If you use cylindrical coordinates $(r,\theta, z) $ since $(r,z)$ relation is known,
only $ (r,\theta)$ relation needs to be found out.
The Clairaut's Law is especially suited to finding geodesics on surfaces of revolution.
The procedure can be used to find geodesics on any surface of revolution when
meridian is given.
Choose any one of the two convex sheets of the hyperboloid given.
$ z^2 - r^2 = 1 $
Differentiate with respect to r; $ z = r* \tan(\phi). $ (1*)
Choose Clairaut's constant $ a = r \sin(\psi). $ (2*)
$(r=a)$ is where all lines are tangent at minimum radius.
From differential geometry, $ dr/ \sin(\phi) = r d\theta * \cot(\psi).$ (3*)
Equns (1*) to (3*) are adequate to find $ r=f(\theta) $, after eliminating $z,\phi,\psi$ and integrating.