Lines in coordinate system of Hyperbolic Plane

Question

An orthogonal coordinate system of the hyperbolic plane 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..

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$For what it's worth, here's a plot of two such families of "equally-spaced" lines in the Poincaré model. (That is, the spacing along the "horizontal axis" of two adjacent "vertical" lines is the same for each pair, and similarly for the second family.) As expected, a line in one family fails to intersect some lines from the other family.

For plotting, these lines were described by intersecting planes through the origin with the "unit sphere" of future timelike vectors in $\Reals^{2,1}$, obtaining the families of lines $$ \gamma_{x, c}(t) = (c\cosh t, \sinh t, \sqrt{1 + c^{2}} \cosh t),\qquad \gamma_{y, c}(t) = (\sinh t, c\cosh t, \sqrt{1 + c^{2}} \cosh t), $$ then mapping to the Poincaré disk (the unit disk in the plane $z = 0$) via $$ (x, y, z) \mapsto \frac{(x, y)}{1 + z}, $$ i.e., via stereographic projection from $(0, 0, -1)$.