Functions and mappings | Hyperbolic geometry | Angle
Angle of parallelism
In hyperbolic geometry, the angle of parallelism , 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 be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, There are five equivalent expressions that relate and a: where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function. (Wikipedia).
