matrices – $PSL(2,mathbb{R})$, $PSO(2)$ and Hyperbolic Distance

Let $PSL(2,mathbb{R})$ be the Projective Special Linear Group and $PSO(2)$ be the Projective Special Orthogonal Group.

It is well-known that $PSL(2,mathbb{R})/PSO(2)$ can be identified with the upper half-plane $mathbb{H}$.

Let $g,h$ be two elements of $PSL(2,mathbb{R})$, How can the hyperbolic distance between $gPSO(2)$ and $hPSO(2)$ be computed?