-language-autonym="Саха тыла" data-language-local-name="jakucki" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sfera" title="Sfera – albański" lang="sq" hreflang="sq" data-title="Sfera" data-language-autonym="Shqip" data-language-local-name="albański" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Sfera" title="Sfera – sycylijski" lang="scn" hreflang="scn" data-title="Sfera" data-language-autonym="Sicilianu" data-language-local-name="sycylijski" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B7%9D%E0%B6%BD%E0%B6%BA" title="ගෝලය – syngaleski" lang="si" hreflang="si" data-title="ගෝලය" data-language-autonym="සිංහල" data-language-local-name="syngaleski" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Sphere" title="Sphere – Simple English" lang="en-simple" hreflang="en-simple" data-title="Sphere" 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/Gu%C4%BEa_(matematika)" title="Guľa (matematika) – słowacki" lang="sk" hreflang="sk" data-title="Guľa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="słowacki" 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/Sfera" title="Sfera – słoweński" lang="sl" hreflang="sl" data-title="Sfera" data-language-autonym="Slovenščina" data-language-local-name="słoweński" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Kubad" title="Kubad – somalijski" lang="so" hreflang="so" data-title="Kubad" data-language-autonym="Soomaaliga" data-language-local-name="somalijski" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%DB%86" title="گۆ – sorani" lang="ckb" hreflang="ckb" data-title="گۆ" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – serbski" lang="sr" hreflang="sr" data-title="Сфера" data-language-autonym="Српски / srpski" data-language-local-name="serbski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sfera" title="Sfera – serbsko-chorwacki" lang="sh" hreflang="sh" data-title="Sfera" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbsko-chorwacki" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Buleudan" title="Buleudan – sundajski" lang="su" hreflang="su" data-title="Buleudan" data-language-autonym="Sunda" data-language-local-name="sundajski" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pallo_(geometria)" title="Pallo (geometria) – fiński" lang="fi" hreflang="fi" data-title="Pallo (geometria)" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sf%C3%A4r" title="Sfär – szwedzki" lang="sv" hreflang="sv" data-title="Sfär" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espera" title="Espera – tagalski" lang="tl" hreflang="tl" data-title="Espera" data-language-autonym="Tagalog" data-language-local-name="tagalski" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%B3%E0%AE%AE%E0%AF%8D" title="கோளம் – tamilski" lang="ta" hreflang="ta" data-title="கோளம்" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tasegla" title="Tasegla – kabylski" lang="kab" hreflang="kab" data-title="Tasegla" data-language-autonym="Taqbaylit" data-language-local-name="kabylski" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – tatarski" lang="tt" hreflang="tt" data-title="Сфера" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarski" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%97%E0%B1%8B%E0%B0%B3%E0%B0%82" title="గోళం – telugu" lang="te" hreflang="te" data-title="గోళం" data-language-autonym="తెలుగు" data-language-local-name="telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A3%E0%B8%87%E0%B8%81%E0%B8%A5%E0%B8%A1" title="ทรงกลม – tajski" lang="th" hreflang="th" data-title="ทรงกลม" data-language-autonym="ไทย" data-language-local-name="tajski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-chr mw-list-item"><a href="https://chr.wikipedia.org/wiki/%E1%8E%A6%E1%8F%90%E1%8F%86%E1%8E%B8" title="ᎦᏐᏆᎸ – czirokeski" lang="chr" hreflang="chr" data-title="ᎦᏐᏆᎸ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="czirokeski" class="interlanguage-link-target"><span>ᏣᎳᎩ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCre" title="Küre – turecki" lang="tr" hreflang="tr" data-title="Küre" data-language-autonym="Türkçe" data-language-local-name="turecki" 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%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – ukraiński" lang="uk" hreflang="uk" data-title="Сфера" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_c%E1%BA%A7u" title="Mặt cầu – wietnamski" lang="vi" hreflang="vi" data-title="Mặt cầu" data-language-autonym="Tiếng Việt" data-language-local-name="wietnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%90%83" title="球 – chiński klasyczny" lang="lzh" hreflang="lzh" data-title="球" data-language-autonym="文言" data-language-local-name="chiński klasyczny" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Espira" title="Espira – waraj" lang="war" hreflang="war" data-title="Espira" data-language-autonym="Winaray" data-language-local-name="waraj" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – wu" lang="wuu" hreflang="wuu" data-title="球面" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%90%83%E9%AB%94" title="球體 – kantoński" lang="yue" hreflang="yue" data-title="球體" data-language-autonym="粵語" data-language-local-name="kantoński" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – chiński" lang="zh" hreflang="zh" data-title="球面" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://rsk.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B1%D0%B4%D0%B0_(%D2%91%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F)" title="Лабда (ґеометрия) – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Лабда (ґеометрия)" data-language-autonym="Руски" data-language-local-name="Pannonian Rusyn" class="interlanguage-link-target"><span>Руски</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12507#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> 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class="vector-dropdown-checkbox " aria-label="Narzędzia" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Narzędzia</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Narzędzia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ukryj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Więcej opcji" > <div class="vector-menu-heading"> Działania </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Sfera"><span>Czytaj</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfera&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfera&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfera&action=history"><span>Wyświetl historię</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ogólne </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Specjalna:Linkuj%C4%85ce/Sfera" title="Pokaż listę wszystkich stron linkujących do tej strony [j]" accesskey="j"><span>Linkujące</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Specjalna:Zmiany_w_linkowanych/Sfera" rel="nofollow" title="Ostatnie zmiany w stronach, do których ta strona linkuje [k]" accesskey="k"><span>Zmiany w linkowanych</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pl.wikipedia.org/wiki/Wikipedia:Prześlij_plik" title="Prześlij pliki [u]" accesskey="u"><span>Prześlij plik</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Specjalna:Strony_specjalne" title="Lista wszystkich stron specjalnych [q]" accesskey="q"><span>Strony specjalne</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Sfera&oldid=75120235" title="Stały link do tej wersji tej strony"><span>Link do tej wersji</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Sfera&action=info" title="Więcej informacji na temat tej strony"><span>Informacje o tej stronie</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Cytuj&page=Sfera&id=75120235&wpFormIdentifier=titleform" title="Informacja o tym jak należy cytować tę stronę"><span>Cytowanie tego artykułu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Skr%C3%B3%C4%87_adres_URL&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FSfera"><span>Zobacz skrócony adres URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Kod_QR&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FSfera"><span>Pobierz kod QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Drukuj lub eksportuj </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Ksi%C4%85%C5%BCka&bookcmd=book_creator&referer=Sfera"><span>Utwórz książkę</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Specjalna:DownloadAsPdf&page=Sfera&action=show-download-screen"><span>Pobierz jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Sfera&printable=yes" title="Wersja do wydruku [p]" accesskey="p"><span>Wersja do druku</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> W innych projektach </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Sphere" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12507" title="Link do powiązanego elementu w repozytorium danych [g]" accesskey="g"><span>Element Wikidanych</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Wygląd</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedii, wolnej encyklopedii</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="pl" dir="ltr"><div class="noprint noexcerpt disambig navigation-not-searchable" style="line-height:1.5em; padding: 3px 6px; background-color: var(--background-color-interactive-subtle, #f8f9fa); color: inherit; border-bottom: 1px solid var(--border-color-subtle, #c8ccd1); font-size: 95%; margin-bottom: 1em; display: flex; gap: 4px; align-items: center;"><span class="notpageimage" typeof="mw:File"><a href="/wiki/Wikipedia:Strona_ujednoznaczniaj%C4%85ca" title="Inne znaczenia"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/25px-Disambig.svg.png" decoding="async" width="25" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/38px-Disambig.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/72/Disambig.svg/50px-Disambig.svg.png 2x" data-file-width="230" data-file-height="183" /></a></span><span>Ten artykuł dotyczy pojęcia matematycznego. Zobacz też: <a href="/wiki/Sfera_(ujednoznacznienie)" class="mw-disambig" title="Sfera (ujednoznacznienie)">inne znaczenia</a>.</span></div> <table class="infobox" style="text-align:left;"> <tbody><tr> <td style="font-family: Linux Libertine,Georgia,Times,serif; font-size: 17px; line-height: 1.3; padding: 0.3em 10px; border-bottom: 1px solid #aaa; border-bottom: 0;">Definicja intuicyjna </td></tr> <tr> <td style="padding: 0.3em 10px;">Sfera to powierzchnia <a href="/wiki/Kula" title="Kula">kuli</a>. </td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/Plik:Sphere_wireframe_10deg_6r.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/270px-Sphere_wireframe_10deg_6r.svg.png" decoding="async" width="270" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/405px-Sphere_wireframe_10deg_6r.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Sphere_wireframe_10deg_6r.svg/540px-Sphere_wireframe_10deg_6r.svg.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Sfera</figcaption></figure> <p><b>Sfera</b> (z <a href="/wiki/J%C4%99zyk_grecki" title="Język grecki">gr.</a> σφαῖρα <i>sphaîra</i> „kula, piłka”) – uogólnienie pojęcia <a href="/wiki/Okr%C4%85g" title="Okrąg">okręgu</a> na więcej <a href="/wiki/Wymiar_(matematyka)" title="Wymiar (matematyka)">wymiarów</a>. Jest to zbiór wszystkich <a href="/wiki/Punkt_(geometria)" title="Punkt (geometria)">punktów</a> (<a href="/wiki/Miejsce_geometryczne" title="Miejsce geometryczne">miejsce geometryczne</a>) w <a href="/wiki/Przestrze%C5%84_metryczna" title="Przestrzeń metryczna">przestrzeni metrycznej</a> oddalonych o ustaloną odległość od wybranego punktu. Ustalona odległość nazywa się <i>promieniem sfery</i>, wybrany punkt nazywa się <i>środkiem sfery</i>. Zwykle przyjmuje się dodatkowo, że promień musi być dodatni<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup>. Tak zdefiniowany zbiór jest <a href="/wiki/Brzeg_(matematyka)" title="Brzeg (matematyka)">brzegiem</a> <a href="/wiki/Kula" title="Kula">kuli</a> o tym samym środku i promieniu<sup id="cite_ref-epwn_2-0" class="reference"><a href="#cite_note-epwn-2">[2]</a></sup>. Zazwyczaj jako przestrzeń metryczną rozpatruje się <a href="/wiki/Przestrze%C5%84_euklidesowa" title="Przestrzeń euklidesowa">przestrzeń euklidesową</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sfera_w_euklidesowej_przestrzeni_trójwymiarowej"><span id="Sfera_w_euklidesowej_przestrzeni_tr.C3.B3jwymiarowej"></span>Sfera w euklidesowej przestrzeni trójwymiarowej</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=1" title="Edytuj sekcję: Sfera w euklidesowej przestrzeni trójwymiarowej" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Sfera w euklidesowej przestrzeni trójwymiarowej"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Najczęściej mówimy o sferze w <a href="/wiki/Przestrze%C5%84_tr%C3%B3jwymiarowa" title="Przestrzeń trójwymiarowa">przestrzeni euklidesowej trójwymiarowej</a>. Taka sfera jest dwuwymiarową powierzchnią opisywaną wzorem: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d48fb8e006a4f8f203cd677f57f927b132b38bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.927ex; height:3.176ex;" alt="{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2},}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0},z_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39177ddeeb9f9a393b664e522bc8e3bf0face153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.59ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})}"></span> to współrzędne <b>środka sfery</b>, a wartość <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> jest nazywana <b>promieniem sfery</b>. Często dodatkowo zakłada się, że <span class="mwe-math-element"><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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cbbcd53bd13620bc53490e3eec42790850b452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r>0}"></span> (sfera z zerowym promieniem to przypadek zdegenerowany, w którym nie wszystkie typowe własności są zachowane). </p><p>W tym samym układzie współrzędnych sfera może być opisana za pomocą <a href="/wiki/R%C3%B3wnanie_parametryczne" title="Równanie parametryczne">równania parametrycznego</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}x(\alpha ,\beta )=x_{0}+r\cos \alpha \cos \beta \\[2pt]y(\alpha ,\beta )=y_{0}+r\sin \beta \\[2pt]z(\alpha ,\beta )=z_{0}+r\sin \alpha \cos \beta \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="0.4em 0.4em 0.2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mi>β</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x(\alpha ,\beta )=x_{0}+r\cos \alpha \cos \beta \\[2pt]y(\alpha ,\beta )=y_{0}+r\sin \beta \\[2pt]z(\alpha ,\beta )=z_{0}+r\sin \alpha \cos \beta \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34985e50eef937b75c5b110890e8e71269b0599e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:29.772ex; height:9.843ex;" alt="{\displaystyle {\begin{cases}x(\alpha ,\beta )=x_{0}+r\cos \alpha \cos \beta \\[2pt]y(\alpha ,\beta )=y_{0}+r\sin \beta \\[2pt]z(\alpha ,\beta )=z_{0}+r\sin \alpha \cos \beta \end{cases}}}"></span></dd></dl> <p>gdzie: </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 \alpha \in [-\pi ,\pi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in [-\pi ,\pi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/081f6bf6acd9dff15bcf6b614ea83bf2136bc8d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.033ex; height:2.843ex;" alt="{\displaystyle \alpha \in [-\pi ,\pi ),}"></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 \beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08a0703e1266f8d74c9a2ab0906f3bd93df4212b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.58ex; height:4.843ex;" alt="{\displaystyle \beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}"></span></li></ul> <p>Parametry <span class="mwe-math-element"><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 ,\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b46b57cfa0011b643037751809904d915c1b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.854ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta }"></span> są odpowiednio <i>długością i szerokością geograficzną</i> w odpowiednim <a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_sferycznych" title="Układ współrzędnych sferycznych">układzie współrzędnych sferycznych</a> związanym ze środkiem sfery </p><p>W układzie współrzędnych sferycznych, równanie sfery o promieniu <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> i środku znajdującym się w środku układu współrzędnych, przyjmuje postać <span class="mwe-math-element"><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(\alpha ,\beta )=r=const}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(\alpha ,\beta )=r=const}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ade3edbd1f03d93bbe8686c001dc43b6ad9890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.416ex; height:2.843ex;" alt="{\displaystyle r(\alpha ,\beta )=r=const}"></span> dla dowolnych kątów <span class="mwe-math-element"><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 \in [-\pi ,\pi ),\beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>π</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in [-\pi ,\pi ),\beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0c3742cde5e9431b787d56d72ff849224d1d74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27ex; height:4.843ex;" alt="{\displaystyle \alpha \in [-\pi ,\pi ),\beta \in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right].}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Związane_pojęcia"><span id="Zwi.C4.85zane_poj.C4.99cia"></span>Związane pojęcia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=2" title="Edytuj sekcję: Związane pojęcia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Związane pojęcia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Cięciwa sfery</b> to <a href="/wiki/Odcinek" title="Odcinek">odcinek</a> o końcach na sferze. </p><p><b>Średnica sfery</b> to: </p> <ul><li>cięciwa przechodząca przez środek sfery</li> <li>długość tej cięciwy, czyli podwojona wartość promienia sfery.</li></ul> <p><b><a href="/wiki/Pole_powierzchni" title="Pole powierzchni">Pole powierzchni</a> sfery</b> wyraża się wzorem: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=4\pi r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=4\pi r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb967f4667bb5161ae3f431fcfacfbcae255e421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.842ex; height:2.676ex;" alt="{\displaystyle S=4\pi r^{2}.}"></span></dd></dl> <p><b><a href="/wiki/Ko%C5%82o_wielkie" title="Koło wielkie">Okrąg wielki</a></b> sfery to okrąg o promieniu tej sfery, o środku w jej środku. </p><p><a href="/wiki/Krzywizna_Gaussa" title="Krzywizna Gaussa">Krzywizna Gaussa</a> sfery w każdym jej punkcie wynosi: </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 K={\frac {1}{r^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {1}{r^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22163db53efdbfaae004bc72143ce76631a5707a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.75ex; height:5.509ex;" alt="{\displaystyle K={\frac {1}{r^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sfera_w_n-wymiarowej_przestrzeni_euklidesowej">Sfera w n-wymiarowej przestrzeni euklidesowej</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=3" title="Edytuj sekcję: Sfera w n-wymiarowej przestrzeni euklidesowej" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Sfera w n-wymiarowej przestrzeni euklidesowej"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Zobacz też: <a href="/wiki/Hipersfera" title="Hipersfera">hipersfera</a>.</i></div> <p>Pojęcie sfery może być zdefiniowane w <a href="/wiki/Przestrze%C5%84_euklidesowa" title="Przestrzeń euklidesowa">przestrzeni euklidesowej</a> dowolnego <a href="/wiki/Wymiar_(matematyka)" title="Wymiar (matematyka)">wymiaru</a>. Wówczas w przestrzeni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-wymiarowej sfera może być opisana następującym wzorem: </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 \sum _{j=1}^{n}(x_{j}-s_{j})^{2}=r^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{n}(x_{j}-s_{j})^{2}=r^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb9a8705be956ded0834f8aabb8d5374c1da48c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.147ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{n}(x_{j}-s_{j})^{2}=r^{2},}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db47cb3d2f9496205a17a6856c91c1d3d363ccd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-ta <a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych" title="Układ współrzędnych">współrzędna</a> <a href="/wiki/Punkt_(geometria)" title="Punkt (geometria)">punktu</a> na sferze, <span class="mwe-math-element"><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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a350c64508aef872d4e72ee677746ef7a20f72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2ex; height:2.343ex;" alt="{\displaystyle s_{j}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-ta współrzędna jej środka, <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> to promień sfery. W tym ujęciu <a href="/wiki/Okr%C4%85g" title="Okrąg">okrąg</a> jest szczególnym przypadkiem sfery w przestrzeni dwuwymiarowej, a zbiór dwóch punktów jest sferą w przestrzeni jednowymiarowej. </p><p>Sfera w przestrzeni <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-wymiarowej jest czasem nazywana <i>sferą m-wymiarową</i> i oznaczana <span class="mwe-math-element"><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^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05df06fa718d61de26bd116bf4ce583b2c4ba37b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.844ex; height:2.676ex;" alt="{\displaystyle S^{m},}"></span> gdzie <span class="mwe-math-element"><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=n-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7597fb84c531e5197107c0ea0f0c05bc60bb1266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.183ex; height:2.509ex;" alt="{\displaystyle m=n-1,}"></span> ponieważ taka sfera jest powierzchnią <span class="mwe-math-element"><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/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>-wymiarową. Dla przykładu, zwykłą sferę rozpatruje się w przestrzeni trójwymiarowej, ale ona jest zwykłą powierzchnią, czyli obiektem dwuwymiarowym; dlatego to sfera dwuwymiarowa, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4bacfddb1688e9b20496c51bbde675848b8a6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.223ex; height:2.676ex;" alt="{\displaystyle S^{2}.}"></span> Jeżeli <span class="mwe-math-element"><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>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e4ce1c04edd8f9602e60f0ec4457b7ac12fcd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>2}"></span> (tzn. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257030caae597fd034c2cbcff2cff9dfc4272d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>3}"></span>), to taka uogólniona sfera jest nazywana też <b>hipersferą</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Uogólnienia"><span id="Uog.C3.B3lnienia"></span>Uogólnienia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=4" title="Edytuj sekcję: Uogólnienia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Uogólnienia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sfera jest też pojęciem <a href="/wiki/Topologia" title="Topologia">topologii</a>, w której oznacza <a href="/wiki/Przestrze%C5%84_topologiczna" title="Przestrzeń topologiczna">przestrzeń topologiczną</a> <a href="/wiki/Homeomorfizm" title="Homeomorfizm">homeomorficzną</a> z <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-wymiarową hipersferą. Sfera rozpatrywana w topologii ma więc te same topologiczne własności jak hipersfera, tzn. jest to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-wymiarowa <a href="/wiki/Rozmaito%C5%9B%C4%87_topologiczna" title="Rozmaitość topologiczna">rozmaitość</a> bez <a href="/wiki/Brzeg_(matematyka)" title="Brzeg (matematyka)">brzegu</a>, <a href="/wiki/Przestrze%C5%84_zwarta" title="Przestrzeń zwarta">zwarta</a> i jest <a href="/wiki/Homotopia#Homotopijna_równoważność" title="Homotopia">homotopijnie równoważna</a> z <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-hipersferą. </p><p><a href="/wiki/Hipoteza_Poincar%C3%A9go" title="Hipoteza Poincarégo">Uogólniona hipoteza Poincarégo</a> (włącznie z potwierdzonym już przypadkiem 3-wymiarowym) stwierdza, że jest też odwrotnie – każda <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-wymiarowa rozmaitość bez brzegu, zwarta i mająca typ homotopijny <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-hipersfery jest homeomorficzna z <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>-hipersferą. </p> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=5" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint plainlinks" cellpadding="4" role="presentation"> <tbody><tr> <td style="vertical-align:middle; text-align:center; width:30px;"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/28px-WiktionaryPl_nodesc.svg.png" decoding="async" width="28" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/42px-WiktionaryPl_nodesc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/56px-WiktionaryPl_nodesc.svg.png 2x" data-file-width="122" data-file-height="117" /></span></span> </td> <td style="line-height:normal; vertical-align:middle; text-align:center; flex:unset;"><a href="https://pl.wiktionary.org/wiki/sfera" class="extiw" title="wikt:sfera"><strong>Zobacz hasło</strong> <em>sfera</em> w Wikisłowniku</a> </td></tr></tbody></table> <table class="infobox noprint plainlinks" cellpadding="4" role="presentation"> <tbody><tr> <td style="vertical-align:middle; text-align:center; width:30px;"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/21px-Commons-logo.svg.png" decoding="async" width="21" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/31px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/42px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> </td> <td style="line-height:normal; vertical-align:middle; text-align:center; flex:unset;"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Spheres?uselang=pl"><strong>Zobacz multimedia</strong> związane z tematem: <em>Sfera</em></a> </td></tr></tbody></table> <ul><li><a href="/wiki/Pas_sferyczny" title="Pas sferyczny">pas sferyczny</a></li> <li><a href="/wiki/Pseudosfera" title="Pseudosfera">pseudosfera</a></li> <li><a href="/wiki/Rogata_sfera_Alexandera" title="Rogata sfera Alexandera">rogata sfera Alexandera</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=6" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=6" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Fritz Reinhardt, Heinrich Soeder: <i>Atlas matematyki</i>. Prószyński i S-ka, 2003, s. 201. <a href="/wiki/Specjalna:Ksi%C4%85%C5%BCki/8374691891" title="Specjalna:Książki/8374691891">ISBN <span class="isbn">83-7469-189-1</span></a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atlas+matematyki&rft.aulast=Reinhardt&rft.aufirst=Fritz&rft.pub=Pr%C3%B3szy%C5%84ski+i+S-ka&rft.pages=201&rft.isbn=83-7469-189-1"></span></cite></span> </li> <li id="cite_note-epwn-2"><span class="mw-cite-backlink"><a href="#cite_ref-epwn_2-0">↑</a></span> <span class="reference-text"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3974401"><i>sfera</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-10-03]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=sfera&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3974401" style="display:none"> </span>.</cite></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=7" title="Edytuj sekcję: Bibliografia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=7" title="Edytuj kod źródłowy sekcji: Bibliografia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book" id="CITEREFCiesielskiPogoda2005"><a href="/wiki/Krzysztof_Ciesielski_(matematyk)" title="Krzysztof Ciesielski (matematyk)">Krzysztof Ciesielski</a>, <a href="/wiki/Zdzis%C5%82aw_Pogoda" title="Zdzisław Pogoda">Zdzisław Pogoda</a>: <i>Bezmiar Matematycznej Wyobraźni</i>. Warszawa: <a href="/wiki/Pr%C3%B3szy%C5%84ski_i_S-ka" title="Prószyński i S-ka">Prószyński i S-ka</a>, 2005. <a href="/wiki/Specjalna:Ksi%C4%85%C5%BCki/8373379320" title="Specjalna:Książki/8373379320">ISBN <span class="isbn">83-7337-932-0</span></a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bezmiar+Matematycznej+Wyobra%C5%BAni&rft.aulast=Ciesielski&rft.aufirst=Krzysztof&rft.pub=%5B%5BPr%C3%B3szy%C5%84ski+i+S-ka%5D%5D&rft.place=Warszawa&rft.isbn=83-7337-932-0"></span></cite></li></ul> <div class="mw-heading mw-heading2"><h2 id="Linki_zewnętrzne"><span id="Linki_zewn.C4.99trzne"></span>Linki zewnętrzne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfera&veaction=edit&section=8" title="Edytuj sekcję: Linki zewnętrzne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfera&action=edit&section=8" title="Edytuj kod źródłowy sekcji: Linki zewnętrzne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation open-access"><span class="cite-name-before"><span class="cite-name-full">Michał</span><span class="cite-name-initials" title="Michał" style="display:none">M.</span> </span><span class="cite-lastname">Kieza</span><span class="cite-name-after" style="display:none"> <span class="cite-name-full">Michał</span><span class="cite-name-initials" title="Michał">M.</span></span>, <a rel="nofollow" class="external text" href="http://www.deltami.edu.pl/2010/03/kacik-przestrzenny-2-najmocniejsze-twierdzenie-stereometrii/"><i>Kącik przestrzenny (2): Najmocniejsze twierdzenie stereometrii</i></a>, „<a href="/wiki/Delta_(miesi%C4%99cznik)" title="Delta (miesięcznik)">Delta</a>”, marzec 2010<span class="issn">, <a href="/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0137-3005">0137-3005</a></span><span class="accessdate"> [dostęp 2024-11-02]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=book&rft.aufirst=Micha%C5%82&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.btitle=K%C4%85cik+przestrzenny+%282%29%3A+Najmocniejsze+twierdzenie+stereometrii&rft.issn=0137-3005&rft.date=2010-03&rft.aulast=Kieza&rft_id=http%3A%2F%2Fwww.deltami.edu.pl%2F2010%2F03%2Fkacik-przestrzenny-2-najmocniejsze-twierdzenie-stereometrii%2F" style="display:none"> </span>.</cite></li></ul> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r74983602">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output 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plainlinks hlist"><li><a href="/wiki/Szablon:Bry%C5%82y_obrotowe" title="Szablon:Bryły obrotowe"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/w/index.php?title=Dyskusja_szablonu:Bry%C5%82y_obrotowe&action=edit&redlink=1" class="new" title="Dyskusja szablonu:Bryły obrotowe (strona nie istnieje)"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Bry%C5%82y_obrotowe&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Bry%C5%82a_obrotowa" title="Bryła obrotowa">Bryły obrotowe</a></div><div class="mw-collapsible-content flex"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row">przykłady<br />i ich <a href="/wiki/Podzbi%C3%B3r" title="Podzbiór">części</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Walec_(bry%C5%82a)" title="Walec (bryła)">walec obrotowy<br />(kołowy prosty)</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Powierzchnia_walcowa" title="Powierzchnia walcowa">powierzchnia walcowa</a></li> <li><a href="/wiki/Rura_cylindryczna" title="Rura cylindryczna">rura cylindryczna</a></li></ul> </td></tr><tr class="a1_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Sto%C5%BCek_(bry%C5%82a)" title="Stożek (bryła)">stożek obrotowy<br />(kołowy prosty)</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Powierzchnia_sto%C5%BCkowa" title="Powierzchnia stożkowa">powierzchnia stożkowa</a></li> <li><a href="/wiki/Sto%C5%BCek_%C5%9Bci%C4%99ty" title="Stożek ścięty">stożek ścięty</a></li></ul> </td></tr><tr class="a1_3"><th class="navbox-group opis" scope="row"><a href="/wiki/Kula" title="Kula">kula</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_3_1"><th class="navbox-group opis" scope="row"><a class="mw-selflink selflink">sfera</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Czasza_kuli" title="Czasza kuli">czasza kuli</a></li> <li><a href="/wiki/Pas_sferyczny" title="Pas sferyczny">pas sferyczny</a></li></ul> </td></tr><tr class="a1_3_2"><th class="navbox-group opis" scope="row">inne części</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Odcinek_kuli" title="Odcinek kuli">odcinek kuli</a> <ul><li><a href="/wiki/P%C3%B3%C5%82kula" title="Półkula">półkula</a></li></ul></li> <li><a href="/wiki/Warstwa_kulista" title="Warstwa kulista">warstwa kulista</a></li> <li><a href="/wiki/Wycinek_kuli" title="Wycinek kuli">wycinek kuli</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a1_4"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Beczka_(bry%C5%82a_geometryczna)" title="Beczka (bryła geometryczna)">beczka</a></li> <li><a href="/wiki/R%C3%B3g_Gabriela" title="Róg Gabriela">róg Gabriela</a></li> <li><a href="/wiki/Toroid" title="Toroid">toroid</a> <ul><li><a href="/wiki/Pe%C5%82ny_torus" title="Pełny torus">pełny torus</a></li></ul></li></ul> </td></tr></tbody></table></td></tr><tr class="a2"><th class="navbox-group opis" scope="row"><a href="/wiki/Relacja_dwuargumentowa" title="Relacja dwuargumentowa">relacje</a> między kulą<br />a innymi <a href="/wiki/Bry%C5%82a_geometryczna" title="Bryła geometryczna">bryłami</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Kula_opisana" title="Kula opisana">kula opisana</a></li> <li><a href="/wiki/Kula_wpisana" title="Kula wpisana">kula wpisana</a></li> <li><a href="/wiki/Sfera_p%C3%B3%C5%82wpisana" title="Sfera półwpisana">kula półwpisana</a></li></ul> </td></tr><tr class="a3"><th class="navbox-group opis" scope="row"><a href="/wiki/Krzywa" title="Krzywa">krzywe</a> tworzone<br /><a href="/wiki/Cz%C4%99%C5%9B%C4%87_wsp%C3%B3lna" title="Część wspólna">przekrojami</a><br />brył obrotowych</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a3_1"><th class="navbox-group opis" scope="row">stożkiem obrotowym<br />i <a href="/wiki/P%C5%82aszczyzna" title="Płaszczyzna">płaszczyzną</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Krzywa_sto%C5%BCkowa" title="Krzywa stożkowa">krzywe stożkowe</a> <ul><li><a href="/wiki/Okr%C4%85g" title="Okrąg">okrąg</a></li> <li><a href="/wiki/Elipsa" title="Elipsa">elipsa</a></li> <li><a href="/wiki/Parabola_(matematyka)" title="Parabola (matematyka)">parabola</a></li> <li><a href="/wiki/Hiperbola_(matematyka)" title="Hiperbola (matematyka)">hiperbola</a></li></ul></li> <li><a href="/wiki/Prosta" title="Prosta">prosta</a></li></ul> </td></tr><tr class="a3_2"><th class="navbox-group opis" scope="row">sferą<br />i płaszczyzną</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Ko%C5%82o_ma%C5%82e" title="Koło małe">okrąg mały</a></li> <li><a href="/wiki/Ko%C5%82o_wielkie" title="Koło wielkie">okrąg wielki</a> <ul><li><a href="/wiki/Ortodroma" title="Ortodroma">ortodroma</a></li> <li><a href="/wiki/Po%C5%82udnik" title="Południk">południk</a></li> <li><a href="/wiki/R%C3%B3wnik" title="Równik">równik</a></li></ul></li> <li><a href="/wiki/R%C3%B3wnole%C5%BCnik" title="Równoleżnik">równoleżnik</a> <ul><li><a href="/wiki/R%C3%B3wnik" title="Równik">równik</a></li></ul></li></ul> </td></tr><tr class="a3_3"><th class="navbox-group opis" scope="row">walcem obrotowym<br />i sferą</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Krzywa_Vivianiego" title="Krzywa Vivianiego">krzywa Vivianiego</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a4"><th class="navbox-group opis" scope="row">inne krzywe na<br />bryłach obrotowych</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a4_1"><th class="navbox-group opis" scope="row">na walcu obrotowym</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Linia_%C5%9Brubowa" title="Linia śrubowa">linia śrubowa</a>, in. helisa</li></ul> </td></tr><tr class="a4_2"><th class="navbox-group opis" scope="row">na sferze</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Dwuk%C4%85t_sferyczny" title="Dwukąt sferyczny">dwukąt sferyczny</a></li> <li><a href="/wiki/Tr%C3%B3jk%C4%85t_sferyczny" title="Trójkąt sferyczny">trójkąt sferyczny</a> <ul><li><a href="/wiki/Tr%C3%B3jk%C4%85t_eulerowski" title="Trójkąt eulerowski">trójkąt eulerowski</a></li></ul></li> <li><a href="/wiki/Loksodroma" title="Loksodroma">loksodroma</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a5"><th class="navbox-group opis" scope="row">powiązane <a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych" title="Układ współrzędnych">układy<br />współrzędnych</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_walcowych" title="Układ współrzędnych walcowych">walcowy</a></li> <li><a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_sferycznych" title="Układ współrzędnych sferycznych">sferyczny</a></li></ul> </td></tr><tr class="a6"><th class="navbox-group opis" scope="row">powiązane<br /><a href="/wiki/Powierzchnia" title="Powierzchnia">powierzchnie</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a6_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Kwadryka" title="Kwadryka">kwadryki</a> obrotowe</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Elipsoida_obrotowa" title="Elipsoida obrotowa">elipsoida o.</a></li> <li><a href="/wiki/Paraboloida_obrotowa" title="Paraboloida obrotowa">paraboloida o.</a></li> <li><a href="/wiki/Hiperboloida_jednopow%C5%82okowa_obrotowa" title="Hiperboloida jednopowłokowa obrotowa">hiperboloida jednopowłokowa o.</a></li> <li><a href="/wiki/Hiperboloida_dwupow%C5%82okowa_obrotowa" title="Hiperboloida dwupowłokowa obrotowa">hiperboloida dwupowłokowa o.</a></li></ul> </td></tr><tr class="a6_2"><th class="navbox-group opis" scope="row">inne powierzchnie<br />obrotowe</th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Pseudosfera" title="Pseudosfera">pseudosfera</a></li> <li><a href="/wiki/Torus_(matematyka)" title="Torus (matematyka)">torus</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a7"><th class="navbox-group opis" scope="row">powiązane<br /><a href="/wiki/Nauka" title="Nauka">nauki</a></th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a7_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Algebra" title="Algebra">algebra</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Algebra_liniowa" title="Algebra liniowa">algebra liniowa</a></li></ul> </td></tr><tr class="a7_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Analiza_matematyczna" title="Analiza matematyczna">analiza matematyczna</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Analiza_rzeczywista" title="Analiza rzeczywista">analiza rzeczywista</a></li></ul> </td></tr><tr class="a7_3"><th class="navbox-group opis" scope="row"><a href="/wiki/Geometria" title="Geometria">geometria</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Stereometria" title="Stereometria">stereometria</a> <ul><li><a href="/wiki/Geometria_sferyczna" title="Geometria sferyczna">geometria sferyczna</a></li></ul></li> <li><a href="/wiki/Geometria_analityczna" title="Geometria analityczna">geometria analityczna</a></li></ul> </td></tr></tbody></table></td></tr></tbody></table><div class="navbox-after after"> <p><span typeof="mw:File"><a href="/wiki/Plik:01-Volumenvergleich_Kugel_-_Zylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/01-Volumenvergleich_Kugel_-_Zylinder.svg/150px-01-Volumenvergleich_Kugel_-_Zylinder.svg.png" decoding="async" width="150" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/01-Volumenvergleich_Kugel_-_Zylinder.svg/225px-01-Volumenvergleich_Kugel_-_Zylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/01-Volumenvergleich_Kugel_-_Zylinder.svg/300px-01-Volumenvergleich_Kugel_-_Zylinder.svg.png 2x" data-file-width="341" data-file-height="419" /></a></span> </p> </div></div></div> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r74983602"><ul class="tnavbar noprint plainlinks hlist"><li><a href="/wiki/Szablon:Kwadryki" title="Szablon:Kwadryki"><span title="Pokaż ten szablon">p</span></a></li><li><a href="/w/index.php?title=Dyskusja_szablonu:Kwadryki&action=edit&redlink=1" class="new" title="Dyskusja szablonu:Kwadryki (strona nie istnieje)"><span title="Dyskusja na temat tego szablonu">d</span></a></li><li title="Możesz edytować ten szablon. Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Kwadryki&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Kwadryka" title="Kwadryka">Kwadryki</a></div><div class="mw-collapsible-content flex"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row">typy</th><td class="navbox-list spis"><table class="inner-standard"><tbody><tr class="a1_1"><th class="navbox-group opis" scope="row"><a href="/wiki/Elipsoida" title="Elipsoida">elipsoidy</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Elipsoida_obrotowa" title="Elipsoida obrotowa">obrotowe</a> <ul><li><a class="mw-selflink selflink">sfera</a></li></ul></li></ul> </td></tr><tr class="a1_2"><th class="navbox-group opis" scope="row"><a href="/wiki/Paraboloida" title="Paraboloida">paraboloidy</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Paraboloida_eliptyczna" title="Paraboloida eliptyczna">eliptyczne</a> <ul><li><a href="/wiki/Paraboloida_obrotowa" title="Paraboloida obrotowa">obrotowe</a></li></ul></li> <li><a href="/wiki/Paraboloida_hiperboliczna" title="Paraboloida hiperboliczna">hiperboliczne</a></li></ul> </td></tr><tr class="a1_3"><th class="navbox-group opis" scope="row"><a href="/wiki/Hiperboloida" title="Hiperboloida">hiperboloidy</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Hiperboloida_jednopow%C5%82okowa" title="Hiperboloida jednopowłokowa">jednopowłokowa</a> <ul><li><a href="/wiki/Hiperboloida_jednopow%C5%82okowa_obrotowa" title="Hiperboloida jednopowłokowa obrotowa">obrotowa</a></li></ul></li> <li><a href="/wiki/Hiperboloida_dwupow%C5%82okowa" title="Hiperboloida dwupowłokowa">dwupowłokowa</a> <ul><li><a href="/wiki/Hiperboloida_dwupow%C5%82okowa_obrotowa" title="Hiperboloida dwupowłokowa obrotowa">obrotowa</a></li></ul></li></ul> </td></tr><tr class="a1_4"><th class="navbox-group opis" scope="row">szczególne<br /><a href="/wiki/Powierzchnia_walcowa" title="Powierzchnia walcowa">powierzchnie walcowe</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Walec_eliptyczny" title="Walec eliptyczny">eliptyczna</a></li> <li><a href="/wiki/Walec_paraboliczny" title="Walec paraboliczny">paraboliczna</a></li> <li><a href="/wiki/Walec_hiperboliczny" title="Walec hiperboliczny">hiperboliczna</a></li></ul> </td></tr><tr class="a1_5"><th class="navbox-group opis" scope="row">inne</th><td class="navbox-list spis hlist navbox-odd"> <ul><li>szczególne <a href="/wiki/Powierzchnia_sto%C5%BCkowa" title="Powierzchnia stożkowa">powierzchnie stożkowe</a></li></ul> </td></tr></tbody></table></td></tr><tr class="a2"><th class="navbox-group opis" scope="row">powiązane <a href="/wiki/Bry%C5%82a_geometryczna" title="Bryła geometryczna">bryły</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Kula" title="Kula">kula</a></li> <li><a href="/wiki/Sto%C5%BCek_(bry%C5%82a)" title="Stożek (bryła)">stożek</a></li> <li><a href="/wiki/Walec_(bry%C5%82a)" title="Walec (bryła)">walec</a></li></ul> </td></tr><tr class="a3"><th class="navbox-group opis" scope="row">inne powiązane<br />pojęcia</th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a href="/wiki/Forma_kwadratowa" title="Forma kwadratowa">forma kwadratowa</a></li> <li><a href="/wiki/Elipsa_szyjna" title="Elipsa szyjna">elipsa szyjna</a></li> <li><a href="/wiki/Punkt_siod%C5%82owy" title="Punkt siodłowy">punkt siodłowy</a></li></ul> </td></tr><tr class="a4"><th class="navbox-group opis" scope="row">występowanie</th><td class="navbox-list spis hlist navbox-even"> <ul><li>elipsoidy: <a href="/wiki/Elipsoida_bezw%C5%82adno%C5%9Bci" title="Elipsoida bezwładności">bezwładności</a>, <a href="/wiki/Elipsoida_ziemska" title="Elipsoida ziemska">ziemska</a>, <a href="/wiki/Elipsoida_Jakobiego" title="Elipsoida Jakobiego">Jacobiego</a>, <a href="/wiki/Elipsoida_Johna" title="Elipsoida Johna">Johna</a></li> <li><a href="/wiki/Sto%C5%BCek_%C5%9Bwietlny" title="Stożek świetlny">stożek świetlny</a></li></ul> </td></tr></tbody></table><div class="navbox-after after"> <p><span typeof="mw:File"><a href="/wiki/Plik:Hyperbolic_paraboloid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Hyperbolic_paraboloid.png/150px-Hyperbolic_paraboloid.png" decoding="async" width="150" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Hyperbolic_paraboloid.png/225px-Hyperbolic_paraboloid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Hyperbolic_paraboloid.png/300px-Hyperbolic_paraboloid.png 2x" data-file-width="450" data-file-height="425" /></a></span> </p> </div></div></div> <style data-mw-deduplicate="TemplateStyles:r74016753">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-fehlt"><div><a href="/wiki/Kontrola_autorytatywna" title="Kontrola autorytatywna">Kontrola autorytatywna</a> (<span class="description"><a href="/wiki/Elipsoida_obrotowa" title="Elipsoida obrotowa">elipsoida obrotowa</a></span>):</div><ul><li><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://lccn.loc.gov/sh85126590">sh85126590</a></span></li><li><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://d-nb.info/gnd/4165914-4">4165914-4</a></span></li><li><a href="/wiki/Biblioth%C3%A8que_nationale" title="Bibliothèque nationale">BnF</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://catalogue.bnf.fr/ark:/12148/cb119812876">119812876</a></span></li><li><a href="/wiki/Centralna_Biblioteka_Narodowa_we_Florencji" title="Centralna Biblioteka Narodowa we Florencji">BNCF</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://thes.bncf.firenze.sbn.it/termine.php?id=38152">38152</a></span></li><li><a href="/wiki/Biblioteka_Narodowa_Izraela" title="Biblioteka Narodowa Izraela">J9U</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007565817805171">987007565817805171</a></span></li></ul></div><div class="normdaten-andere"><div><a href="/wiki/Encyklopedia_internetowa" title="Encyklopedia internetowa">Encyklopedie internetowe</a>:</div> <ul><li><a href="/wiki/Encyklopedia_Britannica" title="Encyklopedia Britannica">Britannica</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/sphere">topic/sphere</a></span></li> <li><a href="/wiki/Enciclopedia_Treccani" title="Enciclopedia Treccani">Treccani</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.treccani.it/enciclopedia/sfera">sfera</a></span></li> <li><a href="/wiki/Store_norske_leksikon" title="Store norske leksikon">SNL</a>: <span class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=4342&url_prefix=https://snl.no/&id=kule">kule</a></span></li> <li><a href="/wiki/Den_Store_Danske_Encyklop%C3%A6di" title="Den Store Danske Encyklopædi">DSDE</a>: <span class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=8313&url_prefix=https://lex.dk/&id=kugle">kugle</a></span></li></ul> </div></div></div><noscript><img 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