Angle of Parallelism

Given a point $P$ and a Line $AB$, draw the Perpendicular through $P$ and call it $PC$. Let $PD$ be any other line from $P$ which meets $CB$ in $D$. In a Hyperbolic Geometry, as $D$ moves off to infinity along $CB$, then the line $PD$ approaches the limiting line $PE$, which is said to be parallel to $CB$ at $P$. The angle $\angle CPE$ which $PE$ makes with $PC$ is then called the angle of parallelism for perpendicular distance $x$, and is given by

$$\Pi(x)=2\tan^{-1} (e^{-x}).$$

This is known as Lobachevsky's Formula.

See also Hyperbolic Geometry, Lobachevsky's Formula

References

Manning, H. P. Introductory Non-Euclidean Geometry. New York: Dover, pp. 31-32 and 58, 1963.