Okay, so I know the cases will probably have to divided up depending on if the geodesic segments are vertical lines or arcs of circles centered on the x-axis, and there has to be some inversion(s), but after that point I am just not sure.
Yes, the upper half plane, so the hyperbolic rigid motions we have covered so far are inversions where the center is located on the x-axis, horizontal translations, and reflections across vertical lines.