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## Main Question or Discussion Point

Hello,

it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere.

What is the geometrical construction that corresponds to a 1-1 mapping between the Poincaré disk and the points on the 2d Cartesian plane?

My attempt was to consider a point (x,y) on the Cartesian plane, then make a parallel projection on the upper sheet of the hyperboloid, thus obtaining (x,y,h), and finally apply a projective transformation with focal point (0,0,-1) and focal plane coincident with the Cartesian plane. This way all the points on the hyperboloid (upper sheet) are mapped onto the unit ball of the Cartesian plane, with the points on the unit circle representing the points at infininity.

Is this the correct way of doing it?

it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere.

What is the geometrical construction that corresponds to a 1-1 mapping between the Poincaré disk and the points on the 2d Cartesian plane?

My attempt was to consider a point (x,y) on the Cartesian plane, then make a parallel projection on the upper sheet of the hyperboloid, thus obtaining (x,y,h), and finally apply a projective transformation with focal point (0,0,-1) and focal plane coincident with the Cartesian plane. This way all the points on the hyperboloid (upper sheet) are mapped onto the unit ball of the Cartesian plane, with the points on the unit circle representing the points at infininity.

Is this the correct way of doing it?