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Pentru înscriere de articole la concurs (nominalizări), condiții de eligibilitate, punctare și alte detalii, vă rugăm să accesați \u003Cb\u003E\u003Ca href=\"/wiki/Wikipedia:Concurs_de_scriere\" title=\"Wikipedia:Concurs de scriere\"\u003Epagina\u0026#160;concursului\u003C/a\u003E\u003C/b\u003E.\u003C/div\u003E\n\u003Cdiv style=\"clear: both;\"\u003E\u003C/div\u003E\n\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <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="Cuprins" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" 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class="vector-toc-link" href="#Ecuații_în_R3"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ecuații în <b>R<sup>3</sup></b></span> </div> </a> <ul id="toc-Ecuații_în_R3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proprietăți" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietăți"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proprietăți</span> </div> </a> <ul id="toc-Proprietăți-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizări" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizări"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generalizări</span> </div> </a> <ul id="toc-Generalizări-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topologie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Topologie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Topologie</span> </div> </a> <button aria-controls="toc-Topologie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Topologie subsection</span> </button> <ul id="toc-Topologie-sublist" class="vector-toc-list"> <li id="toc-Sfera_din_punct_de_vedere_geometric_si_topologic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sfera_din_punct_de_vedere_geometric_si_topologic"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Sfera din punct de vedere geometric si topologic</span> </div> </a> <ul id="toc-Sfera_din_punct_de_vedere_geometric_si_topologic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" 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="Comută cuprinsul" > <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">Comută cuprinsul</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">Sferă</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="Mergeți la un articol în altă limbă. Disponibil în 106 limbi" > <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 limbi</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 – amharică" lang="am" hreflang="am" data-title="ሉል" data-language-autonym="አማርኛ" data-language-local-name="amharică" 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 – arabă" lang="ar" hreflang="ar" data-title="كرة" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</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 – Moroccan Arabic" lang="ary" hreflang="ary" data-title="كورة" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Esfera" title="Esfera – asturiană" lang="ast" hreflang="ast" data-title="Esfera" data-language-autonym="Asturianu" data-language-local-name="asturiană" 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 – azeră" lang="az" hreflang="az" data-title="Sfera" data-language-autonym="Azərbaycanca" data-language-local-name="azeră" 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-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – bașkiră" lang="ba" hreflang="ba" data-title="Сфера" data-language-autonym="Башҡортса" data-language-local-name="bașkiră" 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 – belarusă" lang="be" hreflang="be" data-title="Сфера" data-language-autonym="Беларуская" data-language-local-name="belarusă" class="interlanguage-link-target"><span>Беларуская</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 – bulgară" lang="bg" hreflang="bg" data-title="Сфера" data-language-autonym="Български" data-language-local-name="bulgară" class="interlanguage-link-target"><span>Български</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-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 – bengaleză" lang="bn" hreflang="bn" data-title="গোলক" data-language-autonym="বাংলা" data-language-local-name="bengaleză" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Sfera" title="Sfera – bosniacă" lang="bs" hreflang="bs" data-title="Sfera" data-language-autonym="Bosanski" data-language-local-name="bosniacă" 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 – catalană" lang="ca" hreflang="ca" data-title="Esfera" data-language-autonym="Català" data-language-local-name="catalană" class="interlanguage-link-target"><span>Català</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 – cherokee" lang="chr" hreflang="chr" data-title="ᎦᏐᏆᎸ" data-language-autonym="ᏣᎳᎩ" data-language-local-name="cherokee" class="interlanguage-link-target"><span>ᏣᎳᎩ</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 – kurdă centrală" lang="ckb" hreflang="ckb" data-title="گۆ" data-language-autonym="کوردی" data-language-local-name="kurdă centrală" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Sf%C3%A9ra_(matematika)" title="Sféra (matematika) – cehă" lang="cs" hreflang="cs" data-title="Sféra (matematika)" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – ciuvașă" lang="cv" hreflang="cv" data-title="Сфера" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Sff%C3%AAr" title="Sffêr – galeză" lang="cy" hreflang="cy" data-title="Sffêr" data-language-autonym="Cymraeg" data-language-local-name="galeză" 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 – daneză" lang="da" hreflang="da" data-title="Kugle" data-language-autonym="Dansk" data-language-local-name="daneză" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kugel" title="Kugel – germană" lang="de" hreflang="de" data-title="Kugel" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%86%CE%B1%CE%AF%CF%81%CE%B1" title="Σφαίρα – greacă" lang="el" hreflang="el" data-title="Σφαίρα" data-language-autonym="Ελληνικά" data-language-local-name="greacă" 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 – engleză" lang="en" hreflang="en" data-title="Sphere" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/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-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Esfera" title="Esfera – spaniolă" lang="es" hreflang="es" data-title="Esfera" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Sf%C3%A4%C3%A4r" title="Sfäär – estonă" lang="et" hreflang="et" data-title="Sfäär" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Esfera" title="Esfera – bască" lang="eu" hreflang="eu" data-title="Esfera" data-language-autonym="Euskara" data-language-local-name="bască" 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) – persană" lang="fa" hreflang="fa" data-title="کره (هندسه)" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pallo_(geometria)" title="Pallo (geometria) – finlandeză" lang="fi" hreflang="fi" data-title="Pallo (geometria)" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fj mw-list-item"><a href="https://fj.wikipedia.org/wiki/Vuravura_(Geometry)" title="Vuravura (Geometry) – fijiană" lang="fj" hreflang="fj" data-title="Vuravura (Geometry)" data-language-autonym="Na Vosa Vakaviti" data-language-local-name="fijiană" class="interlanguage-link-target"><span>Na Vosa Vakaviti</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sph%C3%A8re" title="Sphère – franceză" lang="fr" hreflang="fr" data-title="Sphère" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Kuugel" title="Kuugel – frizonă nordică" lang="frr" hreflang="frr" data-title="Kuugel" data-language-autonym="Nordfriisk" data-language-local-name="frizonă nordică" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sf%C3%A9ar" title="Sféar – irlandeză" lang="ga" hreflang="ga" data-title="Sféar" data-language-autonym="Gaeilge" data-language-local-name="irlandeză" class="interlanguage-link-target"><span>Gaeilge</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 – chineză gan" lang="gan" hreflang="gan" data-title="球面" data-language-autonym="贛語" data-language-local-name="chineză gan" class="interlanguage-link-target"><span>贛語</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-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Cruinne" title="Cruinne – gaelică scoțiană" lang="gd" hreflang="gd" data-title="Cruinne" data-language-autonym="Gàidhlig" data-language-local-name="gaelică scoțiană" 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 – galiciană" lang="gl" hreflang="gl" data-title="Esfera" data-language-autonym="Galego" data-language-local-name="galiciană" class="interlanguage-link-target"><span>Galego</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 – gujarati" lang="gu" hreflang="gu" data-title="ગોળો" data-language-autonym="ગુજરાતી" data-language-local-name="gujarati" class="interlanguage-link-target"><span>ગુજરાતી</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) – ebraică" lang="he" hreflang="he" data-title="ספירה (גאומטריה)" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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 – croată" lang="hr" hreflang="hr" data-title="Sfera" data-language-autonym="Hrvatski" data-language-local-name="croată" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Esf%C3%A8" title="Esfè – haitiană" lang="ht" hreflang="ht" data-title="Esfè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitiană" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6mb" title="Gömb – maghiară" lang="hu" hreflang="hu" data-title="Gömb" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bola_(geometri)" title="Bola (geometri) – indoneziană" lang="id" hreflang="id" data-title="Bola (geometri)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/K%C3%BAla" title="Kúla – islandeză" lang="is" hreflang="is" data-title="Kúla" data-language-autonym="Íslenska" data-language-local-name="islandeză" 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 – italiană" lang="it" hreflang="it" data-title="Sfera" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – japoneză" lang="ja" hreflang="ja" data-title="球面" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Sfier" title="Sfier – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Sfier" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target"><span>Patois</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 – georgiană" lang="ka" hreflang="ka" data-title="სფერო" data-language-autonym="ქართული" data-language-local-name="georgiană" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tasegla" title="Tasegla – kabyle" lang="kab" hreflang="kab" data-title="Tasegla" data-language-autonym="Taqbaylit" data-language-local-name="kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – kazahă" lang="kk" hreflang="kk" data-title="Сфера" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</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-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) – coreeană" lang="ko" hreflang="ko" data-title="구 (기하학)" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Sphaera" title="Sphaera – latină" lang="la" hreflang="la" data-title="Sphaera" data-language-autonym="Latina" data-language-local-name="latină" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Sfera" title="Sfera – lituaniană" lang="lt" hreflang="lt" data-title="Sfera" data-language-autonym="Lietuvių" data-language-local-name="lituaniană" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Sf%C4%93ra" title="Sfēra – letonă" lang="lv" hreflang="lv" data-title="Sfēra" data-language-autonym="Latviešu" data-language-local-name="letonă" class="interlanguage-link-target"><span>Latviešu</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 – moksha" lang="mdf" hreflang="mdf" data-title="Сферась" data-language-autonym="Мокшень" data-language-local-name="moksha" class="interlanguage-link-target"><span>Мокшень</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) – malgașă" lang="mg" hreflang="mg" data-title="Bola (jeômetria)" data-language-autonym="Malagasy" data-language-local-name="malgașă" class="interlanguage-link-target"><span>Malagasy</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 – macedoneană" lang="mk" hreflang="mk" data-title="Сфера" data-language-autonym="Македонски" data-language-local-name="macedoneană" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8B%E0%B4%B3%E0%B4%82" title="ഗോളം – malayalam" lang="ml" hreflang="ml" data-title="ഗോളം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</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 – mongolă" lang="mn" hreflang="mn" data-title="Бөмбөлөг" data-language-autonym="Монгол" data-language-local-name="mongolă" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sfera" title="Sfera – malaeză" lang="ms" hreflang="ms" data-title="Sfera" data-language-autonym="Bahasa Melayu" data-language-local-name="malaeză" class="interlanguage-link-target"><span>Bahasa Melayu</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 – birmană" lang="my" hreflang="my" data-title="စက်လုံး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmană" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Sfeer_(wiskunde)" title="Sfeer (wiskunde) – neerlandeză" lang="nl" hreflang="nl" data-title="Sfeer (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kule" title="Kule – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Kule" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kule" title="Kule – norvegiană bokmål" lang="nb" hreflang="nb" data-title="Kule" data-language-autonym="Norsk bokmål" data-language-local-name="norvegiană bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Esf%C3%A8ra" title="Esfèra – occitană" lang="oc" hreflang="oc" data-title="Esfèra" data-language-autonym="Occitan" data-language-local-name="occitană" 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-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 – punjabi" lang="pa" hreflang="pa" data-title="ਗੋਲ਼ਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Sfera" title="Sfera – poloneză" lang="pl" hreflang="pl" data-title="Sfera" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sfera" title="Sfera – Piedmontese" lang="pms" hreflang="pms" data-title="Sfera" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Esfera" title="Esfera – portugheză" lang="pt" hreflang="pt" data-title="Esfera" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Lunq%27u" title="Lunq'u – quechua" lang="qu" hreflang="qu" data-title="Lunq'u" data-language-autonym="Runa Simi" data-language-local-name="quechua" class="interlanguage-link-target"><span>Runa Simi</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><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 – rusă" lang="ru" hreflang="ru" data-title="Сфера" data-language-autonym="Русский" data-language-local-name="rusă" 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 – sakha" lang="sah" hreflang="sah" data-title="Сфера" data-language-autonym="Саха тыла" data-language-local-name="sakha" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Sfera" title="Sfera – siciliană" lang="scn" hreflang="scn" data-title="Sfera" data-language-autonym="Sicilianu" data-language-local-name="siciliană" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Sfera" title="Sfera – sârbo-croată" lang="sh" hreflang="sh" data-title="Sfera" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="sârbo-croată" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</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 – singhaleză" lang="si" hreflang="si" data-title="ගෝලය" data-language-autonym="සිංහල" data-language-local-name="singhaleză" 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) – slovacă" lang="sk" hreflang="sk" data-title="Guľa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovacă" 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 – slovenă" lang="sl" hreflang="sl" data-title="Sfera" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/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-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Kubad" title="Kubad – somaleză" lang="so" hreflang="so" data-title="Kubad" data-language-autonym="Soomaaliga" data-language-local-name="somaleză" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sfera" title="Sfera – albaneză" lang="sq" hreflang="sq" data-title="Sfera" data-language-autonym="Shqip" data-language-local-name="albaneză" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%84%D0%B5%D1%80%D0%B0" title="Сфера – sârbă" lang="sr" hreflang="sr" data-title="Сфера" data-language-autonym="Српски / srpski" data-language-local-name="sârbă" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Buleudan" title="Buleudan – sundaneză" lang="su" hreflang="su" data-title="Buleudan" data-language-autonym="Sunda" data-language-local-name="sundaneză" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sf%C3%A4r" title="Sfär – suedeză" lang="sv" hreflang="sv" data-title="Sfär" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Tufe" title="Tufe – swahili" lang="sw" hreflang="sw" data-title="Tufe" data-language-autonym="Kiswahili" data-language-local-name="swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%8B%E0%AE%B3%E0%AE%AE%E0%AF%8D" title="கோளம் – tamilă" lang="ta" hreflang="ta" data-title="கோளம்" data-language-autonym="தமிழ்" data-language-local-name="tamilă" class="interlanguage-link-target"><span>தமிழ்</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 – thailandeză" lang="th" hreflang="th" data-title="ทรงกลม" data-language-autonym="ไทย" data-language-local-name="thailandeză" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Espera" title="Espera – tagalog" lang="tl" hreflang="tl" data-title="Espera" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCre" title="Küre – turcă" lang="tr" hreflang="tr" data-title="Küre" data-language-autonym="Türkçe" data-language-local-name="turcă" class="interlanguage-link-target"><span>Türkçe</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 – tătară" lang="tt" hreflang="tt" data-title="Сфера" data-language-autonym="Татарча / tatarça" data-language-local-name="tătară" class="interlanguage-link-target"><span>Татарча / tatarça</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 – ucraineană" lang="uk" hreflang="uk" data-title="Сфера" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sfera" title="Sfera – uzbecă" lang="uz" hreflang="uz" data-title="Sfera" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbecă" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%B7t_c%E1%BA%A7u" title="Mặt cầu – vietnameză" lang="vi" hreflang="vi" data-title="Mặt cầu" data-language-autonym="Tiếng Việt" data-language-local-name="vietnameză" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Espira" title="Espira – waray" lang="war" hreflang="war" data-title="Espira" data-language-autonym="Winaray" data-language-local-name="waray" 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 – chineză wu" lang="wuu" hreflang="wuu" data-title="球面" data-language-autonym="吴语" data-language-local-name="chineză wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%83%E9%9D%A2" title="球面 – chineză" lang="zh" hreflang="zh" data-title="球面" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%90%83" title="球 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="球" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ki%C3%BB-b%C4%ABn" title="Kiû-bīn – chineză min nan" lang="nan" hreflang="nan" data-title="Kiû-bīn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="chineză min nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%90%83%E9%AB%94" title="球體 – cantoneză" lang="yue" hreflang="yue" data-title="球體" data-language-autonym="粵語" data-language-local-name="cantoneză" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12507#sitelinks-wikipedia" title="Modifică legăturile interlinguale" class="wbc-editpage">Modifică legăturile</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Spații de nume"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sfer%C4%83" title="Vedeți conținutul paginii [a]" accesskey="a"><span>Articol</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Discu%C8%9Bie:Sfer%C4%83&action=edit&redlink=1" rel="discussion" class="new" title="Discuții despre această pagină — pagină inexistentă [t]" accesskey="t"><span>Discuție</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">română</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vizualizări"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sfer%C4%83"><span>Lectură</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&action=history" title="Versiunile anterioare ale paginii și autorii lor. [h]" accesskey="h"><span>Istoric</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Unelte" > <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">Unelte</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">Unelte</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ascunde</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Mai multe opțiuni" > <div class="vector-menu-heading"> Acțiuni </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/Sfer%C4%83"><span>Lectură</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&veaction=edit" title="Modificați această pagină cu EditorulVizual [v]" accesskey="v"><span>Modificare</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&action=edit" title="Modificați codul sursă al acestei pagini [e]" accesskey="e"><span>Modificare sursă</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&action=history"><span>Istoric</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:Ce_se_leag%C4%83_aici/Sfer%C4%83" title="Lista tuturor paginilor wiki care conduc spre această pagină [j]" accesskey="j"><span>Ce trimite aici</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:Modific%C4%83ri_corelate/Sfer%C4%83" rel="nofollow" title="Schimbări recente în legătură cu această pagină [k]" accesskey="k"><span>Schimbări corelate</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:Trimite_fi%C8%99ier" title="Încărcare fișiere [u]" accesskey="u"><span>Trimite fișier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:Pagini_speciale" title="Lista tuturor paginilor speciale [q]" accesskey="q"><span>Pagini speciale</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&oldid=16312988" title="Legătură permanentă către această versiune a acestei pagini"><span>Legătură permanentă</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&action=info" title="Mai multe informații despre această pagină"><span>Informații despre pagină</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:Citeaz%C4%83&page=Sfer%C4%83&id=16312988&wpFormIdentifier=titleform" title="Informații cu privire la modul de citare a acestei pagini"><span>Citează acest articol</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fro.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DSfer%25C4%2583%26oldid%3D16312988"><span>Obține URL scurtat</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fro.wikipedia.org%2Fw%2Findex.php%3Ftitle%3DSfer%25C4%2583%26oldid%3D16312988"><span>Descărcați codul 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"> Tipărire/exportare </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=Special:Carte&bookcmd=book_creator&referer=Sfer%C4%83"><span>Creare carte</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Sfer%C4%83&action=show-download-screen"><span>Descărcare ca PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Sfer%C4%83&printable=yes" title="Versiunea de tipărit a acestei pagini [p]" accesskey="p"><span>Versiune de tipărit</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"> În alte proiecte </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="Legătură către elementul asociat din depozitul de date [g]" accesskey="g"><span>Element Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspect"> <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">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="cdx-message cdx-message--block cdx-message--warning mw-revision"><span class="cdx-message__icon"></span><div class="cdx-message__content"><div id="mw-revision-info"><div id="viewingold-warning" class="plainlinks" style="background: #FFC; border: 1px solid #C88; color: #000000; font-weight: bold; margin: 2em 0 .5em; padding: .5em 1em; vertical-align: middle; clear: both;">Aceasta este o <a class="external text" href="https://ro.wikipedia.org/w/index.php?title=Sfer%C4%83&action=history">versiune arhivată</a> a paginii, în urma modificării de către <span id="mw-revision-name"><a href="/wiki/Utilizator:Turbojet" class="mw-userlink" title="Utilizator:Turbojet" data-mw-revid="16312988"><bdi>Turbojet</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/Discu%C8%9Bie_Utilizator:Turbojet" class="mw-usertoollinks-talk" title="Discuție Utilizator:Turbojet">discuție</a> | <a href="/wiki/Special:Contribu%C8%9Bii/Turbojet" class="mw-usertoollinks-contribs" title="Special:Contribuții/Turbojet">contribuții</a>)</span></span> la <span id="mw-revision-date">30 mai 2024 11:22</span>. Diferențele față de <a class="external text" href="https://ro.wikipedia.org/wiki/Sfer%C4%83">versiunea curentă</a> pot fi semnificative.</div> <div id="viewingold-plain" style="display:none;">Versiune creată de <a href="/wiki/Utilizator:Turbojet" class="mw-userlink" title="Utilizator:Turbojet" data-mw-revid="16312988"><bdi>Turbojet</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/Discu%C8%9Bie_Utilizator:Turbojet" class="mw-usertoollinks-talk" title="Discuție Utilizator:Turbojet">discuție</a> | <a href="/wiki/Special:Contribu%C8%9Bii/Turbojet" class="mw-usertoollinks-contribs" title="Special:Contribuții/Turbojet">contribuții</a>)</span> la 30 mai 2024 11:22</div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=Sfer%C4%83&diff=prev&oldid=16312988" title="Sferă">dif</a>) <a href="/w/index.php?title=Sfer%C4%83&direction=prev&oldid=16312988" title="Sferă">← Versiunea anterioară</a> | afișează versiunea curentă (dif) | Versiunea următoare → (dif)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ro" dir="ltr"><div role="note" class="dezambiguizare">Pentru alte sensuri, vedeți <a href="/wiki/Sfer%C4%83_(dezambiguizare)" class="mw-disambig" title="Sferă (dezambiguizare)">Sferă (dezambiguizare)</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Sphere_and_Ball.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/250px-Sphere_and_Ball.png" decoding="async" width="250" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/375px-Sphere_and_Ball.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/Sphere_and_Ball.png/500px-Sphere_and_Ball.png 2x" data-file-width="1548" data-file-height="1536" /></a><figcaption>O sferă în care raza este notată „r”</figcaption></figure> <p><b>Sfera</b> (din <a href="/wiki/Greac%C4%83" class="mw-redirect" title="Greacă">greacă</a> <i>σφαίρα</i> - <i>sphaira</i>) este suprafața unei <a href="/wiki/Bil%C4%83" class="mw-disambig" title="Bilă">bile</a>. În spațiul euclidian 3-dimensional, sfera este mulțimea punctelor care se află la o distanță <i>r</i> (<a href="/wiki/Raz%C4%83" title="Rază">raza</a> sferei) de un punct <i>c</i> (<a href="/wiki/Centru_(geometrie)" title="Centru (geometrie)">centrul</a> sferei), unde <i>r</i> este un <a href="/wiki/Num%C4%83r_real" title="Număr real">număr real</a> pozitiv. În cazul particular în care <i>r</i>=1 sfera se numește <i><a href="/wiki/Sfer%C4%83_unitate" title="Sferă unitate">sferă unitate</a></i>. </p><p>În limbaj colocvial, noțiunea de sferă se folosește adesea pentru un corp geometric mărginit de sferă. În limbaj matematic un astfel de obiect se numește <a href="/wiki/Bil%C4%83" class="mw-disambig" title="Bilă">bilă</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ecuații_în_R3"><span id="Ecua.C8.9Bii_.C3.AEn_R3"></span>Ecuații în <b>R<sup>3</sup></b></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=1" title="Modifică secțiunea: Ecuații în R3" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=1" title="Edit section's source code: Ecuații în R3"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În <a href="/wiki/Geometrie_analitic%C4%83" title="Geometrie analitică">geometria analitică</a> sfera de centrul <i>c</i>=(<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>z</i><sub>0</sub>) și rază <i>r</i>>0 este <a href="/wiki/Loc_geometric" title="Loc geometric">locul geometric</a> al punctelor care satisfac ecuația (implicită) </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"> <mspace width="thinmathspace" /> <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/92e46913a082425988391f11c493162ce78bc708" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.314ex; height:3.176ex;" alt="{\displaystyle \,(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}"></span></dd></dl> <p>Dacă considerăm <a href="/wiki/Metric%C4%83" title="Metrică">metrica</a> euclidiană din <b>R<sup>3</sup></b> atunci ecuația de mai sus nu inseamnă altceva decât că toate punctele sferei se află la aceași distanță <i>r</i> de punctul <i>c</i>. </p><p>Considerând un sistem ortonormat de coordonate, sfera (ca suprafață 2-dimensională) poate fi exprimată prin ecuațiile parametrice </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=x_{0}+r\cos \varphi \;\sin \theta \\y=y_{0}+r\sin \varphi \;\sin \theta \qquad (0\leq \varphi \leq 2\pi {\text{ si }}0\leq \theta \leq \pi )\\z=z_{0}+r\cos \theta \end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <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>φ</mi> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <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>φ</mi> <mspace width="thickmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <mi>φ</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> si </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>θ</mi> <mo>≤</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>=</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</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=x_{0}+r\cos \varphi \;\sin \theta \\y=y_{0}+r\sin \varphi \;\sin \theta \qquad (0\leq \varphi \leq 2\pi {\text{ si }}0\leq \theta \leq \pi )\\z=z_{0}+r\cos \theta \end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f6c19faabf5654b5dee4194ee40abf7c289bfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:54.004ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}x=x_{0}+r\cos \varphi \;\sin \theta \\y=y_{0}+r\sin \varphi \;\sin \theta \qquad (0\leq \varphi \leq 2\pi {\text{ si }}0\leq \theta \leq \pi )\\z=z_{0}+r\cos \theta \end{cases}}}"></span></dd></dl> <p>Pentru fiecare <a href="/wiki/Valoare_(matematic%C4%83)" title="Valoare (matematică)">valoare</a> a parametrului θ se obține un cerc de pe sferă - astfel de cercuri se numesc <b>paralele</b>. Asemănător, pentru parametrul φ se obțin cercuri numite <b>meridiane</b>. Pentru θ=0 respectiv θ=π cercurile obținute sunt <a href="/wiki/Degenerare_(matematic%C4%83)" title="Degenerare (matematică)">degenerate</a> - aceste două puncte sunt <b>polul nord</b> (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>z</i><sub>0</sub> + <i>r</i>) respectiv <b>polul sud</b> (<i>x</i><sub>0</sub>, <i>y</i><sub>0</sub>, <i>z</i><sub>0</sub> - <i>r</i>). </p><p>Pentru o sferă cu raza <i>r</i>>0 <a href="/wiki/Arie" title="Arie">aria</a> suprafeței este </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 A=4\pi r^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi r^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70bc31016cef341b9728ccfb2e8b89adfe4081f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.826ex; height:2.676ex;" alt="{\displaystyle A=4\pi r^{2}\,}"></span></dd></dl> <p>iar <a href="/wiki/Volum" class="mw-redirect" title="Volum">volumul</a> este </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 V={\frac {4}{3}}\pi r^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {4}{3}}\pi r^{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c143ddb445538122a38612d2f43fe82eed286a97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.966ex; height:5.176ex;" alt="{\displaystyle V={\frac {4}{3}}\pi r^{3}.}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Proprietăți"><span id="Propriet.C4.83.C8.9Bi"></span>Proprietăți</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=2" title="Modifică secțiunea: Proprietăți" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=2" title="Edit section's source code: Proprietăți"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Prin secțiuni plane ale sferei se obțin cercuri</i> </p> <dl><dd>Dacă se consideră un plan <a href="/wiki/Tangent%C4%83_(geometrie)" title="Tangentă (geometrie)">tangent</a> la sferă se obține un cerc degenerat, adică un punct.</dd></dl> <p><i>Toate Geodezicele sferei sunt drumuri închise</i> </p> <dl><dd><a href="/wiki/Geodezic%C4%83" title="Geodezică">Geodezicele</a> sferei sunt <b>cercurile mari</b>, adică cercurile obținute din secțiuni cu plane care conțin centrul sferei.</dd></dl> <p><i>Dintre toate solidele cu un volum dat, sfera are cea mai mică arie a suprafeței</i> </p> <dl><dd>Pentru o arie dată, sfera de acea arie înconjoară cel mai mare volum.</dd></dl> <p><i>Sfera este invariată în grupul de rotații</i> </p> <dl><dd>Considerând o sferă cu centrul în origine, grupul de rotații <b>SO(3)</b> transformă sfera în ea însăși.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Generalizări"><span id="Generaliz.C4.83ri"></span>Generalizări</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=3" title="Modifică secțiunea: Generalizări" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=3" title="Edit section's source code: Generalizări"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Având în vedere <a href="/w/index.php?title=Spa%C8%9Biu_ambient&action=edit&redlink=1" class="new" title="Spațiu ambient — pagină inexistentă">spațiul ambient</a> al sferei, cât și noțiunea de distanță se pot obține următoarele generalizări </p> <ul><li>Sfera <b>S<sup>n</sup></b> din <a href="/wiki/Spa%C8%9Biu_euclidian" title="Spațiu euclidian">spațiul euclidian</a> (n+1)-dimensional <b>R<sup>n+1</sup></b> de centru <i>c</i>=(<i>c</i><sub>1</sub>, <i>c</i><sub>2</sub>,..., <i>c</i><sub>n+1</sub>) și rază <i>r</i> este mulțimea punctelor din <b>R<sup>n+1</sup></b> care satisfac ecuația</li></ul> <dl><dd><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_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}+...+(x_{n+1}-c_{n+1})^{2}=r^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</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_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}+...+(x_{n+1}-c_{n+1})^{2}=r^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af1de0fd5a0c708cfda15b195bd3dbe882512f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.77ex; height:3.176ex;" alt="{\displaystyle \,(x_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}+...+(x_{n+1}-c_{n+1})^{2}=r^{2}.}"></span>.</dd></dl></dd></dl> <ul><li>Într-un <a href="/wiki/Spa%C8%9Biu_metric" title="Spațiu metric">spațiu metric</a> oarecare (X,d) sfera de centru <i>c</i> și rază <i>r</i> este</li></ul> <dl><dd><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\in X:d(x,c)=r\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>X</mi> <mo>:</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\in X:d(x,c)=r\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94f2cb206b00d4a8c7073b5675f9e2d41a75ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.955ex; height:2.843ex;" alt="{\displaystyle \{x\in X:d(x,c)=r\}}"></span>.</dd></dl></dd></dl> <ul><li>Într-un <a href="/wiki/Spa%C8%9Biu_topologic" title="Spațiu topologic">spațiu topologic</a> oarecare, o <b>n-sferă</b> este o submulțime a spațiului <a href="/w/index.php?title=Homeomorfism&action=edit&redlink=1" class="new" title="Homeomorfism — pagină inexistentă">homeomorfă</a> cu <b>S<sup>n</sup></b> pentru un număr natural <i>n</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Topologie">Topologie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=4" title="Modifică secțiunea: Topologie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=4" title="Edit section's source code: Topologie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Topologia este o ramură a matematicii, mai precis o extensie a geometriei, care studiază deformările spațiului prin transformări continue.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><style data-mw-deduplicate="TemplateStyles:r16584850">.mw-parser-output .citationNeeded{background-color:#ffeaea;color:#444444}</style> Vom considera spațiul euclidian 3-dimensional, notat cu E<sub>3</sub>. </p> <div class="mw-heading mw-heading3"><h3 id="Sfera_din_punct_de_vedere_geometric_si_topologic">Sfera din punct de vedere geometric si topologic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=5" title="Modifică secțiunea: Sfera din punct de vedere geometric si topologic" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=5" title="Edit section's source code: Sfera din punct de vedere geometric si topologic"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Definiție</b>:Fie O є E<sub>3</sub> și r є R.Se numește <b>sfera cu centrul</b> O și <b>raza</b> r figura S(O,r):= {M є E<sub>3</sub> / δ(O;M)=r};<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Se numește <b>corpul (discul) sferic</b> sau <b>bila</b> cu centrul O și raza r, figura B(O,r):= {M є E<sub>3</sub> / δ(O;M)≤r};<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Se numește <b>interiorul corpului sferic</b> B(O,r), figura (B(O,r)):= {M є E<sub>3</sub> / δ(O;M)<r};<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Se numește <b>exteriorul corpului sferic</b> B(O,r), figura (B(O,r)):= {M є E<sub>3</sub> / δ(O;M)>r};<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Orice sferă din S(O,r) din E<sub>3</sub> este o figură nevidă; fiecare semidreaptă [OX conține exact un punct al lui S(O,r), iar o dreaptă care conține centrul O (<b><a href="/wiki/Normal%C4%83" title="Normală">normala</a>, dreapta diametrala</b>) intersectează sfera S(O,r) în doua puncte (diametral opuse). Sfera nu este o figură convexa, corpul sferic și interiorul său sunt figuri convexe.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Dacă S(O,r) este o sferă și α є<i>P</i> este un <b>plan diametral</b> sau <b>normal</b> al lui S(O,r), (O є α), atunci S(O,r) intersectat cu α=:С(O;r) este un cerc, numit <b>cerc mare</b>(ecuator) al lui S(O,r).<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Observație: 1.Fie A,B є S(O,r) și α Є <i>P</i>. Două din condițiile următoare implică pe cea de-a treia:<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p> <ul><li>1. α este perpendicular pe coarda [AB];<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>2. α este plan diametral;<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>3. α conține mijlocul lui [AB];<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li></ul> <p>Orice dreaptă diametrală (respectiv plan diametral) este o <a href="/wiki/Ax%C4%83_de_simetrie" class="mw-redirect" title="Axă de simetrie">axă de simetrie</a> (respectiv plan de simetrie) a sferei S(O;r).<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Exista trei poziții relative posibile ale unui cuplu sferă-dreaptă.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Fie sfera S(O;r) și dreapta d Є <i>D</i>. d se numește <b><a href="/wiki/Tangent%C4%83_(geometrie)" title="Tangentă (geometrie)">tangenta</a></b>, respectiv <b>secanta</b>, respectiv <b>exterioara</b> la <i>C</i>(O;r), dacă d intersectează <i>C</i>(O;r) conține un punct, respectiv conține doua puncte, respectiv este mulțimea vidă.<br /><span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Teorema 1.(Sferă-dreaptă).Fie sfera S(O;r) și dreapta d Є <i>D</i>.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p> <ol><li>d este secanta la S(O;r)<=> δ(O,d)< r<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>d este exterioara la S(O;r)<=> δ(O,d)> r<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>d este tangenta la S(O;r)<=> δ(O,d)= r<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li></ol> <p>Observații 2. </p> <dl><dd>a) O tangentă la sferă este perpendiculară pe baza sferei în punctul de contact.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></dd> <dd>b) O dreaptă este tangentă într-un punct la sferă <a href="/wiki/Dac%C4%83_%C8%99i_numai_dac%C4%83" title="Dacă și numai dacă">dacă și numai dacă</a> ea este tangentă la un <a href="/wiki/Cerc_mare" title="Cerc mare">cerc mare</a> al sferei, în punctul respectiv.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></dd> <dd>c) Fiecare punct al sferei este centrul unui fascicul de drepte <a href="/wiki/Coplanaritate" title="Coplanaritate">coplanare</a>, care sunt tangente la sfera în acel punct.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></dd></dl> <p>Exista, de asemenea, trei poziții posibile ale unui plan în raport cu o sferă.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Fie sfera S(o,r) și planul α Є <i>P</i>, α se numește <b>plan tangent</b>, respectiv <b>plan secant</b>, respectiv <b>plan exterior</b> la S(O,r), daca α intersectat cu S(O,r) este un punct, respectiv un cerc, respectiv mulțimea vidă.<br /><span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Teorema 2.(Sferă-plan).Fie sfera S(O,r) și planul α Є <i>P</i>. </p> <ol><li>α este un plan secant la S(O,r) <=> δ(O,α)< r;<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>α este un plan exterior la S(O,r) <=> δ(O,α)> r;<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li> <li>α este un plan tangent la S(O,r) <=> δ(O,α)= r;<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"></li></ol> <p>Observație 3. In fiecare punct al sferei exista un plan tangent unic la sferă; acesta conține toate tangentele la sferă în punctul respectiv. Perpendiculara pe planul tangent la sfera în punctul de contact este <b>normala sferei în punctul de contact.</b><span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Observație 4. Dacă o sferă conține trei puncte, atunci ea conține cercul determinat de aceste puncte.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Într-adevăr, dacă punctele aparțin sferei, atunci ele sunt necoliniare și determină un plan care intersectează sfera după cercul determinat de cele trei puncte.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Iată câteva moduri în care poate fi determinata o sfera.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Observație 5. Date trei puncte necoliniare, A,B,C, locul geometric al centrelor sferelor care conțin pe A,B,C este perpendiculara pe planul <b>ABC</b> în punctul de intersecție al mediatoarelor triunghiului ABC.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Teorema 3. Locul geometric al centrelor sferelor care conțin un cerc dat este normala pe planul cercului în centrul acestuia.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Teorema 4. Doua cercuri necoplanare, care se intersectează, determina o sferă unică.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Corolar 1. Un cerc și un punct exterior planului său determină o sferă unică.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Corolar 2. Exista o sferă unică, care conține patru puncte necoplanare date.<br /><span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Spațiul euclidian E<sub>3</sub> este un spațiu metric, cu metrica (distanța) δ : E<sub>3</sub> X E<sub>3</sub> -> R , introdusă prin axiomatica geometriei euclidiene în spațiu.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> Proprietățile distanței, precum și maniera în care poate fi calculată au fost stabilite ulterior prin: axioma riglei, existenta sistemelor de coordinate carteziene ortogonale în plan și în spațiu, teorema lui Pitagora.<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p><p>Daca E<sub>3</sub> este raportat la un s.c.c.o OXYZ și S(O,r)={ Mє E<sub>3</sub> / δ(O,M)=r} este sfera cu centrul O și de raza r > 0, atunci se poate considera S(O,r) ca o suprafață în spațiul euclidian. O parametrizare a lui S(O,r) poate fi definită prin relațiile: </p><p>(u,v) Є ( -Π/2;Π/2 X [0,2Π) </p><p>care se numesc ecuațiile parametrice ale sferei S(O,r).<span class="citationNeeded skin-invert"></span><sup class="noprint"><span style="color:red;">[<a href="/wiki/Wikipedia:Citarea_surselor" title="Wikipedia:Citarea surselor"><i>necesită citare</i></a>]</span></sup><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16584850"> </p> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sfer%C4%83&veaction=edit&section=6" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sfer%C4%83&action=edit&section=6" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="metadata plainlinks mbox-small" style="padding:0.25em 0.5em 0.5em 0.75em;border:1px solid #aaa; color:var(--color-base, #000);"> <tbody><tr style="height:25px;"> <td colspan="2" style="padding-bottom:0.5em;border-bottom:1px solid #aaa;margin:auto;text-align:center;">Puteți găsi mai multe informații despre <b>Sferă</b> prin căutarea în proiectele similare ale Wikipediei, grupate sub denumirea generică de <a href="https://en.wikipedia.org/wiki/Wikipedia:Sister_projects" class="extiw" title="en:Wikipedia:Sister projects">„proiecte surori”</a>: </td></tr> <tr style="height:25px;"> <td style="padding-top:0.75em;"><span typeof="mw:File"><a href="https://ro.wiktionary.org/wiki/Special:Search/Sfer%C4%83" title="Search Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/23px-Wiktionary-logo-en.svg.png" decoding="async" width="23" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/35px-Wiktionary-logo-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Wiktionary-logo-en.svg/46px-Wiktionary-logo-en.svg.png 2x" data-file-width="1000" data-file-height="1089" /></a></span> </td> <td style="padding-top:0.75em;"><a href="https://ro.wiktionary.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="wikt:Special:Search/Sferă">Definiții și traduceri</a> în Wikționar </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/Special:Search/Sfer%C4%83" title="Search Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/28px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/37px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> </td> <td><a href="https://commons.wikimedia.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="commons:Special:Search/Sferă">Imagini și media</a> la Commons </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="https://ro.wikiquote.org/wiki/Special:Search/Sfer%C4%83" title="Search Wikiquote"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/21px-Wikiquote-logo.svg.png" decoding="async" width="21" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/32px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/42px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span> </td> <td><a href="https://ro.wikiquote.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="q:Special:Search/Sferă">Citate</a> la Wikicitat </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="https://ro.wikisource.org/wiki/Special:Search/Sfer%C4%83" title="Search Wikisource"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/24px-Wikisource-logo.svg.png" decoding="async" width="24" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/36px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/48px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></a></span> </td> <td><a href="https://ro.wikisource.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="s:Special:Search/Sferă">Texte sursă</a> la Wikisursă </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="https://ro.wikibooks.org/wiki/Special:Search/Sfer%C4%83" title="Search Wikibooks"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/25px-Wikibooks-logo.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/38px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/50px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> </td> <td><a href="https://ro.wikibooks.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="b:Special:Search/Sferă">Manuale</a> la Wikimanuale </td></tr> <tr style="height:25px;"> <td><span typeof="mw:File"><a href="https://en.wikiversity.org/wiki/Special:Search/Sfer%C4%83" title="Search Wikiversity"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/25px-Wikiversity-logo-en.svg.png" decoding="async" width="25" height="23" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/38px-Wikiversity-logo-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Wikiversity-logo-en.svg/50px-Wikiversity-logo-en.svg.png 2x" data-file-width="1000" data-file-height="900" /></a></span> </td> <td><a href="https://en.wikiversity.org/wiki/Special:Search/Sfer%C4%83" class="extiw" title="wikiversity:Special:Search/Sferă">Resurse de studiu</a> la Wikiversitate </td></tr> </tbody></table> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Sphere.html">Wolfram MathWorld</a></li> <li><a rel="nofollow" class="external text" href="https://www.descopera.ro/stiinta/929461-hronicul-si-cantecul-sferelor">Hronicul si cantecul sferelor</a>, 9 august 2007, Andreea Zaporojanu, <i>Descoperă</i></li></ul> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist 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