Geodesic
Diffeomorphisms Taking Lines to Circles, and Quaternionic Hopf Fibrations
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 33-43.

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We list all diffeomorphisms between an open subset of the four-dimensional projective space and an open subset of the four-dimensional sphere that take all line segments to arcs of round circles. These diffeomorphisms are restrictions of quaternionic Hopf fibrations and radial projections from hyperplanes to spheres. 
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V. A. Timorin. Diffeomorphisms Taking Lines to Circles, and Quaternionic Hopf Fibrations. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 2, pp. 33-43. http://geodesic.mathdoc.fr/item/FAA_2006_40_2_a3/

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