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.

常曲率曲面上的渐近线之间的角度——通常是切比雪夫网络的线之间——通常被解析为二阶偏微分方程的解。对于欧几里得空间中恒定负曲率的曲面，这是正弦-戈登方程。相反，恒定负曲率的曲面用于构造和解释正弦-戈登方程的解。这篇文章表明，欧几里得空间和伪欧几里得空间的伪球面上的渐近线之间的夹角可以有不同的解释，即罗巴切夫斯基平面或其理想区域的平行度的两倍角，局部具有德西特几何分别是飞机。