Geodesic
Circles and Clifford Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 56-64

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Consider a smooth map of a neighborhood of the origin in a real vector space into a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra. We also describe a connection between our result and the Hurwitz–Radon theorem about sums of squares.
Keywords: line, circle, Clifford algebra, Hopf map. 
@article{FAA_2004_38_1_a4,
     author = {V. A. Timorin},
     title = {Circles and {Clifford} {Algebras}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {56--64},
     publisher = {mathdoc},
     volume = {38},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a4/}
}
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V. A. Timorin. Circles and Clifford Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 1, pp. 56-64. http://geodesic.mathdoc.fr/item/FAA_2004_38_1_a4/