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Let $(r,\varphi)$ be a point of the Poincaré disk. Consider the transformation $$(r,\varphi)\mapsto \left(2\ln\left(\frac{1+r}{1-r}\right),\varphi\right)$$ of the Euclidean plane, i.e., change the distance coordinate to the hyperbolic distance. What are the images of hyperbolic lines under this transformation?

I need this to determine the lines of the so-called native model of the hyperbolic plane. I determined the equations of the transformed hyperbolic lines, but they are very complicated and doesn't seem to be "nice" curves. Othervise, my Geogebra experiments show that they look like hyperbolas with asymptotes $OA$ and $OB$, where $A$ and $B$ are the endpoints of the hyperbolic line.

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I think the factor 2 in your formula is incorrect. It should be 1.

No, they are not hyperbolas. Take two points on the line which connects the ideal points (1,0,0) and (0,1,0) in the Poincaré disk, e.g., $(1-\sqrt{2}/2, 1-\sqrt{2}/2)$ and $(0.2,0.4)$, and transform them. I get $(0.6232, 0.6232)$ and $(0.4304, 0.8608)$. If they were on a hyperbola with asymptotes OX and OY, $xy$ should be constant, and it is not (0.3884 vs 0.3705).

Hyperbolic straight lines are mapped to hyperbolas in the Gans model (i.e., the orthogonal projection of the Minkowski hyperboloid model). I have a sketch of a proof that there can be only one azimuthal model with this property (up to scaling).

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