In Chapter II of his book Non-Euclidean Geometry (1961; first published in Polish, 1956), Stefan Kulczycki defines a mapping of the hyperbolic plane onto the interior of a disk. Its construction begins by specifying a center O and an acute angle α. Given those, the image P’ of an arbitrary point P is found as follows: drop a perpendicular PQ from P to a radial line making the angle α with OP, then mark off the distance OP’ = OQ on OP. Thus P' lies on OP between O and P. Using Hjelmslev's theorem in absolute geometry (1907), one shows that the mapping preserves straightness.
Where did this mapping, together with its property of preserving straightness, first appear?