Suppose a surface with constant curvature $C$ in which are two points $P_1$ and $P_2$, described using polar coordinates from origin $O$.
As I draw a line from the origin at some angle $\theta$ (such that $\theta_1 \geq \theta \geq \theta_2$) this line will intersect the line segment $\bar{P_1 P_2}$. I am interested in finding the coordinates of this intersection.