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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 http://en.wikipedia.org/wiki/Hyperboloid_model#History2012-09-23
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    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).2012-09-23
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    Probably meant "plane" instead of "plain" in the question.2013-08-22
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    Is this [paper](http://www.math.sunysb.edu/~bishop/classes/math627.S13/Beardon-Minda.pdf) relevant?2013-08-22
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    Also look at the [hyperbolic metric](http://mathworld.wolfram.com/Cayley-Klein-HilbertMetric.html) Fleix Klein developed.2013-08-22
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    Try using hyperbolic trigonometry2015-06-29

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