I have Monge patch surface given as $h=h(u,v)$ and analyze it in points of interest using the usual formulas for coefficients of the first and the second fundamental forms: \begin{align} E&=1+h_u^2, & e&=\frac{h_{uu}}{(1+h_u^2+h_v^2)^{1/2}}, \\ F&=h_u h_v, & f&=\frac{h_{uv}}{(1+h_u^2+h_v^2)^{1/2}}, \\ F&=1+h_v^2, & g&=\frac{h_{vv}}{(1+h_u^2+h_v^2)^{1/2}}. \\ \end{align} The values of $h_u$, $h_v$, $h_{uu}$, $h_{uv}$, $h_{vv}$ are known numerically from polynomial approximation of point cloud.
I understand principal curvatures $\kappa_i$ are eigenvalues of $$\begin{pmatrix}E & F \\ F & G\end{pmatrix}^{-1}\begin{pmatrix}e&f\\ f&g\end{pmatrix}=\mathbf{I}^{-1}\mathbf{II}.$$
I need to compute gradient of the major principal curvature $\kappa_2$ (more specifically its slope perpendicular to its direction, i.e. in the first principal direction $e_1$).
I tried to express the gradient analytically from the eigenvalue problem but got lost in the expressions. Was I on the right track?
I am not an expert in differential geometry & might be missing something obvious; I hope at least I got the question stated correctly :)
