I am trying to find the non trivial geodesics on an oblique helicoid(non-minimal) surface apart from the rulings. Instead of using the usual geodesic equations and also because I was interested in seeing the correspondence between geodesics through isometry in this case, I was looking at using the hyperboloid of one sheet(of revolution) which is isometric to the oblique helicoid. By the hyperboloid I mean the surface given by the equation $$\dfrac{x^2+y^2}{a^2}-\dfrac{z^2}{b^2} = 1$$
The parametrisation of the oblique helicoid considered is this:
\begin{equation} M(u,v) = \bigg\lbrace\dfrac{ub}{\Delta}\cos{\frac{v}{b}}, \dfrac{ub}{\Delta}\sin{\frac{v}{b}}, \dfrac{au}{\Delta}+v\bigg\rbrace \end{equation}
where $\Delta = \sqrt{a^2+b^2}$. The central circle on the hyperboloid is a geodesic which corresponds to the straight axis of the oblique helicoid. But as the hyperboloid is also a surface of revolution, the meridians of which are hyperbolas and are also geodesics. I am trying to see what the images of these geodesics are on the oblique helicoid, but haven't had any luck. Any hints or suggestions are welcome. Thanks in advance.
