In the Poincaré representation of the hyperbolic plane a lines is a sections of a circle perpendicular to the unit disk and running through 2 points of interest $x_1$ and $x_2$. I wish to find the analogous curve in the polar hyperbolic representation. I know that this curve should have some dent toward the origin however I could not find one or come up with an equation for it myself.
Given two points $x_1=(r_1, \phi_1)$ and $x_2=(r_2, \phi_2)$ what is the equation of the projection of hyperbolic line running through these points in the Euclidean plane?