Asymptotic Lines on Pseudospheres and the Angle of Parallelism

Abstract

The angle between the asymptotic lines―and generally between the lines of the Chebyshev network―on surfaces of constant curvature is usually analytically interpreted as a solution of the second-order partial differential equation. For surfaces of constant negative curvature in Euclidean space, this is the sine-Gordon equation. Conversely, surfaces of constant negative curvature are used to construct and interpret solutions to the sine-Gordon equation. This article shows that the angle between the asymptotic lines on the pseudospheres of Euclidean and pseudo-Euclidean spaces can be interpreted differently, namely, as the doubled angle of parallelism of the Lobachevsky plane or its ideal region, locally having the geometry of the de Sitter plane, respectively.