Angle of parallelism

Summary

Take a line of the hyperbolic plane and a point at distance $a$ from it. The angle of parallelism $\Pi(a)$ is the angle between the perpendicular to the line and a limiting parallel through the point. It depends only on the distance $a$, and in curvature $-1$ it is given by $\tan(\Pi(a)/2) = e^{-a}$. It decreases from $\pi/2$ to $0$ as $a$ grows, which measures how far hyperbolic geometry departs from Euclid's parallel postulate. The notation $\Pi(a)$ goes back to Lobachevsky.