Si una superficie es de la forma z = f(x,y) entonces su parametrización es la siguiente:
$$ {\large\begin{equation*} S(x,y) = (x, y, f(x,y)) \end{equation*}} $$
La primera forma fundamental y la segunda forma fundamental de la superficie S(x,y) son las siguientes:
$$ {\Large \begin{align*} I(x,y) &= \begin{bmatrix} 1 + \left( \frac{\partial f}{\partial x} \right)^2 & \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} & 1 + \left( \frac{\partial f}{\partial y} \right)^2 \end{bmatrix} \\ \\ II(x,y) &= \begin{bmatrix} \frac{\frac{\partial^2 f}{\partial x^2}}{\sqrt{1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}} & \frac{\frac{\partial^2 f}{\partial x \partial y}}{\sqrt{1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}} \\ \frac{\frac{\partial^2 f}{\partial x \partial y}}{\sqrt{1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}} & \frac{\frac{\partial^2 f}{\partial y^2}}{\sqrt{1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2}} \end{bmatrix} \end{align*}} $$
Por definición de Curvatura media.
$$ {\large \begin{align*} H(x,y) &= \frac{EN - 2FM + GL}{2(EG - F^2)} \\ H(x,y) &= \frac{\frac{\partial^2 f}{\partial y^2}\left( 1 + \left( \frac{\partial f}{\partial x} \right)^2 \right) - 2\frac{\partial ^2 f}{\partial x \partial y} \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} + \frac{\partial ^2 f}{\partial x^2}\left( 1 + \left( \frac{\partial f}{\partial y} \right)^2 \right)}{ 2\left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right)^{3/2} } \end{align*}} $$
Expandiendo la Curvatura media sumando y restando términos obtenemos lo siguiente respectivamente.
$$ {\large \begin{align*} H(x,y) &= \frac{\frac{\partial^2 f}{\partial y^2}\left( 1 + \left( \frac{\partial f}{\partial x} \right)^2 \right) - 2\frac{\partial ^2 f}{\partial x \partial y} \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} + \frac{\partial ^2 f}{\partial x^2}\left( 1 + \left( \frac{\partial f}{\partial y} \right)^2 \right)}{ 2\left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right)^{3/2} } \\ &+ \left( \frac{\frac{\partial^2 f}{\partial y^2} \left( \frac{\partial f}{\partial y} \right)^2 - \frac{\partial^2 f}{\partial y^2} \left( \frac{\partial f}{\partial y} \right)^2 + \frac{\partial^2 f}{\partial x^2} \left( \frac{\partial f}{\partial x} \right)^2 - \frac{\partial^2 f}{\partial x^2} \left( \frac{\partial f}{\partial x} \right)^2}{2\left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right)^{3/2}} \right) \end{align*}} $$
La curvatura media adopta la siguiente forma
$$ {\large \begin{align*} H(x,y) &= \frac{\frac{\partial^2 f}{\partial x^2}\left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right) - \frac{\partial f}{\partial x} \left( \frac{\partial f}{\partial x} \frac{\partial^2 f}{\partial x^2} + \frac{\partial f}{\partial y} \frac{\partial^2 f}{\partial y \partial x} \right) }{2 \left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right)^{3/2}} \\ &+ \frac{\frac{\partial^2 f}{\partial y^2}\left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right) - \frac{\partial f}{\partial y} \left( \frac{\partial f}{\partial y} \frac{\partial^2 f}{\partial y^2} + \frac{\partial f}{\partial x} \frac{\partial^2 f}{\partial x \partial y} \right) }{2 \left( 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 \right)^{3/2}} \\ H(x,y) &= \frac{1}{2}\left[ \frac{\partial }{\partial x} \left( \frac{\frac{\partial f}{\partial x}}{ \sqrt{ 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 } }\right) + \frac{\partial }{\partial y} \left( \frac{\frac{\partial f}{\partial y}}{ \sqrt{ 1 + \left( \frac{\partial f}{\partial x}\right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 } }\right) \right] \\ H(x,y) &= \frac{1}{2} \left( \nabla \cdot \left( \frac{\nabla f(x,y)}{\sqrt{ 1 + \|\nabla f(x,y) \|^2 }} \right) \right) \end{align*}} $$
Superficie Mínima de Schrek
Si queremos encontrar una superficie mínima de la forma z = f(x,y) = P(x)+ Q(y) entonces la curvatura media es igual a cero. Por lo cual partiendo de la ecuación diferencial parcial de Lagrange - Euler.
$$ {\large \begin{align*} 0 &= \frac{\partial^2 f}{\partial y^2}\left( 1 + \left( \frac{\partial f}{\partial x} \right)^2 \right) - 2\frac{\partial ^2 f}{\partial x \partial y} \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} + \frac{\partial ^2 f}{\partial x^2}\left( 1 + \left( \frac{\partial f}{\partial y} \right)^2 \right) \\ 0 &= \frac{\partial^2 Q}{\partial y^2}\left( 1 + \left( \frac{\partial P}{\partial x} \right)^2 \right) + \frac{\partial ^2 P}{\partial x^2}\left( 1 + \left( \frac{\partial Q}{\partial y} \right)^2 \right) \\ \frac{\partial^2 Q}{\partial y^2}\left( 1 + \left( \frac{\partial P}{\partial x} \right)^2 \right) &= - \frac{\partial ^2 P}{\partial x^2}\left( 1 + \left( \frac{\partial Q}{\partial y} \right)^2 \right) \\ \frac{\partial^2 Q}{\partial y^2}\left( 1 + \left( \frac{\partial Q}{\partial x} \right)^2 \right)^{-1} &= - \frac{\partial ^2 P}{\partial x^2}\left( 1 + \left( \frac{\partial P}{\partial y} \right)^2 \right)^{-1} \end{align*}} $$
Igualando a una constante $\beta$
$$ {\large \begin{equation*}\frac{\partial^2 Q}{\partial y^2} = \beta \left( 1 + \left( \frac{\partial Q}{\partial x} \right)^2 \right) \hspace{1cm},\hspace{1cm} \frac{\partial^2 P}{\partial y^2} = -\beta \left( 1 + \left( \frac{\partial P}{\partial x} \right)^2 \right) \end{equation*}} $$
Si definimos las funciones siguientes:
$$ {\large \begin{equation*} h = \frac{\partial Q}{\partial y} \hspace{1cm},\hspace{1cm} w = \frac{\partial P}{\partial x} \end{equation*}} $$
Las soluciones de las ecuaciones diferenciales son las siguientes:
$$ {\large \begin{equation*} Q(y) = \frac{-\ln{\cos{(\beta y + C)}}}{\beta} \hspace{1cm},\hspace{1cm} P(x) = \frac{\ln{\cos{(\beta x + C)}}}{\beta} \end{equation*}} $$
Por consiguiente la función es:
$$ {\large \begin{align*} f(x,y) &= P(x) + Q(y) \\ f(x,y) &= \frac{1}{\beta}\left( \ln{\cos{(\beta x + C)}} - \ln{\cos{(\beta x + C)}} \right) \\ f(x,y) &= \frac{1}{\beta} \left( \ln{\left(\frac{\cos{(\beta x + C)}}{\cos{(\beta y + C)}}\right)} \right) \end{align*}} $$
Por lo tanto la parametrización de la superficie mínima es la siguiente:
$${\large \begin{equation*} S(u,v) = \left( u, v, \frac{1}{\beta} \left( \ln{\left(\frac{\cos{(\beta u + C)}}{\cos{(\beta v + C)}}\right)} \right) \right) \end{equation*} }$$
Donde ${\large [u, v]}$ = ${\large\mathbb{R}^2}$
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| Superficie de Schrek para $\beta$ = 1 y C= 0 en el cuadrado $ [-4\pi, 4\pi] \times [-4\pi, 4\pi]$ |
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