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Angle of parallelism

In hyperbolic geometry, the angle of parallelism $\Pi\left(a\right)$, is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism.
Given a point off of a line, if we drop a perpendicular to the line from the point, then ''a'' is the distance along this perpendicular segment, and ''φ'' or $\Pi\left(a\right)$ is the least angle such that the line drawn through the point at that angle does not intersect the given line. Since two sides are asymptotic parallel,
: $\lim_ \Pi\left(a\right) = \tfrac\pi\quad\text\quad\lim_ \Pi\left(a\right) = 0.$
There are five equivalent expressions that relate '' $\Pi\left(a\right)$'' and ''a'':

: $\sin\Pi\left(a\right) = \operatorname a = \frac ,$

: $\cos\Pi\left(a\right) = \tanh a ,$

: $\tan\Pi\left(a\right) = \operatorname a = \frac,$

: $\tan\left(\tfrac\Pi\left(a\right)\right) = e^,$

: $\Pi\left(a\right) = \tfrac\pi - \operatorname\left(a\right),$
where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function.
==History==

The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nicolai Lobachevsky.
This publication became widely known in English after the Texas professor G. B. Halsted produced a translation in 1891. (''Geometrical Researches on the Theory of Parallels'')
The following passages define this pivotal concept in hyperbolic geometry:
:''The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism) which we will here designate by Π(p) for AD = p''.〔Nicholaus Lobatschewsky (1840) G.B. Halsted translator (1891) (Geometrical Researches on the Theory of Parallels ), link from Google Books

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