Geometría diferencial de curvas - Wikipedia, la enciclopedia libre

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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Longitud_de_arco"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Longitud de arco</span> </div> </a> <ul id="toc-Longitud_de_arco-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vectores_tangente,_normal_y_binormal:_Triedro_de_Frênet-Serret" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vectores_tangente,_normal_y_binormal:_Triedro_de_Frênet-Serret"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Vectores tangente, normal y binormal: Triedro de Frênet-Serret</span> </div> </a> <ul id="toc-Vectores_tangente,_normal_y_binormal:_Triedro_de_Frênet-Serret-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curvatura_y_torsión" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Curvatura_y_torsión"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Curvatura y torsión</span> </div> </a> <ul id="toc-Curvatura_y_torsión-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plano_osculador" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Plano_osculador"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Plano osculador</span> </div> </a> <ul id="toc-Plano_osculador-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Centro_de_curvatura" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Centro_de_curvatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Centro de curvatura</span> </div> </a> <ul id="toc-Centro_de_curvatura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_fundamental_de_curvas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a 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id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Geometría diferencial de curvas</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. 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class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Differentiable_curve" title="Differentiable curve (inglés)" lang="en" hreflang="en" data-title="Differentiable curve" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria_differenziale_delle_curve" title="Geometria differenziale delle curve (italiano)" lang="it" hreflang="it" data-title="Geometria differenziale delle curve" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Curva_diferenci%C3%A1vel" title="Curva diferenciável (portugués)" lang="pt" hreflang="pt" 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class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="es" dir="ltr"><p>En matemáticas, la <b>geometría diferencial de curvas</b> propone definiciones y métodos para analizar <a href="/wiki/Curva" title="Curva">curvas</a> simples en <a href="/wiki/Variedad_de_Riemann" title="Variedad de Riemann">Variedades de Riemann</a>, y en particular, en el <a href="/wiki/Espacio_Eucl%C3%ADdeo" class="mw-redirect" title="Espacio Euclídeo">Espacio Euclídeo</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Longitud_de_arco">Longitud de arco</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=1" title="Editar sección: Longitud de arco"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint AP rellink"><span style="font-size:88%">Artículo principal:</span> <i><a href="/wiki/Longitud_de_arco" title="Longitud de arco"> Longitud de arco</a></i></div> <p>Dada una curva suficientemente <a href="/wiki/Curva#curva_suave" title="Curva">suave</a> (diferenciable y de clase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{2}(\mathrm {I} )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">I</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{2}(\mathrm {I} )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3ac38d84991006522dcb73c3288caf295265356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.888ex; height:3.176ex;" alt="{\displaystyle C^{2}(\mathrm {I} )\,}"></span>), en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> y dado su vector de posición <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} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7216e69d28fdb11c29c859f015a284c1ec6f06ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.751ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)}"></span> expresado mediante el parámetro <i>t</i>; </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 \mathbf {r} (t)=x(t)\mathbf {i} +y(t)\mathbf {j} +z(t)\mathbf {k} \qquad t\in [a,b]\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mspace width="2em" /> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (t)=x(t)\mathbf {i} +y(t)\mathbf {j} +z(t)\mathbf {k} \qquad t\in [a,b]\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b241899d3a43740404e9505d6d5069ab22008dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.287ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} (t)=x(t)\mathbf {i} +y(t)\mathbf {j} +z(t)\mathbf {k} \qquad t\in [a,b]\,}"></span></dd></dl> <p>se define el llamado <a href="/wiki/Longitud_de_arco" title="Longitud de arco">parámetro de arco</a> <i>s</i> como:<br /> <br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\phi (t)=\int _{a}^{t}{\sqrt {\left[x'(\tau )\right]^{2}+\left[y'(\tau )\right]^{2}+\left[z'(\tau )\right]^{2}}}\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>[</mo> <mrow> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\phi (t)=\int _{a}^{t}{\sqrt {\left[x'(\tau )\right]^{2}+\left[y'(\tau )\right]^{2}+\left[z'(\tau )\right]^{2}}}\,d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf8ce2b8166eef1410d586e92613dc1aff870d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.521ex; height:6.176ex;" alt="{\displaystyle s=\phi (t)=\int _{a}^{t}{\sqrt {\left[x'(\tau )\right]^{2}+\left[y'(\tau )\right]^{2}+\left[z'(\tau )\right]^{2}}}\,d\tau }"></span></dd></dl> <p>La cual se puede expresar también de la siguiente forma en la cual resulta más fácil de recordar </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\phi (t)=\int _{a}^{t}{\left\Vert \mathbf {r} '(\tau )\right\|}d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\phi (t)=\int _{a}^{t}{\left\Vert \mathbf {r} '(\tau )\right\|}d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe7d5d2aab3cdca46b477d0144642402ca3777d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.54ex; height:6.176ex;" alt="{\displaystyle s=\phi (t)=\int _{a}^{t}{\left\Vert \mathbf {r} '(\tau )\right\|}d\tau }"></span></dd></dl> <p>Lo cual permite reparametrizar la curva de la siguiente manera: <br /> </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 \mathbf {\tilde {r}} (s)=\left({\tilde {x}}(s),{\tilde {y}}(s),{\tilde {z}}(s)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tilde {r}} (s)=\left({\tilde {x}}(s),{\tilde {y}}(s),{\tilde {z}}(s)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11dad3e7ad99cc2bd267072c4836022192691ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.839ex; height:2.843ex;" alt="{\displaystyle \mathbf {\tilde {r}} (s)=\left({\tilde {x}}(s),{\tilde {y}}(s),{\tilde {z}}(s)\right)}"></span></dd></dl> <p><br /> donde <br /> </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 {\tilde {x}}(\phi (t))=x(t),\qquad {\tilde {y}}(\phi (t))=y(t),\qquad {\tilde {z}}(\phi (t))=z(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {x}}(\phi (t))=x(t),\qquad {\tilde {y}}(\phi (t))=y(t),\qquad {\tilde {z}}(\phi (t))=z(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87214c32f903ee3c45d18c9eee22779a2b7c9e66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.633ex; height:2.843ex;" alt="{\displaystyle {\tilde {x}}(\phi (t))=x(t),\qquad {\tilde {y}}(\phi (t))=y(t),\qquad {\tilde {z}}(\phi (t))=z(t)}"></span></dd></dl> <p><br /> son las relaciones entre las dos parametrizaciones. </p> <div class="mw-heading mw-heading2"><h2 id="Vectores_tangente,_normal_y_binormal:_Triedro_de_Frênet-Serret"><span id="Vectores_tangente.2C_normal_y_binormal:_Triedro_de_Fr.C3.AAnet-Serret"></span>Vectores tangente, normal y binormal: Triedro de Frênet-Serret</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=2" title="Editar sección: Vectores tangente, normal y binormal: Triedro de Frênet-Serret"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="VT rellink"><span style="font-size:88%">Véase también:</span> <i><a href="/wiki/F%C3%B3rmulas_de_Frenet-Serret" title="Fórmulas de Frenet-Serret">Fórmulas de Frenet-Serret</a></i></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Archivo:Frenet_trihedron.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Frenet_trihedron.svg/350px-Frenet_trihedron.svg.png" decoding="async" width="350" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Frenet_trihedron.svg/525px-Frenet_trihedron.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Frenet_trihedron.svg/700px-Frenet_trihedron.svg.png 2x" data-file-width="520" data-file-height="310" /></a><figcaption>Vista esquemática del <span style="color:#0000FF">vector tangente</span>, <span style="color:#00FF00">vector normal</span> y <span style="color:#FF0000">vector binormal</span> de una curva <a href="/wiki/H%C3%A9lice_(geometr%C3%ADa)" title="Hélice (geometría)">hélice</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/Archivo:Frenet-Serret-frame_helix_around_torus.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Frenet-Serret-frame_helix_around_torus.gif/350px-Frenet-Serret-frame_helix_around_torus.gif" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Frenet-Serret-frame_helix_around_torus.gif/525px-Frenet-Serret-frame_helix_around_torus.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Frenet-Serret-frame_helix_around_torus.gif/700px-Frenet-Serret-frame_helix_around_torus.gif 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>Triedro de Frenet-Serret. Hélice alrededor de un <a href="/wiki/Toro_(geometr%C3%ADa)" title="Toro (geometría)">toro</a>.</figcaption></figure> <p>Dada una curva parametrizada <b>r</b>(<i>t</i>) según un parámetro cualquiera <i>t</i> se define el llamado vector tangente, normal y binormal como:<br /> <br /> </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 \mathbf {T} (t)=\mathbf {N} (t)\times \mathbf {B} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} (t)=\mathbf {N} (t)\times \mathbf {B} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2ef030a9f43c57b0345ce41fef86e725976ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.737ex; height:2.843ex;" alt="{\displaystyle \mathbf {T} (t)=\mathbf {N} (t)\times \mathbf {B} (t)}"></span> o bien <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 {T} (t)={\frac {\mathbf {r} '(t)}{\left\Vert \mathbf {r} '(t)\right\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\left\Vert \mathbf {r} '(t)\right\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572697fb9d510317b656348f1c495e015fab9b4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.203ex; height:6.509ex;" alt="{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\left\Vert \mathbf {r} '(t)\right\|}}}"></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 \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f7ab0ea089a65798c6dd1a3282666d6cf1bbdb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.737ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)}"></span> o bien <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/013a1d6abd6f53513e8d16349952f1c9d2848f8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.974ex; height:6.509ex;" alt="{\displaystyle \mathbf {B} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}}}"></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 \mathbf {N} (t)=\mathbf {B} (t)\times \mathbf {T} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} (t)=\mathbf {B} (t)\times \mathbf {T} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2306b8819bedaa724ab90439608a25262933f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.737ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} (t)=\mathbf {B} (t)\times \mathbf {T} (t)}"></span> o bien <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} (t)={\frac {[\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)}{\left\Vert [\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)\right\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} (t)={\frac {[\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)}{\left\Vert [\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)\right\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e71aa3e3a5873f30bd8bac396cd285259bc92042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.734ex; height:6.509ex;" alt="{\displaystyle \mathbf {N} (t)={\frac {[\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)}{\left\Vert [\mathbf {r} '(t)\times \mathbf {r} ''(t)]\times \mathbf {r} '(t)\right\|}}}"></span></dd></dl> <p><br /> Estos tres vectores son unitarios y perpendiculares entre sí, juntos configuran un <a href="/wiki/Marco_m%C3%B3vil" title="Marco móvil">sistema de referencia móvil</a> conocido como <b>Triedro de Frênet-Serret</b> a raíz del estudio de <a href="/wiki/Jean_Frenet" title="Jean Frenet">Jean Frenet</a> y <a href="/wiki/Joseph_Serret" class="mw-redirect" title="Joseph Serret">Joseph Serret</a>. Es interesante que para una <a href="/wiki/Part%C3%ADcula_puntual" class="mw-redirect" title="Partícula puntual">partícula</a> física desplazándose en el espacio, el vector tangente es paralelo a la velocidad, mientras que el vector normal da el cambio dirección por unidad de tiempo de la velocidad o <a href="/wiki/Aceleraci%C3%B3n#Componentes_intrínsecas_de_la_aceleración:_aceleraciones_tangencial_y_normal" title="Aceleración">aceleración normal</a>. </p><p>Si la curva está parametrizada según la longitud de arco, como se explicó en la sección anterior las fórmulas anteriores pueden simplificarse notablemente: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} (s)={\frac {d{\tilde {\mathbf {r} }}(s)}{ds}}\qquad \mathbf {N} (s)={\frac {1}{\chi }}{\frac {d\mathbf {T} (s)}{ds}}\qquad \mathbf {B} (s)={\frac {1}{\tau }}\left({\frac {d\mathbf {N} (s)}{ds}}+\chi \mathbf {T} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>χ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} (s)={\frac {d{\tilde {\mathbf {r} }}(s)}{ds}}\qquad \mathbf {N} (s)={\frac {1}{\chi }}{\frac {d\mathbf {T} (s)}{ds}}\qquad \mathbf {B} (s)={\frac {1}{\tau }}\left({\frac {d\mathbf {N} (s)}{ds}}+\chi \mathbf {T} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07b9c8e8d9d8b3a776b6d94e87cbfdf2b5f8aa66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.398ex; height:6.343ex;" alt="{\displaystyle \mathbf {T} (s)={\frac {d{\tilde {\mathbf {r} }}(s)}{ds}}\qquad \mathbf {N} (s)={\frac {1}{\chi }}{\frac {d\mathbf {T} (s)}{ds}}\qquad \mathbf {B} (s)={\frac {1}{\tau }}\left({\frac {d\mathbf {N} (s)}{ds}}+\chi \mathbf {T} \right)}"></span> </p> </blockquote> <p>Donde los parámetros χ y τ anteriores designan respectivamente a la curvatura y a la torsión. </p> <div class="mw-heading mw-heading2"><h2 id="Curvatura_y_torsión"><span id="Curvatura_y_torsi.C3.B3n"></span>Curvatura y torsión</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=3" title="Editar sección: Curvatura y torsión"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La <b>curvatura</b> es una medida del cambio de dirección del vector tangente a una curva, cuanto más rápido cambia este a medida que nos desplazamos a lo largo de la curva, se dice que es más grande la curvatura. Para una curva parametrizada cualquiera la curvatura es igual a: <br /> </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 \chi (t)={\frac {\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}{\left\Vert \mathbf {r} '(t)\right\|^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (t)={\frac {\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}{\left\Vert \mathbf {r} '(t)\right\|^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd91f43c0cbf78bbb5a4ff1859f02357a2218056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.528ex; height:6.843ex;" alt="{\displaystyle \chi (t)={\frac {\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|}{\left\Vert \mathbf {r} '(t)\right\|^{3}}}}"></span></dd></dl> <p><br /> Si la curva está parametrizada por el parámetro de longitud de arco, la anterior ecuación se reduce simplemente a: <br /> </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 \chi (s)=\left\Vert \mathbf {\tilde {r}} ''(s)\right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi (s)=\left\Vert \mathbf {\tilde {r}} ''(s)\right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0333efe46d4f6220e06e2b4b71fd4b4fac8979a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.152ex; height:3.176ex;" alt="{\displaystyle \chi (s)=\left\Vert \mathbf {\tilde {r}} ''(s)\right\|}"></span></dd></dl> <p><br /> Además de la curvatura se suele definir el llamado <a href="/wiki/Radio_de_curvatura" title="Radio de curvatura">radio de curvatura</a>, como el inverso de la curvatura. </p><p>La <b>torsión</b> es una medida del cambio de dirección del vector binormal: cuanto más rápido cambia, más rápido gira el vector binormal alrededor del vector tangente y más retorcida aparece la curva. Por lo tanto, para una curva totalmente contenida en el plano la torsión es nula ya que el vector binormal es constantemente perpendicular al plano que la contiene. Para el caso general la torsión viene dada por: <br /> </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 \tau (t)=-{\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (t)=-{\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81525df275cb710b56a6444ea3c8c88c1a64821c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.587ex; height:7.009ex;" alt="{\displaystyle \tau (t)=-{\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {r} '(t)\times \mathbf {r} ''(t)\right\|^{2}}}}"></span></dd></dl> <p><br /> Si la curva está parametrizada por el parámetro de longitud de arco, la anterior ecuación se reduce a: <br /> </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 \tau (s)={\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {\tilde {r}} ''(s)\right\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau (s)={\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {\tilde {r}} ''(s)\right\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c56f0e9d4b34cacb7c573f69bd28bc20055c3ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.03ex; height:7.343ex;" alt="{\displaystyle \tau (s)={\frac {\left(\mathbf {r} '(t)\times \mathbf {r} ''(t)\right)\cdot \mathbf {r} '''(t)}{\left\Vert \mathbf {\tilde {r}} ''(s)\right\|^{2}}}}"></span></dd></dl> <p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Plano_osculador">Plano osculador</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=4" title="Editar sección: Plano osculador"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En cada punto de una curva, el plano osculador es el plano que contiene a su vector tangente y al vector normal a la curva. Para una partícula desplazándose en el espacio tridimensional, el plano osculador coincide con el plano que en cada instante contiene a la aceleración y la velocidad. La ecuación de este plano viene dada por:<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x'_{0}&y'_{0}&z'_{0}\\x''_{0}&y''_{0}&z''_{0}\end{vmatrix}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi>y</mi> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi>z</mi> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>″</mo> </msubsup> </mtd> <mtd> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>″</mo> </msubsup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x'_{0}&y'_{0}&z'_{0}\\x''_{0}&y''_{0}&z''_{0}\end{vmatrix}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc0be438e3d1626eb7f8d69e90ce791c501671a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.126ex; margin-bottom: -0.212ex; width:33.375ex; height:9.843ex;" alt="{\displaystyle \det {\begin{vmatrix}x-x_{0}&y-y_{0}&z-z_{0}\\x'_{0}&y'_{0}&z'_{0}\\x''_{0}&y''_{0}&z''_{0}\end{vmatrix}}=0}"></span> </p> </blockquote> <p>Donde: </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_{0},y_{0},z_{0})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0},z_{0})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af41d2d30beaec96424b7b91092c41abbf3546f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.977ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0},z_{0})\,}"></span>, el punto de la trayectoria.<br /></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 (x'_{0},y'_{0},z'_{0})\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x'_{0},y'_{0},z'_{0})\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61da0af689f44a4c28b5d0b1332873562da50ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.977ex; height:3.009ex;" alt="{\displaystyle (x'_{0},y'_{0},z'_{0})\,}"></span>, el vector velocidad en el punto considerado.<br /></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 (x,y,z)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c95076d93d11e04e9605954265296e6c36fc88c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.838ex; height:2.843ex;" alt="{\displaystyle (x,y,z)\,}"></span>, las coordenadas de un punto genérico del plano osculador.<br /></dd></dl> <p>Si se tiene una partícula en la posición <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b677d5c2321ab34fec1ddcedf73a23deaf7accb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.47ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} _{p}}"></span>, moviéndose con velocidad <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 {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> y sometida a una aceleración <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbd86f21598eb5c1c24152737ed1c22835368e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.735ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} \neq \mathbf {0} }"></span> el plano osculador viene dado por el conjunto de puntos: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {v} \times \mathbf {a} )\cdot (\mathbf {x} -\mathbf {x} _{p})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {v} \times \mathbf {a} )\cdot (\mathbf {x} -\mathbf {x} _{p})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79794b060a6dbeaded87c98fe97092940f06cfbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.831ex; height:3.009ex;" alt="{\displaystyle (\mathbf {v} \times \mathbf {a} )\cdot (\mathbf {x} -\mathbf {x} _{p})=0}"></span> </p> </blockquote> <p>Obviamente si la partícula tiene un <a href="/wiki/Movimiento_rectil%C3%ADneo" title="Movimiento rectilíneo">movimiento rectilíneo</a> el plano osculador no está definido. </p> <div class="mw-heading mw-heading2"><h2 id="Centro_de_curvatura">Centro de curvatura</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=5" title="Editar sección: Centro de curvatura"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Archivo:Osculating_circle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Osculating_circle.svg/280px-Osculating_circle.svg.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Osculating_circle.svg/420px-Osculating_circle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Osculating_circle.svg/560px-Osculating_circle.svg.png 2x" data-file-width="480" data-file-height="360" /></a><figcaption>Ilustración de la circunferencia osculatriz en el punto <i>P</i> de la curva <i>C</i>, en la que se muestra también el radio y centro de curvatura.</figcaption></figure> <p>En un entorno de un punto de una curva puede ser aproximado por un círculo, llamado <a href="/wiki/C%C3%ADrculo_osculador" class="mw-redirect" title="Círculo osculador">círculo osculador</a> por estar contenido en el plano osculador. El radio del círculo osculador coincide con el radio de curvatura (inverso de la curvatura). El centro de dicho círculo puede buscarse como: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} _{c}(t)=\mathbf {r} (t)-{\frac {\|\mathbf {r} '(t)\|^{2}(\mathbf {r} '(t)\cdot \mathbf {r} ''(t))}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} '(t)+{\frac {\|\mathbf {r} '(t)\|^{4}}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} ''(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{c}(t)=\mathbf {r} (t)-{\frac {\|\mathbf {r} '(t)\|^{2}(\mathbf {r} '(t)\cdot \mathbf {r} ''(t))}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} '(t)+{\frac {\|\mathbf {r} '(t)\|^{4}}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} ''(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f811dbb33e232e371a3b1f76edcf95226219932a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.392ex; height:6.676ex;" alt="{\displaystyle \mathbf {r} _{c}(t)=\mathbf {r} (t)-{\frac {\|\mathbf {r} '(t)\|^{2}(\mathbf {r} '(t)\cdot \mathbf {r} ''(t))}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} '(t)+{\frac {\|\mathbf {r} '(t)\|^{4}}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}\mathbf {r} ''(t)}"></span> </p> </blockquote> <p>O más sencillamente en función del parámetro de arco como: </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\tilde {r}} _{c}(s)=\mathbf {\tilde {r}} (s)+{\frac {\mathbf {\tilde {r}} ''(s)}{\|\mathbf {\tilde {r}} ''(s)\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\tilde {r}} _{c}(s)=\mathbf {\tilde {r}} (s)+{\frac {\mathbf {\tilde {r}} ''(s)}{\|\mathbf {\tilde {r}} ''(s)\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0deb25c72a86547609fa6cbd1fbf4e2aa75460f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.945ex; height:6.843ex;" alt="{\displaystyle \mathbf {\tilde {r}} _{c}(s)=\mathbf {\tilde {r}} (s)+{\frac {\mathbf {\tilde {r}} ''(s)}{\|\mathbf {\tilde {r}} ''(s)\|^{2}}}}"></span> </p> </blockquote> <div class="mw-heading mw-heading2"><h2 id="Teorema_fundamental_de_curvas">Teorema fundamental de curvas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=6" title="Editar sección: Teorema fundamental de curvas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El teorema fundamental de curvas que enunciamos a continuación nos dice que conocido un punto de una curva y su vector tangente, la curva queda totalmente especificada si se conoce la función de curvatura y de torsión. Su enunciado es el siguiente: </p><p>Sea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\subset \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\subset \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02638c7f85d66abbf916acb4ec91c26122d81438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.248ex; height:2.176ex;" alt="{\displaystyle J\subset \mathbb {R} }"></span> un intervalo. Dadas dos funciones continuas χ y τ de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74cf25dd030038def6817d413bfcf9250294f7c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:2.176ex;" alt="{\displaystyle J\,}"></span> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 \mathbb {R} }"></span> y dado un sistema de referencia fijo (<a href="/wiki/Ortonormal" title="Ortonormal">ortonormal</a>) de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, {<i>x</i><sub>0</sub>; <b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, <b>e</b><sub>3</sub>}, entonces existe una única curva parametrizada de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} :J\to \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>:</mo> <mi>J</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} :J\to \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875d269a30749463adf7810684c12a40584ea76b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.166ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} :J\to \mathbb {R} ^{3}}"></span> y tales que: </p> <ol><li>La curva pasa por <i>x</i><sub>0</sub>, y el vector tangente <b>T</b> a la curva en ese punto coincide con <b>e</b><sub>1</sub>.</li> <li>A lo largo de la curva pueden definirse tres campos vectoriales <b>T</b>(s), <b>N</b>(s) y <b>B</b>(s) llamados respectivamente vector tangente, normal y binormal, perpendiculares entre sí y tales que en el punto inicial coinciden con <b>e</b><sub>1</sub>, <b>e</b><sub>2</sub>, <b>e</b><sub>3</sub> (es decir, <b>T</b>(0) = <b>e</b><sub>1</sub>, <b>N</b>(0) = <b>e</b><sub>2</sub>, <b>B</b>(0) = <b>e</b><sub>3</sub>).</li> <li>Se cumplen las siguientes ecuaciones:</li></ol> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}{\cfrac {d\mathbf {x} (s)}{ds}}=\mathbf {T} (s)&{\cfrac {d\mathbf {T} (s)}{ds}}=\chi (s)\mathbf {N} (s)\\{\cfrac {d\mathbf {N} (s)}{ds}}=-\chi (s)\mathbf {T} (s)+\tau (s)\mathbf {B} (s)&{\cfrac {d\mathbf {B} (s)}{ds}}=-\tau (s)\mathbf {N} (s)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>s</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}{\cfrac {d\mathbf {x} (s)}{ds}}=\mathbf {T} (s)&{\cfrac {d\mathbf {T} (s)}{ds}}=\chi (s)\mathbf {N} (s)\\{\cfrac {d\mathbf {N} (s)}{ds}}=-\chi (s)\mathbf {T} (s)+\tau (s)\mathbf {B} (s)&{\cfrac {d\mathbf {B} (s)}{ds}}=-\tau (s)\mathbf {N} (s)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd52600502a970cd86b1cc92e14fa8b691afd552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:58.799ex; height:13.509ex;" alt="{\displaystyle {\begin{cases}{\cfrac {d\mathbf {x} (s)}{ds}}=\mathbf {T} (s)&{\cfrac {d\mathbf {T} (s)}{ds}}=\chi (s)\mathbf {N} (s)\\{\cfrac {d\mathbf {N} (s)}{ds}}=-\chi (s)\mathbf {T} (s)+\tau (s)\mathbf {B} (s)&{\cfrac {d\mathbf {B} (s)}{ds}}=-\tau (s)\mathbf {N} (s)\end{cases}}}"></span> </p> </blockquote> <p>O bien escrito matricialmente </p> <blockquote style="padding: 5px 10px; background-color: var(--background-color-base, #fff); color: var(--color-base, #202122); text-align: left; margin-left:30px; margin-bottom: 0.4em; margin-top:0.2em; min-width:50%;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\dot {T}}\\{\dot {N}}\\{\dot {B}}\\\end{bmatrix}}={\begin{bmatrix}0&\chi &0\\-\chi &0&\tau \\0&-\tau &0\\\end{bmatrix}}{\begin{bmatrix}T\\N\\B\\\end{bmatrix}}}"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>T</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>χ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>χ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>τ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>T</mi> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\dot {T}}\\{\dot {N}}\\{\dot {B}}\\\end{bmatrix}}={\begin{bmatrix}0&\chi &0\\-\chi &0&\tau \\0&-\tau &0\\\end{bmatrix}}{\begin{bmatrix}T\\N\\B\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c14fcdece989fb87e87e251976106f34ea56e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:30.903ex; height:10.176ex;" alt="{\displaystyle {\begin{bmatrix}{\dot {T}}\\{\dot {N}}\\{\dot {B}}\\\end{bmatrix}}={\begin{bmatrix}0&\chi &0\\-\chi &0&\tau \\0&-\tau &0\\\end{bmatrix}}{\begin{bmatrix}T\\N\\B\\\end{bmatrix}}}"></span> </p> </blockquote> <p>donde el punto es la derivada con respecto al arcoparámetro s. </p><p>Esto tiene implicaciones físicas interesantes, por ejemplo, la trayectoria de una partícula queda especificada si se conocen la posición inicial, la velocidad inicial y la variación en el tiempo de las derivadas segundas (que están relacionadas con la curvatura y la torsión). Es por eso por lo que las <a href="/wiki/Leyes_de_Newton" title="Leyes de Newton">leyes de Newton</a> o las <a href="/wiki/Ecuaciones_de_Euler-Lagrange" title="Ecuaciones de Euler-Lagrange">ecuaciones de Euler-Lagrange</a> se expresan en términos de derivadas de segundo orden (que es necesario complementar con la posición y velocidades iniciales). </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=7" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Geometr%C3%ADa_diferencial_de_superficies" title="Geometría diferencial de superficies">Geometría diferencial de superficies</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=8" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Spiegel & Abellanas, p. 120</span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Bibliografía"><span id="Bibliograf.C3.ADa"></span>Bibliografía</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=9" title="Editar sección: Bibliografía"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Girbau, J.: "<i>Geometria diferencial i relativitat</i>", Ed. <a href="/wiki/Universidad_Aut%C3%B3noma_de_Barcelona" title="Universidad Autónoma de Barcelona">Universidad Autónoma de Barcelona</a>, 1993. <a href="/wiki/Especial:FuentesDeLibros/847929776X" class="internal mw-magiclink-isbn">ISBN 84-7929-776-X</a>.</li> <li>Spiegel, M. & Abellanas, L.: "<i>Fórmulas y tablas de matemática aplicada</i>", Ed. McGraw-Hill, 1988. <a href="/wiki/Especial:FuentesDeLibros/8476151977" class="internal mw-magiclink-isbn">ISBN 84-7615-197-7</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Enlaces_externos">Enlaces externos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Geometr%C3%ADa_diferencial_de_curvas&action=edit&section=10" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Archivo:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description" title="Ver el portal sobre Matemática"><img alt="Ver el portal sobre Matemática" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/20px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/30px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/40px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Matem%C3%A1tica" title="Portal:Matemática">Portal:Matemática</a>. 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Geometría diferencial de curvas - Wikipedia, la enciclopedia libre