The problem is as follows:
Suppose that a geodesic $\alpha$ is intersecting a ruling of the helicoid S given by $$x(u,v) = (u \cos(v),u \sin(v), v) $$ at a point $p$. The angle at $p$ between the ruling and $\alpha$ is $\theta \in (0, \pi/2)$. Furthermore, $p$ does not lie on the z axis and we call the relative distance $\delta > 0$.
(a) Show that $\delta > \cot(\theta)$
(b) Show that if $\delta < \cot(\theta)$, then $\delta$ does not intersect the $z$-axis
(c) Find the geodesics for $\delta = \cot(\theta )$
I'm struggling with figuring out how to start on this. I don't understand how can I express the angle $\theta$ and the distance $\delta$. If you could help me explain that, I think I would be able to continue on my own.
Any help appreciated!
P.S.: This question is slightly connected to Geodesics of Helicoid , the results obtained there might be useful in this problem as well.