Lobachevsky's Formula

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

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

which is called Lobachevsky's formula.

See also Angle of Parallelism, Hyperbolic Geometry

References

Manning, H. P. Introductory Non-Euclidean Geometry. New York: Dover, p. 58, 1963.