Consider two points $P$ and $Q$ inside the Poincare disk, at distances $r_1$ and $r_2$ respectively from the origin $O$. Furthermore, $P$ and $Q$ form an angle of $\theta$ at the origin. Is there any closed form result for the radius of the orthogonal circle that is used to construct the geodesic through $P$ and $Q$?
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3If you want to construct the geodesic, consider this approach: let $P'$ be the reflection of $P$ at the Poincaré boundary, then the circle through $P, Q, P'$ is the geodesic you're looking for. – Magma May 07 '23 at 00:31