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?



Mapping the hyperbolic plane onto the interior of a disk
According to the ["Hjelmslev transformation" page](https://en.wikipedia.org/wiki/Hjelmslev_transformation)
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](https://es.wikipedia.org/wiki/Transformaci%C3%B3n_de_Hjelmslev)
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](https://books.google.com/books?id=hK64AAAAIAAJ&q=Hjelmslev&hl=en)
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:


* [zbMATH: author is Johannes Hjelmslev](https://zbmath.org/authors/?q=ai%3Ahjelmslev.johannes)
* [zbMATH: title contains Hjelmslev](https://zbmath.org/?q=ti%3Ahjelmslev)
* [Johannes Hjelmslev on Google Scholar](https://scholar.google.com/scholar?q=hjelmslev%2C+johannes)
* [Johannes Hjelmslev on the Web archive](https://archive.org/search.php?query=creator%3A%22Hjelmslev%2C+Johannes+Trolle%2C+1873-%22)
* ["Johannes Hjelmslev" by Jakob Nielsen (in Danish)](http://publ.royalacademy.dk/backend/web/uploads/2019-09-04/AFL%203/O_1949_00_00_1950_4994/O_1949_04_00_1950_4992.pdf)
* ["Johannes Hjelmslev in memoriam: 1873–1950", by Harald Bohr](https://projecteuclid.org/euclid.acta/1485888580)


