Beltrami interpretation

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A realization of a part of the Lobachevskii plane on a pseudo-sphere — a surface of constant negative curvature. In the Beltrami interpretation, the geodesic lines (cf. Geodesic line) and their segments on the pseudo-sphere play the role of straight lines and their segments on the Lobachevskii plane. The isometric mappings of the pseudo-sphere onto itself represent the movements in the Lobachevskii plane, with preservation of horocircles. The lengths, angles and areas on the pseudo-sphere correspond to the lengths, angles and areas on the Lobachevskii plane. Under these conditions, to each theorem of Lobachevskii planimetry which refers to a part of the Lobachevskii plane there corresponds an analogous theorem of the intrinsic geometry of the pseudo-sphere. The Beltrami interpretation realizes a part of the Lobachevskii plane, but the entire Lobachevskii plane cannot be realized in three-dimensional Euclidean space as a regular surface (Hilbert's theorem).

The Beltrami interpretation was proposed by E. Beltrami in 1868 [1]. This publication represented the first realization of Lobachevskii's "imaginary geometry" in three-dimensional Euclidean space.

For other interpretations of the Lobachevskii geometry see Klein interpretation; Poincaré model.


[1] E. Beltrami, "Saggio di interpretazione della geometria non-euclidea" , Opere Mat. , 1 (1868) pp. 374–405
[2] V.F. Kagan, "Foundations of geometry" , 1 , Moscow-Leningrad (1949) (In Russian)



[a1] F. Klein, "Vorlesungen über Nicht-Euklidische Geometrie" , Springer (1968)
How to Cite This Entry:
Beltrami interpretation. E.V. Shikin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098