I wonder how difficult it is to compute geodesics on *Dini's Surface*,
a twisted pseudosphere?

Here is one parametrization, from 
Alfred Gray's *[Modern Differential Geometry of Curves and Surfaces][1]*, p.495:
\begin{eqnarray}
x(u,v) &=&
a \cos (u) \sin (v)\\
y(u,v) &=&
a \sin (u) \sin (v)\\
z(u,v) &=&
a \left[\cos (v)+\log \left(\tan
   \frac{v}{2}\right)\right]+b u
\end{eqnarray}
Dini's surface has constant curvature of $\frac{-1}{a^2+b^2}$.

And here is an image, for $a=1,\; b=\frac{1}{12}$, with $u \in [0,8\pi]$
(*Added*: now with Will Sawin's $v=\pi/2$ geodesic shown green):
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![DiniCurve][2]
<br />
What I especially wonder is if there is a geodesic that spirals down through every turn,
which would be kinda cool. :-)
<br />
(*Added*: See Will Sawin's answer.)

  [1]: http://www.crcpress.com/product/isbn/9781584884484
  [2]: https://i.sstatic.net/nKCiM.png