Does anyone know of a relatively simple and easy-to-use graphics app (like GeoGebra) that can graph hyperbolic geodesics on a hyperbolic paraboloid? I'm trying to graph the geodesics on the top image to the surface on the bottom image, specifically to best visualize the hyperbolic case of the parallel postulate. I haven't found a way to do it on GeoGebra.
As Lee Mosher explained, there is no isometry between the two. The natural 3D construction of hyperbolic plane is to use half of a two sheet hyperboloid: $$ x^2+y^2-z^2=1 $$ with the $(+,+,-)$ metric. As a relativistic analogue, it's a surface of constant proper time with respect to the origin. In this case, the geodesics are the intersection of the sheet with a plane passing by the origin. This will let you easily visualise the parallel postulate.
The one sheet paraboloid is the hyperbolic analogue of the unit sphere for the euclidean metric. The construction of the geodesics is the analogue of the great circles of a sphere. You just change the signature of the metric, so the same reasoning applies. The analogue of the rotations are Lorentz transformations. Btw, to get the Poincaré disk from there, you just do a stereographic projection from the "South Pole" at $(0,0,-1)$. Alternatively, you can get the Beltrami-Klein model by a gnomonic projection from the origin at $(0,0,0)$ (this immediately shows why the geodesics are straight).
While the hyperbolic paraboloid is common introductory example for hyperbolic geometry, it is not the simplest mathematical model as it is not maximally symmetric. For example, it is not isotropic. This is why there is no simple geometric construction for finding its geodesics. If you really want to calculate its geodesics, there are different methods outlined in the answers to geodesics on a hyperbolic paraboloid.
Hope this helps.