Angle of parallelism

Remarks

The value does not depend on the side of $PQ$, nor on $\ell$ and $P$, only on $a$, and Lobachevsky's formula gives $$\tan \frac{\Pi(a)}{2} = e^{-a}, \qquad \text{equivalently} \qquad \cos \Pi(a) = \tanh a, \quad \sin \Pi(a) = \frac{1}{\cosh a}.$$ Example: in the Beltrami–Klein model, take $P$ the centre and $\ell$ the chord $x = r$ with $0 < r < 1$. Lines through the centre are diameters, so the foot is $Q = (r, 0)$ and $a = \operatorname{artanh} r$. The limiting parallels are the chords from $P$ to the endpoints $(r, \pm\sqrt{1 - r^2})$, which make angle $\theta$ with $\cos \theta = r$ at the centre; angles at the centre are Euclidean in this model. So $\cos \Pi(a) = r = \tanh a$, as the formula says. Non-example: in the Euclidean plane every ray from $P$ at angle less than $\pi/2$ with $PQ$ meets $\ell$, and the ray at angle $\pi/2$ is the unique parallel. There the supremum is $\pi/2$ for every $a$, so the angle carries no information about distance. In curvature $-k^2$, the formula holds with $a$ replaced by $ka$.