Los coeficientes de Weingarten para una superficie S(u,v) = (X(u,v), Y(u,v), Z(u,v)) son las funciones $a_{ij}$ del siguiente sistema de ecuaciones respectivamente.
$$ {\large \begin{align*} \frac{\partial \hat{n}}{\partial u} &= a_{11}\frac{\partial S}{\partial u} + a_{12}\frac{\partial S}{\partial v} \\ \frac{\partial \hat{n}}{\partial v} &= a_{21}\frac{\partial S}{\partial u} + a_{22}\frac{\partial S}{\partial v} \end{align*}} $$
Para deducirse esas funciones correspondientes partimos de la definición del vector normal para una superficie S(u,v) respectivamente.
- Caso 1
$$ {\large \begin{align*} 0 &= \left< \frac{\partial S}{\partial u}, \hat{n} \right> \\ 0 &= \frac{\partial }{\partial u} \left( \left< \frac{\partial S}{\partial u}, \hat{n} \right> \right) \\ 0 &= \left< \frac{\partial^2 S}{\partial u^2}, \hat{n} \right> + \left< \frac{\partial S}{\partial u}, \frac{\partial \hat{n}}{\partial u} \right> \hspace{1cm}...\hspace{0.2cm}(1) \end{align*}} $$
Reemplazando la primera ecuación de Weingarten en la ecuación (1)
$$ {\large \begin{align*} 0 &= \left< \frac{\partial^2 S}{\partial u^2}, \hat{n} \right> + \left< \frac{\partial S}{\partial u}, \frac{\partial \hat{n}}{\partial u} \right> \\ 0 &= L + \left< \frac{\partial S}{\partial u}, a_{11}\frac{\partial S}{\partial u} + a_{12}\frac{\partial S}{\partial v} \right> \\ -L &= a_{11}E + a_{12}F \end{align*}} $$
- Caso 2
$$ {\large \begin{align*} 0 &= \left< \frac{\partial S}{\partial u}, \hat{n} \right> \\ 0 &= \frac{\partial }{\partial v} \left( \left< \frac{\partial S}{\partial u}, \hat{n} \right> \right) \\ 0 &= \left< \frac{\partial^2 S}{\partial u \partial v}, \hat{n} \right> + \left< \frac{\partial S}{\partial u}, \frac{\partial \hat{n}}{\partial v} \right> \hspace{1cm}...\hspace{0.2cm}(2) \end{align*}} $$
Reemplazando la segunda ecuación de Weingarten en la ecuación (2)
$$ {\large \begin{align*} 0 &= \left< \frac{\partial^2 S}{\partial u \partial v}, \hat{n} \right> + \left< \frac{\partial S}{\partial u}, \frac{\partial \hat{n}}{\partial v} \right> \\ 0 &= M + \left< \frac{\partial S}{\partial u}, a_{21}\frac{\partial S}{\partial u} + a_{22}\frac{\partial S}{\partial v} \right> \\ -M &= a_{21}E + a_{22}F \end{align*}} $$
- Caso 3
$$ {\large \begin{align*}0 &= \left< \frac{\partial S}{\partial v}, \hat{n} \right> \\ 0 &= \frac{\partial }{\partial v} \left( \left< \frac{\partial S}{\partial v}, \hat{n} \right> \right) \\ 0 &= \left< \frac{\partial^2 S}{\partial v^2}, \hat{n} \right> + \left< \frac{\partial S}{\partial v}, \frac{\partial \hat{n}}{\partial v} \right> \hspace{1cm}...\hspace{0.2cm}(3) \end{align*}} $$
Reemplazando la segunda ecuación de Weingarten en la ecuación (3)
$${\large \begin{align*}0 &= \left< \frac{\partial^2 S}{\partial v^2}, \hat{n} \right> + \left< \frac{\partial S}{\partial v}, \frac{\partial \hat{n}}{\partial v} \right> \\ 0 &= N + \left< \frac{\partial S}{\partial v}, a_{21}\frac{\partial S}{\partial u} + a_{22}\frac{\partial S}{\partial v} \right> \\ -N &= a_{21}F + a_{22}G \end{align*}} $$
- Caso 4
$$ {\large \begin{align*} 0 &= \left< \frac{\partial S}{\partial v}, \hat{n} \right> \\ 0 &= \frac{\partial }{\partial u} \left( \left< \frac{\partial S}{\partial v}, \hat{n} \right> \right) \\ 0 &= \left< \frac{\partial^2 S}{\partial v \partial u}, \hat{n} \right> + \left< \frac{\partial S}{\partial v}, \frac{\partial \hat{n}}{\partial u} \right> \hspace{1cm}...\hspace{0.2cm}(4) \end{align*}} $$
Reemplazando la primera ecuación de Weingarten en la ecuación (4)
$$ {\large \begin{align*} 0 &= \left< \frac{\partial^2 S}{\partial v \partial u}, \hat{n} \right> + \left< \frac{\partial S}{\partial v}, \frac{\partial \hat{n}}{\partial u} \right> \\ 0 &= M + \left< \frac{\partial S}{\partial v}, a_{11}\frac{\partial S}{\partial u} + a_{12}\frac{\partial S}{\partial v} \right> \\ -M &= a_{11}F + a_{12}G \end{align*}} $$
Agrupando las ecuaciones obtenidas
$$ {\large \begin{align*} -L &= a_{11}E + a_{12}F \hspace{1cm}...\hspace{0.2cm}(5) \\ -M &= a_{21}E + a_{22}F \hspace{1cm}...\hspace{0.2cm}(6) \\ -M &= a_{11}F + a_{12}G \hspace{1cm}...\hspace{0.2cm}(7) \\ -N &= a_{21}F + a_{22}G \hspace{1cm}...\hspace{0.2cm}(8) \end{align*}} $$
Realizando la operación (7) + (8) como también (6) + (5)
$${\large \begin{align*} \begin{bmatrix} -M -N \\ -M-L \end{bmatrix} &= \begin{bmatrix} (a_{11} + a_{21})F + (a_{12} + a_{22})G \\ (a_{11} + a_{21})E + (a_{12} + a_{22})F \end{bmatrix} \\ \begin{bmatrix} -M -N \\ -M-L \end{bmatrix} &= \begin{bmatrix} F & G \\ E & F \end{bmatrix} \cdot \begin{bmatrix} a_{11} + a_{21} \\ a_{12} + a_{22} \end{bmatrix} \\ \begin{bmatrix} a_{11} + a_{21} \\ a_{12} + a_{22} \end{bmatrix} &= \frac{1}{F^2 - EG} \begin{bmatrix} F & - G \\ -E & F \end{bmatrix} \cdot \begin{bmatrix}- M - N \\ - M - L \end{bmatrix} \\ \begin{bmatrix} a_{11} + a_{21} \\ a_{12} + a_{22} \end{bmatrix} &= \frac{1}{EG - F^2 }\begin{bmatrix} F(M+N) - G(M+L) \\ F(M+L) - E(M+N) \end{bmatrix} \hspace{1cm}...\hspace{0.2cm}(9) \end{align*}} $$
Realizando la operación (7) - (8) como también (5) - (6)
$$ {\large \begin{align*} \begin{bmatrix} -M +N \\ -L+M \end{bmatrix} &= \begin{bmatrix} (a_{11} - a_{21})F + (a_{12} - a_{22})G \\ (a_{11} - a_{21})E + (a_{12} - a_{22})F \end{bmatrix} \\ \begin{bmatrix} -M + N \\ -L+M \end{bmatrix} &= \begin{bmatrix} F & G \\ E & F \end{bmatrix} \cdot \begin{bmatrix} a_{11} - a_{21} \\ a_{12} - a_{22} \end{bmatrix} \\ \begin{bmatrix} a_{11} - a_{21} \\ a_{12} - a_{22} \end{bmatrix} &= \frac{1}{F^2 - EG} \begin{bmatrix} F & - G \\ -E & F \end{bmatrix} \cdot \begin{bmatrix} - M + N \\ - L + M \end{bmatrix} \\ \begin{bmatrix} a_{11} - a_{21} \\ a_{12} - a_{22} \end{bmatrix} &= \frac{1}{EG - F^2 }\begin{bmatrix} F(M-N) - G(L-M) \\ F(L-M) - E(M-N) \end{bmatrix} \hspace{1cm}...\hspace{0.2cm}(10) \end{align*}} $$
De las ecuaciones (9) y (10) podemos deducir los valores de los coeficientes de Weingarten
$$ {\large \begin{align*} \begin{bmatrix} a_{11} \\ a_{12} \\ a_{21} \\ a_{22} \end{bmatrix} = \frac{1}{EG-F^2} \begin{bmatrix} FM - GL \\ FL - EM \\ FN - GM \\ FM - EN \end{bmatrix} \end{align*}} $$
Por consiguiente
$$ {\large \begin{align*} \frac{\partial \hat{n}}{\partial u} &= \left(\frac{FM - GL}{EG-F^2}\right) \frac{\partial S}{\partial u} + \left( \frac{FL - EM}{EG-F^2} \right) \frac{\partial S}{\partial v} \\ \frac{\partial \hat{n}}{\partial v} &= \left( \frac{FN - GM}{EG-F^2} \right)\frac{\partial S}{\partial u} + \left( \frac{FM - EN}{EG - F^2} \right) \frac{\partial S}{\partial v} \end{align*}} $$
En forma matricial es igual a:
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