View source for Beltrami interpretation

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|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 [[#References|[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|Klein interpretation]]; [[Poincaré model|Poincaré model]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Beltrami,   "Saggio di interpretazione della geometria non-euclidea" , ''Opere Mat.'' , '''1'''  (1868)  pp. 374–405</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.F. Kagan,   "Foundations of geometry" , '''1''' , Moscow-Leningrad  (1949)  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Klein,   "Vorlesungen über Nicht-Euklidische Geometrie" , Springer  (1968)</TD></TR></table>