An orthogonal coordinate system of the hyperbolic plain can be set up by fixing an orgio $O$, an $x$-axis, a $y$-axis (intersecting each other at $O$ in angle $90^\circ$), and, from any point $P$ there is a unique line including $P$ orthogonal to the $x$-axis, and similarly to the $y$-axis. (All it is much similar to the Euclidean case, but the angle at $P$ must be now less than $90^\circ$)
I was wondering, what are the equations of the lines in these terms?
On the other side, what curves will the linear coordinate equations give?
I was trying to use orthogonal circles in the Poincaré disc modell, but the calculations got too complicated.. At least, by symmetry reasons, I could figure out that $y=x$ and $y=-x$ do give lines in the hyperbolic plane, too..