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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..

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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 en.wikipedia.org/wiki/Hyperboloid_model#History –  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
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