This may be a trivial question, IMO the answer should be yes.
Given a geodesic $\delta$ on the Poincaré Disk's model with $A, B, C \in \delta$
And given that $f(x)$ is an isometry from the Poincaré Disk to the Poincaré Half-Plane.
Do $f(A)$, $f(B)$ and $f(C)$ lie on the same geodesic $\gamma$ on the Poincaré Half-Plane ?
My implementation shows me that this is not true, but I feel that my implementation is wrong.
Does 3 points on Poincaré Disk geodesic lie on the same Poincaré Half Plane geodesic ?