Given any two points on a hyperbolic paraboloid ($xy = z$ or $z = (x^2 - y^2)/2$) how does one find the geodesic between them?


I know that since the hyperbolic paraboloid is doubly ruled, some of the geodesics are lines. However, I have very little idea of how to find the geodesics between arbitrary points. 


If an exact answer cannot be given, then I would be interested in a technique of approximating the geodesics. 


Finally, I'd like to add that this is outside my area of expertise (I'm an algebraist!) so you'll have to explain it to me.


Thanks!



Geodesics on a hyperbolic paraboloid
It is standard differential geometry to find the differential equation for the geodesics on this surface. (But I could easily have made a mistake in the calculation anyway.) Since it is a complete negatively curved surface, there is exactly one geodesic connecting any two points. You have a curve
$\vec{p}(t) = (x(t),y(t),z(t))$ on the surface $z = xy$. The geodesic equation is
$$\vec{p}''(t) \propto \vec{\nabla}z \oplus -1$$
(the acceleration is perpendicular to the surface), which expands to
$$(x'',y'',x''y+2x'y' + xy'') \propto (y,x,-1).$$
Thus
$$\frac{x''}{y} = \frac{y''}{x} = -(x''y+2x'y'+xy'').$$
That is an equation that Mathematica can solve. The only tricky part is to solve it with boundary conditions at both ends, where you may as well assume that the endpoints are at $t=0$ and $t=1$. For that purpose it could be better to minimize the curve's energy, by definition
$$E[\vec{p}] = \int\_0^1 |\vec{p}'(t)|^2 dt = \int\_0^1 (x'^2+y'^2 + (x'y+xy')^2) dt.$$
I don't know the most convenient way to do this numerically, but somehow it should be possible. Again, since the surface is negatively curved, this energy functional is well-behaved.


I don't know whether there is a closed form solution in elementary functions. There is a closed form solution for a [round paraboloid](https://mathworld.wolfram.com/ParaboloidGeodesic.html), but it's messy.


