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A surface in Lobachevskii space, orthogonal to (hyperbolic) parallel straight lines in a certain direction. A horosphere can be considered as a sphere with centre at infinity. A Euclidean geometry can be realized on a horosphere if straight lines are taken to be horocycles, the order of the points is defined through the order of the straight lines in the pencil of parallels defining the horocycle, and a motion is taken to be a motion in Lobachevskii space that takes the horosphere onto itself.


[a1] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a2] A.P. Norden, "Elementare Einführung in die Lobatschewskische Geometrie" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
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Horosphere. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article