@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-TX92VB3N-7
  skos:prefLabel "differential geometry of surfaces"@en, "géométrie différentielle des surfaces"@fr ;
  a skos:Concept ;
  skos:narrower psr:-R8P8T2NQ-K .

psr: a skos:ConceptScheme .
psr:-R8P8T2NQ-K
  skos:definition """En géométrie différentielle, le <b>théorème d'Euler</b>  relatif aux rayons de courbure des courbes tracées sur une surface S deux fois différentiable fournit la valeur des courbures des courbes de cette surface passant par un même point M, sous la forme&nbsp;:<center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{X}=\\\\kappa _{1}\\\\cos ^{2}(\\	heta )+\\\\kappa _{2}\\\\sin ^{2}(\\	heta )}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>cos</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>θ<!-- θ --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>+</mo>
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>sin</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>θ<!-- θ --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{X}=\\\\kappa _{1}\\\\cos ^{2}(\\	heta )+\\\\kappa _{2}\\\\sin ^{2}(\\	heta )}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a91cb347717d70618eab25c24a7701e58e37a704" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:28.346ex; height:3.176ex;" alt="{\\\\displaystyle \\\\kappa _{X}=\\\\kappa _{1}\\\\cos ^{2}(\\	heta )+\\\\kappa _{2}\\\\sin ^{2}(\\	heta )}"></span>.</center>
<br/>où :
<br/>
<br/><ul><li>- <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{X}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{X}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55acd353c2bfa85ad0383b7dd9b792d404d34ddb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.971ex; height:2.009ex;" alt="{\\\\displaystyle \\\\kappa _{X}}"></span> est la courbure normale d'une courbe tracée sur la surface S, et admettant X comme vecteur tangent au point M. C'est aussi la courbure de la courbe obtenue comme intersection de la surface S avec le plan perpendiculaire au plan tangent en M à S, et contenant le vecteur X&nbsp;;</li>
<br/><li>- <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{1}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{1}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78471d428eed0c2e016ad82fb2f30af4cedc86eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.009ex;" alt="{\\\\displaystyle \\\\kappa _{1}}"></span> et <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{2}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{2}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c55c9239a3520d5ef55ae207eaebb7bb518d81f3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.009ex;" alt="\\\\kappa_2"></span> sont les courbures principales de la surface au point considéré&nbsp;;</li>
<br/><li>- <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\	heta }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>θ<!-- θ --></mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\	heta }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="\\	heta "></span> est l'angle entre la direction principale de la courbure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{1}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{1}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78471d428eed0c2e016ad82fb2f30af4cedc86eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.009ex;" alt="{\\\\displaystyle \\\\kappa _{1}}"></span> et le vecteur X.</li> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_d%27Euler_(courbure_des_surfaces)">https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_d%27Euler_(courbure_des_surfaces)</a>)"""@fr, """In the mathematical field of differential geometry, <b>Euler's theorem</b> is a result on the curvature of curves on a surface.  The theorem establishes the existence of principal curvatures and associated <i>principal directions</i> which give the directions in which the surface curves the most and the least.  The theorem is named for Leonhard Euler who proved the theorem in (Euler 1760).
<br/>More precisely, let <i>M</i> be a surface in three-dimensional Euclidean space, and <i>p</i> a point on <i>M</i>.  A <i>normal plane</i> through <i>p</i> is a plane passing through the point <i>p</i> containing the <i>normal vector</i> to <i>M</i>.  Through each (unit) tangent vector to <i>M</i> at <i>p</i>, there passes a normal plane <i>P</i><sub><i>X</i></sub> which cuts out a curve in <i>M</i>.  That curve has a certain curvature κ<sub><i>X</i></sub> when regarded as a curve inside <i>P</i><sub><i>X</i></sub>.  Provided not all κ<sub><i>X</i></sub> are equal, there is some unit vector <i>X</i><sub>1</sub> for which <i>k</i><sub>1</sub>&nbsp;=&nbsp;κ<sub><i>X</i><sub>1</sub></sub> is as large as possible, and another unit vector <i>X</i><sub>2</sub> for which <i>k</i><sub>2</sub>&nbsp;=&nbsp;κ<sub><i>X</i><sub>2</sub></sub> is as small as possible.  Euler's theorem asserts that <i>X</i><sub>1</sub> and <i>X</i><sub>2</sub> are perpendicular and that, moreover, if <i>X</i> is any vector making an angle θ with <i>X</i><sub>1</sub>, then
<br/><dl><dd>
<br/><table role="presentation" style="border-collapse:collapse; margin:0; border:none;">
<br/><tbody><tr>
<br/><td style="vertical-align:middle; border:none; padding:0.08em;" class="nowrap"><div style="margin:0;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\kappa _{X}=k_{1}\\\\cos ^{2}\\	heta +k_{2}\\\\sin ^{2}\\	heta .\\\\,}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>κ<!-- κ --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>X</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>k</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>cos</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>θ<!-- θ --></mi>
<br/>        <mo>+</mo>
<br/>        <msub>
<br/>          <mi>k</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msup>
<br/>          <mi>sin</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>⁡<!-- ⁡ --></mo>
<br/>        <mi>θ<!-- θ --></mi>
<br/>        <mo>.</mo>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\kappa _{X}=k_{1}\\\\cos ^{2}\\	heta +k_{2}\\\\sin ^{2}\\	heta .\\\\,}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1da104b6afd01bb21b3297b568707b0fddd53ae0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:26.28ex; height:3.009ex;" alt="{\\\\displaystyle \\\\kappa _{X}=k_{1}\\\\cos ^{2}\\	heta +k_{2}\\\\sin ^{2}\\	heta .\\\\,}"> 
<br/>
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_theorem_(differential_geometry)">https://en.wikipedia.org/wiki/Euler%27s_theorem_(differential_geometry)</a>)"""@en ;
  a skos:Concept ;
  skos:inScheme psr: ;
  skos:broader psr:-TX92VB3N-7 ;
  skos:prefLabel "Euler's theorem"@en, "théorème d'Euler"@fr ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_d%27Euler_(courbure_des_surfaces)>, <https://en.wikipedia.org/wiki/Euler%27s_theorem_(differential_geometry)> .