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Plocha – Wikipedie
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<main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Plocha</span></h1> <div class="tagline"> 2-dimenzionální oblast </div> </div> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"><a role="button" href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#p-lang" data-mw="interface" data-event-name="menu.languages" title="Jazyk" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> <span class="minerva-icon minerva-icon--language"></span> <span>Jazyk</span> </a></li> <li id="page-actions-watch" class="page-actions-menu__list-item"><a role="button" id="ca-watch" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Speci%C3%A1ln%C3%AD:P%C5%99ihl%C3%A1sit&returnto=Plocha&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> <span class="minerva-icon minerva-icon--star"></span> <span>Sledovat</span> </a></li> <li id="page-actions-edit" class="page-actions-menu__list-item"><a role="button" id="ca-edit" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Editovat</span> </a></li> </ul> </nav><!-- version 1.0.2 (change every time you update a partial) --> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <div class="uvodni-upozorneni hatnote noprint"> Tento článek je o geometrické ploše. O pracovní ploše v počítači pojednává článek <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Desktopov%C3%A9_prost%C5%99ed%C3%AD?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Desktopové prostředí">Desktopové prostředí</a>. </div> <p><b>Plocha</b> označuje v matematice a fyzice dvojrozměrný <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Geometrick%C3%BD_%C3%BAtvar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometrický útvar">geometrický útvar</a>. Příkladem ploch jsou <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Rovina?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rovina">rovina</a>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Sf%C3%A9ra_(matematika)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Sféra (matematika)">kulová plocha</a>, povrch <a href="https://cs-m-wikipedia-org.translate.goog/wiki/V%C3%A1lec?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Válec">válce</a> nebo <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ku%C5%BEelov%C3%A1_plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kuželová plocha">kuželová plocha</a>. Přesné matematické definice se v různých kontextech a v různých teoriích liší.</p> <figure class="mw-default-size" typeof="mw:File/Thumb"> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Soubor:Surface_generated_by_transformation_(rotation_%2B_scaling)_of_a_segment.png?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Surface_generated_by_transformation_%28rotation_%2B_scaling%29_of_a_segment.png/220px-Surface_generated_by_transformation_%28rotation_%2B_scaling%29_of_a_segment.png" decoding="async" width="220" height="115" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Surface_generated_by_transformation_%2528rotation_%252B_scaling%2529_of_a_segment.png/330px-Surface_generated_by_transformation_%2528rotation_%252B_scaling%2529_of_a_segment.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Surface_generated_by_transformation_%2528rotation_%252B_scaling%2529_of_a_segment.png/440px-Surface_generated_by_transformation_%2528rotation_%252B_scaling%2529_of_a_segment.png 2x" data-file-width="1658" data-file-height="869"></a> <figcaption></figcaption> </figure> <p>Výraz plocha se někdy nesprávně používá nejen pro označení <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Geometrick%C3%BD_%C3%BAtvar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometrický útvar">geometrického útvaru</a>, ale také pro označení <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Obsah?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Obsah">obsahu</a> geometrického <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Geometrick%C3%BD_%C3%BAtvar?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometrický útvar">tělesa</a>.</p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="cs" dir="ltr"> <h2 id="mw-toc-heading">Obsah</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Plochy_v_euklidovsk%C3%A9m_prostoru"><span class="tocnumber">1</span> <span class="toctext">Plochy v euklidovském prostoru</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Implicitn%C3%AD_rovnice_plochy"><span class="tocnumber">1.1</span> <span class="toctext">Implicitní rovnice plochy</span></a></li> <li class="toclevel-2 tocsection-3"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Parametrick%C3%A9_rovnice"><span class="tocnumber">1.2</span> <span class="toctext">Parametrické rovnice</span></a></li> <li class="toclevel-2 tocsection-4"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Explicitn%C3%AD_rovnice_plochy"><span class="tocnumber">1.3</span> <span class="toctext">Explicitní rovnice plochy</span></a></li> </ul></li> <li class="toclevel-1 tocsection-5"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Z%C3%A1kladn%C3%AD_rovnice_plochy"><span class="tocnumber">2</span> <span class="toctext">Základní rovnice plochy</span></a> <ul> <li class="toclevel-2 tocsection-6"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Weingartenovy_rovnice_plochy"><span class="tocnumber">2.1</span> <span class="toctext">Weingartenovy rovnice plochy</span></a></li> <li class="toclevel-2 tocsection-7"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Gaussovy_rovnice_plochy"><span class="tocnumber">2.2</span> <span class="toctext">Gaussovy rovnice plochy</span></a></li> <li class="toclevel-2 tocsection-8"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Codazziho_rovnice_plochy"><span class="tocnumber">2.3</span> <span class="toctext">Codazziho rovnice plochy</span></a></li> </ul></li> <li class="toclevel-1 tocsection-9"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Vlastnosti"><span class="tocnumber">3</span> <span class="toctext">Vlastnosti</span></a></li> <li class="toclevel-1 tocsection-10"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Souvisej%C3%ADc%C3%AD_%C4%8Dl%C3%A1nky"><span class="tocnumber">4</span> <span class="toctext">Související články</span></a></li> <li class="toclevel-1 tocsection-11"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Extern%C3%AD_odkazy"><span class="tocnumber">5</span> <span class="toctext">Externí odkazy</span></a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Plochy_v_euklidovském_prostoru"><span id="Plochy_v_euklidovsk.C3.A9m_prostoru"></span>Plochy v euklidovském prostoru</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Plochy v euklidovském prostoru" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>V dalším předpokládejme, že plocha je podmnožina třírozměrného <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Eukleidovsk%C3%BD_prostor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Eukleidovský prostor">euklidovského prostoru</a>. Můžeme ji definovat jako <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Mno%C5%BEina?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Množina">množinu</a> všech <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Bod?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bod">bodů</a>, jejichž <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Soustava_sou%C5%99adnic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Soustava souřadnic">souřadnice</a> vyhovují <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Rovnice?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rovnice">rovnici</a></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y,z)=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.452ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.452ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" data-alt="{\displaystyle F(x,y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, </dd> </dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> je <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Funkce_(matematika)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Funkce (matematika)">funkce</a>, která má v každém bodě <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Spojitost?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Spojitost">spojitou</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Parci%C3%A1ln%C3%AD_derivace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parciální derivace">parciální derivaci</a> alespoň prvního řádu a na žádné otevřené množině není identicky rovna nule.</p> <p>Body plochy, v nichž je alespoň jedna z těchto parciálních derivací nenulová, se nazývají <i><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Regul%C3%A1rn%C3%AD_bod&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Regulární bod (stránka neexistuje)">regulární body</a></i> plochy, zatímco body, v nichž jsou všechny parciální derivace prvního řádu <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Nula?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nula">nulové</a> označujeme jako <i><a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Singul%C3%A1rn%C3%AD_bod&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Singulární bod (stránka neexistuje)">singulární body</a></i>. Příkladem singulárního bodu je např. vrchol <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ku%C5%BEel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kužel">kužele</a>.</p> <p>Singulární bod, v němž funkce <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"> </noscript><span class="lazy-image-placeholder" style="width: 1.741ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" data-alt="{\displaystyle F}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> má alespoň jednu nenulovou parciální derivaci druhého řádu, se nazývá <i>kónický bod</i> plochy.</p> <p>Plocha určená svojí <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Norm%C3%A1la_plochy?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Normála plochy">normálou</a> se označuje jako <b>orientovaná plocha</b>.</p> <p>Rovnici plochy lze vyjádřit v různých tvarech.</p> <div class="mw-heading mw-heading3"> <h3 id="Implicitní_rovnice_plochy"><span id="Implicitn.C3.AD_rovnice_plochy"></span>Implicitní rovnice plochy</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Implicitní rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Implicitní rovnice plochy má tvar</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x,y,z)=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> F </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo> , </mo> <mi> z </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle F(x,y,z)=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.452ex; height:2.843ex;" alt="{\displaystyle F(x,y,z)=0}"> </noscript><span class="lazy-image-placeholder" style="width: 13.452ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0796f292c93e738b5c7c46a9aea6023ceb2d8ec7" data-alt="{\displaystyle F(x,y,z)=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <div class="mw-heading mw-heading3"> <h3 id="Parametrické_rovnice"><span id="Parametrick.C3.A9_rovnice"></span>Parametrické rovnice</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Parametrické rovnice" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Uvažujme plochu, jejíž souřadnice jsou vyjádřeny <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Soustava_rovnic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Soustava rovnic">soustavou rovnic</a></p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> x </mi> <mo> = </mo> <mi> x </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle x=x(u,v)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.058ex; height:2.843ex;" alt="{\displaystyle x=x(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 11.058ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdd605515e00cc0a45af42ba3df16046defac9b" data-alt="{\displaystyle x=x(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <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 y=y(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> y </mi> <mo> = </mo> <mi> y </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle y=y(u,v)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.71ex; height:2.843ex;" alt="{\displaystyle y=y(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.71ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc932826f9f3a969aee6da3cf579aad8966ffb8" data-alt="{\displaystyle y=y(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <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 z=z(u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> z </mi> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=z(u,v)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.575ex; height:2.843ex;" alt="{\displaystyle z=z(u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.575ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e263aebd2eeba9e55fd1072eb87e251125d01faf" data-alt="{\displaystyle z=z(u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Tato soustava rovnic představuje <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Parametrick%C3%A1_funkce?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Parametrická funkce">parametrické</a> vyjádření plochy, přičemž <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u,v} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}"> </noscript><span class="lazy-image-placeholder" style="width: 3.491ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" data-alt="{\displaystyle u,v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou parametry plochy. Každou dvojici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u,v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> <mo> , </mo> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u,v} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle u,v}"> </noscript><span class="lazy-image-placeholder" style="width: 3.491ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66f4b32a0181923cc1337a5634f38241e5c697" data-alt="{\displaystyle u,v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> z určitého oboru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Ω<!-- Ω --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Omega } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> </noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" data-alt="{\displaystyle \Omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> nazýváme <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Bod?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bod">bodem</a> plochy. Předpokládáme přitom, že tyto rovnice jsou na <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Ω<!-- Ω --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Omega } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> </noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" data-alt="{\displaystyle \Omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> spojité a mají spojité nebo po částech spojité parciální derivace prvního řádu podle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> u </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle u} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> </noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" data-alt="{\displaystyle u}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> v </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle v} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> </noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <div class="mw-heading mw-heading3"> <h3 id="Explicitní_rovnice_plochy"><span id="Explicitn.C3.AD_rovnice_plochy"></span>Explicitní rovnice plochy</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Explicitní rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p>Pokud lze předchozí rovnice plochy převést na tvar</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 z=f(x,y)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> z </mi> <mo> = </mo> <mi> f </mi> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo> , </mo> <mi> y </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle z=f(x,y)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.794ex; height:2.843ex;" alt="{\displaystyle z=f(x,y)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.794ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eefb2840000f404c8c0f3f5d6d72f2624854591" data-alt="{\displaystyle z=f(x,y)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, </dd> </dl> <p>pak hovoříme o explicitní rovnici plochy.</p> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Základní_rovnice_plochy"><span id="Z.C3.A1kladn.C3.AD_rovnice_plochy"></span>Základní rovnice plochy</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Základní rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Vztahy mezi <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Norm%C3%A1la?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Normála">normálou</a> plochy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/R%C3%A1diusvektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rádiusvektor">rádiusvektorem</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> a jejich <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Derivace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Derivace">derivacemi</a> určují tzv. <i>základní rovnice plochy</i>. Tyto <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Rovnice?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rovnice">rovnice</a> lze pro plochu určenou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =\mathbf {r} (u,v)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> <mo stretchy="false"> ( </mo> <mi> u </mi> <mo> , </mo> <mi> v </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} =\mathbf {r} (u,v)} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498915ac0755e189367d0761fde21fee48242767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =\mathbf {r} (u,v)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.603ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498915ac0755e189367d0761fde21fee48242767" data-alt="{\displaystyle \mathbf {r} =\mathbf {r} (u,v)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> uvést v různých tvarech.</p> <div class="noprint ambox labelced labelced-page labelced-page-type-content ambox-content plainlinks"> <div class="labelced_message"> <div class="labelced_message_inner"> <div class="labelced_message_headline ambox-text"> Tento článek potřebuje úpravy. </div> <div class="labelced_message_text hide-when-compact ambox-text"> Můžete Wikipedii pomoci tím, že ho <span class="editlink plainlinks" style=""><a class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/w/index.php?title%3DPlocha%26action%3Dedit">vylepšíte</a></span>. Jak by měly články vypadat, popisují stránky <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikipedie:Vzhled_a_styl?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedie:Vzhled a styl">Vzhled a styl</a>, <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikipedie:Encyklopedick%C3%BD_styl?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedie:Encyklopedický styl">Encyklopedický styl</a> a <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikipedie:Pr%C5%AFvodce_(odkazy)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedie:Průvodce (odkazy)">Odkazy</a>. </div> </div> </div> </div> <div class="mw-heading mw-heading3"> <h3 id="Weingartenovy_rovnice_plochy">Weingartenovy rovnice plochy</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Weingartenovy rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p><b>Weingartenovy rovnice plochy</b> určují vztahy mezi <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Derivace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Derivace">derivacemi</a> <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Vektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vektor">vektorů</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {n} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46" data-alt="{\displaystyle \mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {n} }{\partial u}}={\frac {FM-GL}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FL-EM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> F </mi> <mi> M </mi> <mo> −<!-- − --> </mo> <mi> G </mi> <mi> L </mi> </mrow> <mrow> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> F </mi> <mi> L </mi> <mo> −<!-- − --> </mo> <mi> E </mi> <mi> M </mi> </mrow> <mrow> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial \mathbf {n} }{\partial u}}={\frac {FM-GL}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FL-EM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a320d581a1040431c1e801f3e817e35c1dc259" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.831ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {n} }{\partial u}}={\frac {FM-GL}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FL-EM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 38.831ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95a320d581a1040431c1e801f3e817e35c1dc259" data-alt="{\displaystyle {\frac {\partial \mathbf {n} }{\partial u}}={\frac {FM-GL}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FL-EM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {n} }{\partial v}}={\frac {FN-GM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FM-EN}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> F </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <mi> G </mi> <mi> M </mi> </mrow> <mrow> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> F </mi> <mi> M </mi> <mo> −<!-- − --> </mo> <mi> E </mi> <mi> N </mi> </mrow> <mrow> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial \mathbf {n} }{\partial v}}={\frac {FN-GM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FM-EN}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d887731c7090944d2f45f3763e25c11d0d4f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.793ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {n} }{\partial v}}={\frac {FN-GM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FM-EN}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 39.793ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d887731c7090944d2f45f3763e25c11d0d4f11" data-alt="{\displaystyle {\frac {\partial \mathbf {n} }{\partial v}}={\frac {FN-GM}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {FM-EN}{EG-F^{2}}}{\frac {\partial \mathbf {r} }{\partial v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd></dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {r} }{\partial u}}={\frac {MF-NE}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {ME-LF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> M </mi> <mi> F </mi> <mo> −<!-- − --> </mo> <mi> N </mi> <mi> E </mi> </mrow> <mrow> <mi> L </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <msup> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> M </mi> <mi> E </mi> <mo> −<!-- − --> </mo> <mi> L </mi> <mi> F </mi> </mrow> <mrow> <mi> L </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <msup> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial \mathbf {r} }{\partial u}}={\frac {MF-NE}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {ME-LF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75102f4383d6328f60339fa69c14bbb291cb3c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.618ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {r} }{\partial u}}={\frac {MF-NE}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {ME-LF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 39.618ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75102f4383d6328f60339fa69c14bbb291cb3c31" data-alt="{\displaystyle {\frac {\partial \mathbf {r} }{\partial u}}={\frac {MF-NE}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {ME-LF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {r} }{\partial v}}={\frac {MG-NF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {MF-LG}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> M </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <mi> N </mi> <mi> F </mi> </mrow> <mrow> <mi> L </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <msup> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> M </mi> <mi> F </mi> <mo> −<!-- − --> </mo> <mi> L </mi> <mi> G </mi> </mrow> <mrow> <mi> L </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <msup> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial \mathbf {r} }{\partial v}}={\frac {MG-NF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {MF-LG}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9881a0c4ee495534e864f473ef08ddd6f85f09ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.519ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {r} }{\partial v}}={\frac {MG-NF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {MF-LG}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}"> </noscript><span class="lazy-image-placeholder" style="width: 39.519ex;height: 5.843ex;vertical-align: -2.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9881a0c4ee495534e864f473ef08ddd6f85f09ef" data-alt="{\displaystyle {\frac {\partial \mathbf {r} }{\partial v}}={\frac {MG-NF}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial u}}+{\frac {MF-LG}{LN-M^{2}}}{\frac {\partial \mathbf {n} }{\partial v}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,F,G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> E </mi> <mo> , </mo> <mi> F </mi> <mo> , </mo> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E,F,G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.411ex; height:2.509ex;" alt="{\displaystyle E,F,G}"> </noscript><span class="lazy-image-placeholder" style="width: 7.411ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" data-alt="{\displaystyle E,F,G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základní veličiny plochy prvního řádu</a> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,M,N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <mo> , </mo> <mi> M </mi> <mo> , </mo> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L,M,N} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.157ex; height:2.509ex;" alt="{\displaystyle L,M,N}"> </noscript><span class="lazy-image-placeholder" style="width: 8.157ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" data-alt="{\displaystyle L,M,N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základní veličiny plochy druhého řádu</a>.</p> <div class="mw-heading mw-heading3"> <h3 id="Gaussovy_rovnice_plochy">Gaussovy rovnice plochy</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Gaussovy rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p><b>Gaussovy rovnice plochy</b> umožňují určit druhou derivaci <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Polohov%C3%BD_vektor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polohový vektor">polohového vektoru</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {r} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.102ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" data-alt="{\displaystyle \mathbf {r} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</p> <dl> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u^{2}}}={\frac {G{\frac {\partial E}{\partial u}}-2F{\frac {\partial F}{\partial u}}+F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {-F{\frac {\partial E}{\partial u}}+2E{\frac {\partial F}{\partial u}}-E{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+L\mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <msup> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> −<!-- − --> </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 2 </mn> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u^{2}}}={\frac {G{\frac {\partial E}{\partial u}}-2F{\frac {\partial F}{\partial u}}+F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {-F{\frac {\partial E}{\partial u}}+2E{\frac {\partial F}{\partial u}}-E{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+L\mathbf {n} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadd26a2971a2f6c86961b1a479a9b1561e86d8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.011ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u^{2}}}={\frac {G{\frac {\partial E}{\partial u}}-2F{\frac {\partial F}{\partial u}}+F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {-F{\frac {\partial E}{\partial u}}+2E{\frac {\partial F}{\partial u}}-E{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+L\mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 69.011ex;height: 7.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eadd26a2971a2f6c86961b1a479a9b1561e86d8a" data-alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u^{2}}}={\frac {G{\frac {\partial E}{\partial u}}-2F{\frac {\partial F}{\partial u}}+F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {-F{\frac {\partial E}{\partial u}}+2E{\frac {\partial F}{\partial u}}-E{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+L\mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u\partial v}}={\frac {G{\frac {\partial E}{\partial v}}-F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial u}}-F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+M\mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> M </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u\partial v}}={\frac {G{\frac {\partial E}{\partial v}}-F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial u}}-F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+M\mathbf {n} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5495bf71e69ef4bceaf51858d1bb4117b4d10d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.006ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u\partial v}}={\frac {G{\frac {\partial E}{\partial v}}-F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial u}}-F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+M\mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 52.006ex;height: 7.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5495bf71e69ef4bceaf51858d1bb4117b4d10d4" data-alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial u\partial v}}={\frac {G{\frac {\partial E}{\partial v}}-F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial u}}-F{\frac {\partial E}{\partial v}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+M\mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <dd> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial v^{2}}}={\frac {-F{\frac {\partial G}{\partial v}}+2G{\frac {\partial F}{\partial v}}-G{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial v}}-2F{\frac {\partial F}{\partial v}}+F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+N\mathbf {n} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <msup> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo> −<!-- − --> </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 2 </mn> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mi> G </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> E </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo stretchy="false"> ) </mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> r </mi> </mrow> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mi> N </mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> n </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial v^{2}}}={\frac {-F{\frac {\partial G}{\partial v}}+2G{\frac {\partial F}{\partial v}}-G{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial v}}-2F{\frac {\partial F}{\partial v}}+F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+N\mathbf {n} } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8445e5d4850a7580390645be28ec25743f9d45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.486ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial v^{2}}}={\frac {-F{\frac {\partial G}{\partial v}}+2G{\frac {\partial F}{\partial v}}-G{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial v}}-2F{\frac {\partial F}{\partial v}}+F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+N\mathbf {n} }"> </noscript><span class="lazy-image-placeholder" style="width: 69.486ex;height: 7.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8445e5d4850a7580390645be28ec25743f9d45" data-alt="{\displaystyle {\frac {\partial ^{2}\mathbf {r} }{\partial v^{2}}}={\frac {-F{\frac {\partial G}{\partial v}}+2G{\frac {\partial F}{\partial v}}-G{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial u}}+{\frac {E{\frac {\partial G}{\partial v}}-2F{\frac {\partial F}{\partial v}}+F{\frac {\partial G}{\partial u}}}{2(EG-F^{2})}}{\frac {\partial \mathbf {r} }{\partial v}}+N\mathbf {n} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,F,G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> E </mi> <mo> , </mo> <mi> F </mi> <mo> , </mo> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E,F,G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.411ex; height:2.509ex;" alt="{\displaystyle E,F,G}"> </noscript><span class="lazy-image-placeholder" style="width: 7.411ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" data-alt="{\displaystyle E,F,G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základní veličiny plochy prvního řádu</a> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,M,N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <mo> , </mo> <mi> M </mi> <mo> , </mo> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L,M,N} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.157ex; height:2.509ex;" alt="{\displaystyle L,M,N}"> </noscript><span class="lazy-image-placeholder" style="width: 8.157ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" data-alt="{\displaystyle L,M,N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základní veličiny plochy druhého řádu</a>.</p> <div class="mw-heading mw-heading3"> <h3 id="Codazziho_rovnice_plochy">Codazziho rovnice plochy</h3><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Codazziho rovnice plochy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <p><b>Codazziho</b> (nebo také <b>Mainardiho</b>) <b>rovnice plochy</b> určují vztahy mezi <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základními veličinami plochy prvního řádu</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E,F,G}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> E </mi> <mo> , </mo> <mi> F </mi> <mo> , </mo> <mi> G </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle E,F,G} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.411ex; height:2.509ex;" alt="{\displaystyle E,F,G}"> </noscript><span class="lazy-image-placeholder" style="width: 7.411ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc6395da181061e60cc425cff6ad41453c22ea6" data-alt="{\displaystyle E,F,G}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> a <a href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Z%C3%A1kladn%C3%AD_veli%C4%8Dina_plochy&action=edit&redlink=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="new" title="Základní veličina plochy (stránka neexistuje)">základními veličinami plochy druhého řádu</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L,M,N}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> <mo> , </mo> <mi> M </mi> <mo> , </mo> <mi> N </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L,M,N} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.157ex; height:2.509ex;" alt="{\displaystyle L,M,N}"> </noscript><span class="lazy-image-placeholder" style="width: 8.157ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ec8e5b50816fc40ef75e37a261c1209964cdd4" data-alt="{\displaystyle L,M,N}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</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 (EG-2F^{2}+GE)\left({\frac {\partial L}{\partial v}}-{\frac {\partial M}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial E}{\partial v}}-{\frac {\partial F}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial u}}&L\\F&{\frac {\partial F}{\partial u}}&M\\G&{\frac {\partial G}{\partial u}}&N\end{vmatrix}}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> G </mi> <mi> E </mi> <mo stretchy="false"> ) </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> L </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> M </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> −<!-- − --> </mo> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mi> F </mi> <mi> M </mi> <mo> + </mo> <mi> G </mi> <mi> L </mi> <mo stretchy="false"> ) </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> E </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> L </mi> </mtd> </mtr> <mtr> <mtd> <mi> F </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> M </mi> </mtd> </mtr> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> N </mi> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial L}{\partial v}}-{\frac {\partial M}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial E}{\partial v}}-{\frac {\partial F}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial u}}&L\\F&{\frac {\partial F}{\partial u}}&M\\G&{\frac {\partial G}{\partial u}}&N\end{vmatrix}}=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00bf4992abb46e9e2c4620b36563e94a49d2d468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.345ex; margin-bottom: -0.326ex; width:92.227ex; height:12.509ex;" alt="{\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial L}{\partial v}}-{\frac {\partial M}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial E}{\partial v}}-{\frac {\partial F}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial u}}&L\\F&{\frac {\partial F}{\partial u}}&M\\G&{\frac {\partial G}{\partial u}}&N\end{vmatrix}}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 92.227ex;height: 12.509ex;vertical-align: -5.345ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00bf4992abb46e9e2c4620b36563e94a49d2d468" data-alt="{\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial L}{\partial v}}-{\frac {\partial M}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial E}{\partial v}}-{\frac {\partial F}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial u}}&L\\F&{\frac {\partial F}{\partial u}}&M\\G&{\frac {\partial G}{\partial u}}&N\end{vmatrix}}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> <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 (EG-2F^{2}+GE)\left({\frac {\partial M}{\partial v}}-{\frac {\partial N}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial F}{\partial v}}-{\frac {\partial G}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial v}}&L\\F&{\frac {\partial F}{\partial v}}&M\\G&{\frac {\partial G}{\partial v}}&N\end{vmatrix}}=0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> G </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <msup> <mi> F </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <mi> G </mi> <mi> E </mi> <mo stretchy="false"> ) </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> M </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> N </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> −<!-- − --> </mo> <mo stretchy="false"> ( </mo> <mi> E </mi> <mi> N </mi> <mo> −<!-- − --> </mo> <mn> 2 </mn> <mi> F </mi> <mi> M </mi> <mo> + </mo> <mi> G </mi> <mi> L </mi> <mo stretchy="false"> ) </mo> <mrow> <mo> ( </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> <mo> −<!-- − --> </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi> E </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> E </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> L </mi> </mtd> </mtr> <mtr> <mtd> <mi> F </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> F </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> M </mi> </mtd> </mtr> <mtr> <mtd> <mi> G </mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> G </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi> N </mi> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> <mo> = </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial M}{\partial v}}-{\frac {\partial N}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial F}{\partial v}}-{\frac {\partial G}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial v}}&L\\F&{\frac {\partial F}{\partial v}}&M\\G&{\frac {\partial G}{\partial v}}&N\end{vmatrix}}=0} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/080f6b526db8bdcad704acdf03907069bb76ba46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.345ex; margin-bottom: -0.326ex; width:92.759ex; height:12.509ex;" alt="{\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial M}{\partial v}}-{\frac {\partial N}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial F}{\partial v}}-{\frac {\partial G}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial v}}&L\\F&{\frac {\partial F}{\partial v}}&M\\G&{\frac {\partial G}{\partial v}}&N\end{vmatrix}}=0}"> </noscript><span class="lazy-image-placeholder" style="width: 92.759ex;height: 12.509ex;vertical-align: -5.345ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/080f6b526db8bdcad704acdf03907069bb76ba46" data-alt="{\displaystyle (EG-2F^{2}+GE)\left({\frac {\partial M}{\partial v}}-{\frac {\partial N}{\partial u}}\right)-(EN-2FM+GL)\left({\frac {\partial F}{\partial v}}-{\frac {\partial G}{\partial u}}\right)+{\begin{vmatrix}E&{\frac {\partial E}{\partial v}}&L\\F&{\frac {\partial F}{\partial v}}&M\\G&{\frac {\partial G}{\partial v}}&N\end{vmatrix}}=0}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Vlastnosti" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> <ul> <li>Zavedeme <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Matice?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matice">matici</a></li> </ul> <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{pmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial y}{\partial u}}&{\frac {\partial z}{\partial u}}\\{\frac {\partial x}{\partial v}}&{\frac {\partial y}{\partial v}}&{\frac {\partial z}{\partial v}}\end{pmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> ( </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> x </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> y </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> z </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> u </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> x </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> y </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> z </mi> </mrow> <mrow> <mi mathvariant="normal"> ∂<!-- ∂ --> </mi> <mi> v </mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{pmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial y}{\partial u}}&{\frac {\partial z}{\partial u}}\\{\frac {\partial x}{\partial v}}&{\frac {\partial y}{\partial v}}&{\frac {\partial z}{\partial v}}\end{pmatrix}}} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4792bfb7c2c8a87646b9ebf4d6644f707e59dffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:17.589ex; height:8.843ex;" alt="{\displaystyle {\begin{pmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial y}{\partial u}}&{\frac {\partial z}{\partial u}}\\{\frac {\partial x}{\partial v}}&{\frac {\partial y}{\partial v}}&{\frac {\partial z}{\partial v}}\end{pmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.589ex;height: 8.843ex;vertical-align: -3.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4792bfb7c2c8a87646b9ebf4d6644f707e59dffe" data-alt="{\displaystyle {\begin{pmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial y}{\partial u}}&{\frac {\partial z}{\partial u}}\\{\frac {\partial x}{\partial v}}&{\frac {\partial y}{\partial v}}&{\frac {\partial z}{\partial v}}\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> </dd> </dl> <p>Body plochy, v nichž má tato matice <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Hodnost_matice?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hodnost matice">hodnost</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377419f817c2a1af4abdf5159b27aa21b5de5c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=2}"> </noscript><span class="lazy-image-placeholder" style="width: 5.6ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377419f817c2a1af4abdf5159b27aa21b5de5c2b" data-alt="{\displaystyle h=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> jsou regulárními body. Je-li hodnost matice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h<2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> < </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h<2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992d174114d64fbd2a6f7b2b4e0b1aee79b099a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h<2}"> </noscript><span class="lazy-image-placeholder" style="width: 5.6ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992d174114d64fbd2a6f7b2b4e0b1aee79b099a6" data-alt="{\displaystyle h<2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, pak jde o singulární body.</p> <ul> <li>Máme-li plochu zadanou rovnicemi, které mají všude v <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> Ω<!-- Ω --> </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \Omega } </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"> </noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" data-alt="{\displaystyle \Omega }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> nenulovou parciální derivaci prvního řádu a uvedená matice má v každém bodě hodnost <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=2}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> h </mi> <mo> = </mo> <mn> 2 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle h=2} </annotation> </semantics> </math></span> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377419f817c2a1af4abdf5159b27aa21b5de5c2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=2}"> </noscript><span class="lazy-image-placeholder" style="width: 5.6ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377419f817c2a1af4abdf5159b27aa21b5de5c2b" data-alt="{\displaystyle h=2}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, pak plochu označujeme jako <b>hladkou</b>.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Související články" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Prostorov%C3%A9_geometrick%C3%A9_%C3%BAtvary?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Prostorové geometrické útvary">Prostorové geometrické útvary</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/P%C5%99%C3%ADmkov%C3%A1_plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Přímková plocha">Přímková plocha</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Kvadrika?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kvadrika">Kvadratická plocha</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ku%C5%BEelov%C3%A1_plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Kuželová plocha">Kuželová plocha</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/V%C3%A1lcov%C3%A1_plocha?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Válcová plocha">Válcová plocha</a></li> <li><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Obsah?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Obsah">Obsah</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <span class="indicator mf-icon mf-icon-expand mf-icon--small"></span> <h2 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h2><span class="mw-editsection"> <a role="button" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Editace sekce: Externí odkazy" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>editovat</span> </a> </span> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul> <li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikimedia_Commons?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikimedia Commons"> <noscript> <img alt="Logo Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" data-file-width="1024" data-file-height="1376"> </noscript><span class="lazy-image-placeholder" style="width: 12px;height: 16px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" data-alt="Logo Wikimedia Commons" data-width="12" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-class="mw-file-element"> </span></a></span></span> <span class="sisterproject_text">Obrázky, zvuky či videa k tématu <span class="sisterproject_text_target"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://commons.wikimedia.org/wiki/Category:Surfaces" class="extiw" title="c:Category:Surfaces">plocha</a></span> na <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Wikimedia_Commons?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikimedia Commons">Wikimedia Commons</a></span></span></span><i> </i></li> <li><span class="sisterproject sisterproject-wikiquote"><span class="sisterproject_image"><span typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" decoding="async" width="13" height="16" class="mw-file-element" data-file-width="300" data-file-height="355"> </noscript><span class="lazy-image-placeholder" style="width: 13px;height: 16px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/13px-Wikiquote-logo.svg.png" data-alt="" data-width="13" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/20px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/27px-Wikiquote-logo.svg.png 2x" data-class="mw-file-element"> </span></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Téma </span><span class="sisterproject_text_target"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikiquote.org/wiki/Plocha" class="extiw" title="q:Plocha">Plocha</a></span><span class="sisterproject_text_suffix"> ve Wikicitátech</span></span></span></li> <li><span class="sisterproject sisterproject-wiktionary"><span class="sisterproject_image"><span typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" decoding="async" width="16" height="16" class="mw-file-element" data-file-width="411" data-file-height="411"> </noscript><span class="lazy-image-placeholder" style="width: 16px;height: 16px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/16px-Wiktionary-logo-cs.svg.png" data-alt="" data-width="16" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/24px-Wiktionary-logo-cs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Wiktionary-logo-cs.svg/32px-Wiktionary-logo-cs.svg.png 2x" data-class="mw-file-element"> </span></span></span></span> <span class="sisterproject_text"><span class="sisterproject_text_prefix">Slovníkové heslo </span><span class="sisterproject_text_target"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wiktionary.org/wiki/plocha" class="extiw" title="wikt:plocha">plocha</a></span><span class="sisterproject_text_suffix"> ve Wikislovníku</span></span></span></li> <li><span class="sisterproject-inline"><span typeof="mw:File"><span> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/15px-Wikisource-logo.svg.png" decoding="async" width="15" height="16" class="mw-file-element" data-file-width="410" data-file-height="430"> </noscript><span class="lazy-image-placeholder" style="width: 15px;height: 16px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/15px-Wikisource-logo.svg.png" data-alt="" data-width="15" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/23px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/30px-Wikisource-logo.svg.png 2x" data-class="mw-file-element"> </span></span></span> Encyklopedické heslo <strong><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikisource.org/wiki/Ott%25C5%25AFv_slovn%25C3%25ADk_nau%25C4%258Dn%25C3%25BD/Plocha" class="extiw" title="s:Ottův slovník naučný/Plocha">Plocha</a></strong> v <a href="https://cs-m-wikipedia-org.translate.goog/wiki/Ott%C5%AFv_slovn%C3%ADk_nau%C4%8Dn%C3%BD?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ottův slovník naučný">Ottově slovníku naučném</a> ve Wikizdrojích</span></li> </ul> <style 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Rendering was triggered because: page-view --> </section> </div><!-- MobileFormatter took 0.023 seconds --><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --> <noscript> <img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=mobile" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div class="printfooter" data-nosnippet=""> Citováno z „<a dir="ltr" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/w/index.php?title%3DPlocha%26oldid%3D24116421">https://cs.wikipedia.org/w/index.php?title=Plocha&oldid=24116421</a>“ </div> </div> </div> <div class="post-content" id="page-secondary-actions"> </div> </main> <footer class="mw-footer minerva-footer" role="contentinfo"><a class="last-modified-bar" href="https://cs-m-wikipedia-org.translate.goog/w/index.php?title=Plocha&action=history&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB"> <div class="post-content last-modified-bar__content"><span class="minerva-icon minerva-icon-size-medium minerva-icon--modified-history"></span> <span class="last-modified-bar__text modified-enhancement" data-user-name="Geek-woman" data-user-gender="unknown" data-timestamp="1721891765"> <span>Naposledy editováno 25. 7. 2024 v 08:16</span> </span> <span class="minerva-icon minerva-icon-size-small minerva-icon--expand"></span> </div></a> <div class="post-content footer-content"> <div id="mw-data-after-content"> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Jazyky</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://af.wikipedia.org/wiki/Oppervlak" title="Oppervlak – afrikánština" lang="af" hreflang="af" data-title="Oppervlak" data-language-autonym="Afrikaans" data-language-local-name="afrikánština" class="interlanguage-link-target"><span>Afrikaans</span></a></li> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://als.wikipedia.org/wiki/Fl%25C3%25A4che_(Topologie)" title="Fläche (Topologie) – němčina (Švýcarsko)" lang="gsw" hreflang="gsw" data-title="Fläche (Topologie)" data-language-autonym="Alemannisch" data-language-local-name="němčina (Švýcarsko)" class="interlanguage-link-target"><span>Alemannisch</span></a></li> <li class="interlanguage-link interwiki-an mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://an.wikipedia.org/wiki/Superficie" title="Superficie – aragonština" lang="an" hreflang="an" data-title="Superficie" data-language-autonym="Aragonés" data-language-local-name="aragonština" class="interlanguage-link-target"><span>Aragonés</span></a></li> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ar.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح – arabština" lang="ar" hreflang="ar" data-title="سطح" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></a></li> <li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ast.wikipedia.org/wiki/Superficie" title="Superficie – asturština" lang="ast" hreflang="ast" data-title="Superficie" data-language-autonym="Asturianu" data-language-local-name="asturština" class="interlanguage-link-target"><span>Asturianu</span></a></li> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://az.wikipedia.org/wiki/S%25C9%2599th" title="Səth – ázerbájdžánština" lang="az" hreflang="az" data-title="Səth" data-language-autonym="Azərbaycanca" data-language-local-name="ázerbájdžánština" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li> <li class="interlanguage-link interwiki-be mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://be.wikipedia.org/wiki/%25D0%259F%25D0%25B0%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Паверхня – běloruština" lang="be" hreflang="be" data-title="Паверхня" data-language-autonym="Беларуская" data-language-local-name="běloruština" class="interlanguage-link-target"><span>Беларуская</span></a></li> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bg.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%258A%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582" title="Повърхност – bulharština" lang="bg" hreflang="bg" data-title="Повърхност" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li> <li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bn.wikipedia.org/wiki/%25E0%25A6%25A4%25E0%25A6%25B2_(%25E0%25A6%259F%25E0%25A6%25AA%25E0%25A7%258B%25E0%25A6%25B2%25E0%25A6%259C%25E0%25A6%25BF)" title="তল (টপোলজি) – bengálština" lang="bn" hreflang="bn" data-title="তল (টপোলজি)" data-language-autonym="বাংলা" data-language-local-name="bengálština" class="interlanguage-link-target"><span>বাংলা</span></a></li> <li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bs.wikipedia.org/wiki/Povr%25C5%25A1" title="Površ – bosenština" lang="bs" hreflang="bs" data-title="Površ" data-language-autonym="Bosanski" data-language-local-name="bosenština" class="interlanguage-link-target"><span>Bosanski</span></a></li> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Superf%25C3%25ADcie_(matem%25C3%25A0tiques)" title="Superfície (matemàtiques) – katalánština" lang="ca" hreflang="ca" data-title="Superfície (matemàtiques)" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li> <li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ckb.wikipedia.org/wiki/%25DA%2595%25D9%2588%25D9%2588" title="ڕوو – kurdština (sorání)" lang="ckb" hreflang="ckb" data-title="ڕوو" data-language-autonym="کوردی" data-language-local-name="kurdština (sorání)" class="interlanguage-link-target"><span>کوردی</span></a></li> <li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cv.wikipedia.org/wiki/%25C3%2587%25D0%25B8%25D0%25B9" title="Çий – čuvaština" lang="cv" hreflang="cv" data-title="Çий" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikipedia.org/wiki/Surface_(topology)" title="Surface (topology) – angličtina" lang="en" hreflang="en" data-title="Surface (topology)" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Surfaco" title="Surfaco – esperanto" lang="eo" hreflang="eo" data-title="Surfaco" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Superficie_(topolog%25C3%25ADa)" title="Superficie (topología) – španělština" lang="es" hreflang="es" data-title="Superficie (topología)" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li> <li class="interlanguage-link interwiki-et mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://et.wikipedia.org/wiki/Pind" title="Pind – estonština" lang="et" hreflang="et" data-title="Pind" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Gainazal" title="Gainazal – baskičtina" lang="eu" hreflang="eu" data-title="Gainazal" data-language-autonym="Euskara" data-language-local-name="baskičtina" class="interlanguage-link-target"><span>Euskara</span></a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25D8%25B1%25D9%2588%25DB%258C%25D9%2587_(%25D8%25AA%25D9%2588%25D9%25BE%25D9%2588%25D9%2584%25D9%2588%25DA%2598%25DB%258C)" title="رویه (توپولوژی) – perština" lang="fa" hreflang="fa" data-title="رویه (توپولوژی)" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li> <li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Pinta_(geometria)" title="Pinta (geometria) – finština" lang="fi" hreflang="fi" data-title="Pinta (geometria)" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li> <li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Surface_(g%25C3%25A9om%25C3%25A9trie)" title="Surface (géométrie) – francouzština" lang="fr" hreflang="fr" data-title="Surface (géométrie)" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li> <li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fur.wikipedia.org/wiki/Superficie" title="Superficie – furlanština" lang="fur" hreflang="fur" data-title="Superficie" data-language-autonym="Furlan" data-language-local-name="furlanština" class="interlanguage-link-target"><span>Furlan</span></a></li> <li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ga.wikipedia.org/wiki/Dromchla_(toipeola%25C3%25ADocht)" title="Dromchla (toipeolaíocht) – irština" lang="ga" hreflang="ga" data-title="Dromchla (toipeolaíocht)" data-language-autonym="Gaeilge" data-language-local-name="irština" class="interlanguage-link-target"><span>Gaeilge</span></a></li> <li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://gl.wikipedia.org/wiki/Superficie" title="Superficie – galicijština" lang="gl" hreflang="gl" data-title="Superficie" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%259E%25D7%25A9%25D7%2598%25D7%2597_(%25D7%2598%25D7%2595%25D7%25A4%25D7%2595%25D7%259C%25D7%2595%25D7%2592%25D7%2599%25D7%2594)" title="משטח (טופולוגיה) – hebrejština" lang="he" hreflang="he" data-title="משטח (טופולוגיה)" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></a></li> <li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hi.wikipedia.org/wiki/%25E0%25A4%25AA%25E0%25A5%2583%25E0%25A4%25B7%25E0%25A5%258D%25E0%25A4%259F" title="पृष्ट – hindština" lang="hi" hreflang="hi" data-title="पृष्ट" data-language-autonym="हिन्दी" data-language-local-name="hindština" class="interlanguage-link-target"><span>हिन्दी</span></a></li> <li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hr.wikipedia.org/wiki/Ploha_(geometrija)" title="Ploha (geometrija) – chorvatština" lang="hr" hreflang="hr" data-title="Ploha (geometrija)" data-language-autonym="Hrvatski" data-language-local-name="chorvatština" class="interlanguage-link-target"><span>Hrvatski</span></a></li> <li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hu.wikipedia.org/wiki/Felsz%25C3%25ADn" title="Felszín – maďarština" lang="hu" hreflang="hu" data-title="Felszín" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></a></li> <li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://hy.wikipedia.org/wiki/%25D5%2584%25D5%25A1%25D5%25AF%25D5%25A5%25D6%2580%25D6%2587%25D5%25B8%25D6%2582%25D5%25B5%25D5%25A9" title="Մակերևույթ – arménština" lang="hy" hreflang="hy" data-title="Մակերևույթ" data-language-autonym="Հայերեն" data-language-local-name="arménština" class="interlanguage-link-target"><span>Հայերեն</span></a></li> <li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ia.wikipedia.org/wiki/Superficie" title="Superficie – interlingua" lang="ia" hreflang="ia" data-title="Superficie" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://id.wikipedia.org/wiki/Permukaan_(topologi)" title="Permukaan (topologi) – indonéština" lang="id" hreflang="id" data-title="Permukaan (topologi)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li> <li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://inh.wikipedia.org/wiki/%25D0%25A2%25D3%2580%25D0%25B5%25D1%2585%25D0%25B5" title="ТӀехе – inguština" lang="inh" hreflang="inh" data-title="ТӀехе" data-language-autonym="ГӀалгӀай" data-language-local-name="inguština" class="interlanguage-link-target"><span>ГӀалгӀай</span></a></li> <li class="interlanguage-link interwiki-io mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://io.wikipedia.org/wiki/Surfaco" title="Surfaco – ido" lang="io" hreflang="io" data-title="Surfaco" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://is.wikipedia.org/wiki/Yfirbor%25C3%25B0" title="Yfirborð – islandština" lang="is" hreflang="is" data-title="Yfirborð" data-language-autonym="Íslenska" data-language-local-name="islandština" class="interlanguage-link-target"><span>Íslenska</span></a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Superficie" title="Superficie – italština" lang="it" hreflang="it" data-title="Superficie" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – japonština" lang="ja" hreflang="ja" data-title="曲面" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></a></li> <li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://kk.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%25B3%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (геометрия) – kazaština" lang="kk" hreflang="kk" data-title="Бет (геометрия)" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EA%25B3%25A1%25EB%25A9%25B4" title="곡면 – korejština" lang="ko" hreflang="ko" data-title="곡면" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></a></li> <li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ky.wikipedia.org/wiki/%25D0%2591%25D0%25B5%25D1%2582_(%25D0%2593%25D0%25B5%25D0%25BE%25D0%25BC%25D0%25B5%25D1%2582%25D1%2580%25D0%25B8%25D1%258F)" title="Бет (Геометрия) – kyrgyzština" lang="ky" hreflang="ky" data-title="Бет (Геометрия)" data-language-autonym="Кыргызча" data-language-local-name="kyrgyzština" class="interlanguage-link-target"><span>Кыргызча</span></a></li> <li class="interlanguage-link interwiki-lij mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lij.wikipedia.org/wiki/Superfi%25C3%25A7ie_(matematica)" title="Superfiçie (matematica) – ligurština" lang="lij" hreflang="lij" data-title="Superfiçie (matematica)" data-language-autonym="Ligure" data-language-local-name="ligurština" class="interlanguage-link-target"><span>Ligure</span></a></li> <li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lt.wikipedia.org/wiki/Pavir%25C5%25A1ius" title="Paviršius – litevština" lang="lt" hreflang="lt" data-title="Paviršius" data-language-autonym="Lietuvių" data-language-local-name="litevština" class="interlanguage-link-target"><span>Lietuvių</span></a></li> <li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://lv.wikipedia.org/wiki/Virsma" title="Virsma – lotyština" lang="lv" hreflang="lv" data-title="Virsma" data-language-autonym="Latviešu" data-language-local-name="lotyština" class="interlanguage-link-target"><span>Latviešu</span></a></li> <li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mr.wikipedia.org/wiki/%25E0%25A4%2586%25E0%25A4%25A1" title="आड – maráthština" lang="mr" hreflang="mr" data-title="आड" data-language-autonym="मराठी" data-language-local-name="maráthština" class="interlanguage-link-target"><span>मराठी</span></a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Oppervlak_(topologie)" title="Oppervlak (topologie) – nizozemština" lang="nl" hreflang="nl" data-title="Oppervlak (topologie)" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></a></li> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nn.wikipedia.org/wiki/Flate" title="Flate – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Flate" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li> <li class="interlanguage-link interwiki-no mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://no.wikipedia.org/wiki/Flate" title="Flate – norština (bokmål)" lang="nb" hreflang="nb" data-title="Flate" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li> <li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://oc.wikipedia.org/wiki/Superf%25C3%25ADcia_(matematicas)" title="Superfícia (matematicas) – okcitánština" lang="oc" hreflang="oc" data-title="Superfícia (matematicas)" data-language-autonym="Occitan" data-language-local-name="okcitánština" class="interlanguage-link-target"><span>Occitan</span></a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Powierzchnia" title="Powierzchnia – polština" lang="pl" hreflang="pl" data-title="Powierzchnia" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li> <li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pms.wikipedia.org/wiki/Surfassa" title="Surfassa – piemonština" lang="pms" hreflang="pms" data-title="Surfassa" data-language-autonym="Piemontèis" data-language-local-name="piemonština" class="interlanguage-link-target"><span>Piemontèis</span></a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Superf%25C3%25ADcie" title="Superfície – portugalština" lang="pt" hreflang="pt" data-title="Superfície" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Suprafa%25C8%259B%25C4%2583" title="Suprafață – rumunština" lang="ro" hreflang="ro" data-title="Suprafață" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li> <li class="interlanguage-link interwiki-rsk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://rsk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2586" title="Поверхносц – Pannonian Rusyn" lang="rsk" hreflang="rsk" data-title="Поверхносц" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D0%25BE%25D1%2581%25D1%2582%25D1%258C" title="Поверхность – ruština" lang="ru" hreflang="ru" data-title="Поверхность" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></a></li> <li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Povr%25C5%25A1" title="Površ – srbochorvatština" lang="sh" hreflang="sh" data-title="Površ" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srbochorvatština" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Surface" title="Surface – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sk.wikipedia.org/wiki/Povrch" title="Povrch – slovenština" lang="sk" hreflang="sk" data-title="Povrch" data-language-autonym="Slovenčina" data-language-local-name="slovenština" class="interlanguage-link-target"><span>Slovenčina</span></a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Ploskev" title="Ploskev – slovinština" lang="sl" hreflang="sl" data-title="Ploskev" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></a></li> <li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sn.wikipedia.org/wiki/Chiso_(Chiumbwa)" title="Chiso (Chiumbwa) – šonština" lang="sn" hreflang="sn" data-title="Chiso (Chiumbwa)" data-language-autonym="ChiShona" data-language-local-name="šonština" class="interlanguage-link-target"><span>ChiShona</span></a></li> <li class="interlanguage-link interwiki-so mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://so.wikipedia.org/wiki/Oogo_(dhul)" title="Oogo (dhul) – somálština" lang="so" hreflang="so" data-title="Oogo (dhul)" data-language-autonym="Soomaaliga" data-language-local-name="somálština" class="interlanguage-link-target"><span>Soomaaliga</span></a></li> <li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D1%2580%25D1%2588" title="Површ – srbština" lang="sr" hreflang="sr" data-title="Површ" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Yta" title="Yta – švédština" lang="sv" hreflang="sv" data-title="Yta" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></a></li> <li class="interlanguage-link interwiki-te mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://te.wikipedia.org/wiki/%25E0%25B0%2589%25E0%25B0%25AA%25E0%25B0%25B0%25E0%25B0%25BF%25E0%25B0%25A4%25E0%25B0%25B2%25E0%25B0%2582" title="ఉపరితలం – telugština" lang="te" hreflang="te" data-title="ఉపరితలం" data-language-autonym="తెలుగు" data-language-local-name="telugština" class="interlanguage-link-target"><span>తెలుగు</span></a></li> <li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://tr.wikipedia.org/wiki/Y%25C3%25BCzey" title="Yüzey – turečtina" lang="tr" hreflang="tr" data-title="Yüzey" data-language-autonym="Türkçe" data-language-local-name="turečtina" class="interlanguage-link-target"><span>Türkçe</span></a></li> <li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uk.wikipedia.org/wiki/%25D0%259F%25D0%25BE%25D0%25B2%25D0%25B5%25D1%2580%25D1%2585%25D0%25BD%25D1%258F" title="Поверхня – ukrajinština" lang="uk" hreflang="uk" data-title="Поверхня" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li> <li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ur.wikipedia.org/wiki/%25D8%25B3%25D8%25B7%25D8%25AD" title="سطح – urdština" lang="ur" hreflang="ur" data-title="سطح" data-language-autonym="اردو" data-language-local-name="urdština" class="interlanguage-link-target"><span>اردو</span></a></li> <li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://uz.wikipedia.org/wiki/Sirt" title="Sirt – uzbečtina" lang="uz" hreflang="uz" data-title="Sirt" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečtina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li> <li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vec.wikipedia.org/wiki/Superficie" title="Superficie – benátština" lang="vec" hreflang="vec" data-title="Superficie" data-language-autonym="Vèneto" data-language-local-name="benátština" class="interlanguage-link-target"><span>Vèneto</span></a></li> <li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://vi.wikipedia.org/wiki/M%25E1%25BA%25B7t_(t%25C3%25B4_p%25C3%25B4)" title="Mặt (tô pô) – vietnamština" lang="vi" hreflang="vi" data-title="Mặt (tô pô)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamština" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> <li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://wuu.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – čínština (dialekty Wu)" lang="wuu" hreflang="wuu" data-title="曲面" data-language-autonym="吴语" data-language-local-name="čínština (dialekty Wu)" class="interlanguage-link-target"><span>吴语</span></a></li> <li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – čínština" lang="zh" hreflang="zh" data-title="曲面" data-language-autonym="中文" data-language-local-name="čínština" class="interlanguage-link-target"><span>中文</span></a></li> <li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://zh-yue.wikipedia.org/wiki/%25E6%259B%25B2%25E9%259D%25A2" title="曲面 – kantonština" lang="yue" hreflang="yue" data-title="曲面" data-language-autonym="粵語" data-language-local-name="kantonština" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> </section> </div> <div class="minerva-footer-logo"> <img 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