Angle of parallelism

In hyperbolic geometry, the angle of parallelism ${\displaystyle \Pi (a)}$, 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 ${\displaystyle \Pi (a)}$ 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,

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

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