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Here's an atypical conformal model of the hyperbolic plane.

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Here's an atypical conformal model of the hyperbolic plane. The Poincaré disk is mapped to the entire plane via the Joukowsky transformation (https://www.johndcook.com/blog/2016/01/31/joukowsky-transformation/), compressing the boundary-at-infinity to the interval [-1,1] on the real axis.

Has this model appeared in the literature? If so, what is it named?