Consider a hyperboloid of 1-sheet with the following parametrization:
$x = \sinh(v) \cos(\theta)$
$y = \sinh(v) \sin(\theta)$
$z = \cosh(v)$
where $v>0$ and $\theta \in [0,2\pi]$.
Without any calculation, I would guess that the principal curvatures $\kappa_1$ and $\kappa_2$ are
$\kappa_1 = \frac{1}{\sinh(v)}$, since $\sinh(v)$ is the radius of the circle when fixing $v$
and
$\kappa_1 = 0$, since when fixing $\theta$, the corresponding curve parametrized by $v$ is a straight line
However, this is certainly wrong as I know that the Gauss curvature is given by:
Gauss = $\frac{1}{|1+2\cosh(v)^2|^2}$
which is the product of $\kappa_1$ and $\kappa_2$. So I know that my intuition is wrong, however, I don't understand where my logic fails. Where is my thinking mistake?