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 $u \in [0,8\pi]$: <br /> ![DiniSnap][2] <br /> What I especially wonder is if there is a geodesic that spirals down through every turn, which would be kinda cool. :-) [1]: http://www.crcpress.com/product/isbn/9781584884484 [2]: https://i.sstatic.net/nVhwN.jpg