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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..

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Nothing linear or entirely satisfactory is possible. I recommend using the upper half plane, using the positive $y$-axis and the semicircle $x^2 + y^2 = 1, \; y > 0.$ Meanwhile, look up Weierstrass coordinates, as in – Will Jagy Sep 23 '12 at 20:58
There is also Beltrami's model of real hyperbolic spaces, in which the totally geodesic submanifolds are interesections of the unit ball in $\mathbb R^n$ with ordinary affine subspaces. Unfortunately, I do not recall offhand what the hyperbolic angles are in terms of the Euclidean ones (and distance from the origin). – paul garrett Sep 23 '12 at 21:54
Probably meant "plane" instead of "plain" in the question. – ja72 Aug 22 '13 at 15:02
Is this paper relevant? – ja72 Aug 22 '13 at 16:23
Also look at the hyperbolic metric Fleix Klein developed. – ja72 Aug 22 '13 at 16:31

$\newcommand{\Reals}{\mathbf{R}}$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.

Orthogonal lines in the Poincare disk

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 $(0, 0, -1)$.

Projecting the hyperboloid to the disk

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Thank you, really nice pictures. – Berci Jul 1 '15 at 9:13

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