Editing Angle of parallelism (section)

{{short description|An angle in certain right triangles in the hyperbolic plane}}
[[File:Angleofparallelism.svg|thumb|Angle of parallelism in hyperbolic geometry]]

In [[hyperbolic geometry]], '''angle of parallelism ''' <math> \Pi(a) </math> is the [[angle]] at the non-right angle vertex of a right [[hyperbolic triangle]] having two [[limiting parallel|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 <math> \Pi(a) </math> be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel,

: <math> \lim_{a\to 0} \Pi(a) = \tfrac{1}{2}\pi\quad\text{ and }\quad\lim_{a\to\infty} \Pi(a) =  0.  </math>

There are five equivalent expressions that relate '' <math> \Pi(a)</math>'' and ''a'':

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: <math> \sin\Pi(a) = \operatorname{sech} a  = \frac{1}{\cosh a} =\frac{2}{e^a + e^{-a}} \ , </math>

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: <math> \cos\Pi(a) = \tanh a = \frac {e^a - e^{-a}} {e^a + e^{-a}}  \ , </math>

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: <math> \tan\Pi(a) = \operatorname{csch} a = \frac{1}{\sinh a} = \frac {2}{e^a - e^{-a}} \  , </math>

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: <math> \tan \left( \tfrac{1}{2}\Pi(a) \right) = e^{-a}, </math>

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: <math> \Pi(a) = \tfrac{1}{2}\pi - \operatorname{gd}(a), </math>

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