Angle of parallelism

In hyperbolic geometry, angle of parallelism ${\displaystyle \Pi (a)}$ is the angle at the non-right angle vertex of a right hyperbolic triangle having 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 not on a line, drop a perpendicular to the line from the point. Let a be the length of this perpendicular segment, and ${\displaystyle \Pi (a)}$ be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel,

${\displaystyle \lim _{a\to 0}\Pi (a)={\tfrac {1}{2}}\pi \quad {\text{ and }}\quad \lim _{a\to \infty }\Pi (a)=0.}$

There are five equivalent expressions that relate ${\displaystyle \Pi (a)}$ and a:

${\displaystyle \sin \Pi (a)=\operatorname {sech} a={\frac {1}{\cosh a}}={\frac {2}{e^{a}+e^{-a}}}\ ,}$

${\displaystyle \cos \Pi (a)=\tanh a={\frac {e^{a}-e^{-a}}{e^{a}+e^{-a}}}\ ,}$

${\displaystyle \tan \Pi (a)=\operatorname {csch} a={\frac {1}{\sinh a}}={\frac {2}{e^{a}-e^{-a}}}\ ,}$

${\displaystyle \tan \left({\tfrac {1}{2}}\Pi (a)\right)=e^{-a},}$

${\displaystyle \Pi (a)={\tfrac {1}{2}}\pi -\operatorname {gd} (a),}$

where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function.