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The task is:

Find Principal curvature and Gaussian curvature, mean curvature and main directions. Find types of dots(points) on hyperboloid $f(u,v) = (a \cosh(u) \cos(v), a \cosh(u) \sin(v), c \sinh(u) )$

I'm not asking for a full solution, but any tips are highly appreciated.

Also the problem is that I translated task from another language so there might be some language misunderstandings..

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1 Answer

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Let S be a regular surface, and let p be a point on S. The maximum and minimum normal curvatures $k_1$ and $k_2$ at p are called the principal curvatures of S at p. The corresponding directions, i.e., unit eigenvectors $e_1$ and $e_2$ with $dn_p(e_i) = −k_i e_i$ , are called principal directions at p, where $dn_p= -g^{-1} L$. $g$ is the first fundamental form, while $L$ is the second fundamental form. This means that you first compute $g$ and $L$, then $dn_p$. Since the first fundamental form is positive definite and since $dn_p$ is self-adjoint with respect to this form, then by the Spectral Theorem, $dn_p$ is diagonalizable.You can then compute the eigenvalues of $dn_p$. The Gaussian and mean curvature are related to $k_1$ and $k_2$ as a matter of factthe Gaussian curvature is \begin{equation} K= k_1 k_2 = det(dn_p) \end{equation} while the mean curvature is \begin{equation} H= \frac{k_1+k_2}{2} =-\frac{1}{2} Tr(dn_p) \end{equation} where Tr is the trace of the matrix $dn_p$.

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