I'm working on a visualization project in javascript regarding the curvature of the universe. As part of this I'm drawing a saddle shaped plane (negative curvature) and I'd like to draw a geodesic triangle on top of the plane. For reference, I'm trying to recreate the middle figure in this image
I cannot figure out what the geodesic lines on my saddle shape are, and I'd like help.
The saddle in question is $$z = \dfrac{x^2-y^2}{r}$$
where $r$ is an arbitrary parameter. I'm hoping to find an equation for the geodesic line between two arbitrary points $(x_1,y_1)$ and $(x_2,y_2)$ where the $z$ coordinate is defined by the surface. Ideally, I'd like something in parametric form so I can graph it on a computer easily.
I have a hunch that the geodesic is a hyperbola but I can't figure out its form. Any advice is much appreciated!
EDIT: I ended up finding a workaround by using mathematica to list points on the geodesic line a la (https://mathematica.stackexchange.com/questions/103624/geodesics-on-torus), then loading those points into my webpage from a csv file