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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?

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According to the "Hjelmslev transformation" page on the English language Wikipedia, this mapping is due to Johannes Hjelmslev himself who studied its properties and proved it preserves straightness.

The corresponding "Transformación de Hjelmslev" page on the Spanish language Wikipedia states that too, with additionally a reference to

  • Reports of a mathematical colloquium, vol 4--8, University of Notre Dame, 1943.

Excerpts of those reports on Google Books show hits for "Hjelsmlev transformation".

They are from a paper published in three parts in the "Reports":

  • James C. Abbott. The projective theory of non-euclidean geometry I. In: Reports of a mathematical colloquium, second series, issue 3, edited by Karl Menger, Notre Dame University Press, 1941. Pages 13--27.

  • James C. Abbott. The projective theory of non-euclidean geometry II. In: Reports of a mathematical colloquium, second series, issue 4, edited by Karl Menger, Notre Dame University Press, 1943. Pages 22--30.

  • James C. Abbott. The projective theory of non-euclidean geometry (conclusion). In: Reports of a mathematical colloquium, second series, issue 5-6, edited by Karl Menger, Notre Dame University Press, 1944. Pages 43--52.

and more precisely, they come from the fifth section, which is contained in the last part of the article, in issue 5-6 of the Reports.

Extra links:

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  • $\begingroup$ Thank you for the references. Unfortunately, the Wikipedia article cites no sources. Hjelmslev's 1907 paper does not address the application of his theorem to non-Euclidean geometry. Without further information, there is no reason to attribute that application to him. $\endgroup$
    – Firestone
    Commented Dec 29, 2020 at 15:49
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    $\begingroup$ Thanks again for pusuing this! Google's exiguous snippets give us the phrase, and even an indication that it means the kind of thing I'm interested in, but, alas, no suggestion of the crucial preservation of straightness. So the book is needed. WorldCat finds 5 copies, all (unlike me) in Europe, 2 of them in Paris libraries: at Sorbonne Université, and at Bibliothèque Interuniv Science Jussieu-RCON. If you should happen to visit one of those... Well, perhaps when times are better. $\endgroup$
    – Firestone
    Commented Dec 30, 2020 at 21:42
  • $\begingroup$ Email me for a scanned version of Abbott's paper. $\endgroup$
    – Samuel Lelièvre
    Commented Nov 12, 2021 at 16:15

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