
Angle of parallelism

In hyperbolic geometry, the angle of parallelism formula_1, 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 formula_1 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 " formula_4" and "a":
where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function.
Construction.
János Bolyai discovered a construction which gives the asymptotic parallel "s" to a line "r" passing through a point "A" not on "r". Drop a perpendicular from "A" onto "B" on "r". Choose any point "C" on "r" different from "B". Erect a perpendicular "t" to "r" at "C". Drop a perpendicular from "A" onto "D" on "t". Then length "DA" is longer than "CB", but shorter than "CA". Draw a circle around "C" with radius equal to "DA". It will intersect the segment "AB" at a point "E". Then the angle "BEC" is independent of the length "BC", depending only on "AB"; it is the angle of parallelism. Construct "s" through "A" at angle "BEC" from "AB".
See Trigonometry of right triangles for the formulas used here.
History.
The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nikolai 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:
Demonstration.
In the Poincaré half-plane model of the hyperbolic plane (see Hyperbolic motions), one can establish the relation of "φ" to "a" with Euclidean geometry. Let "Q" be the semicircle with diameter on the "x"-axis that passes through the points (1,0) and (0,"y"), where "y" > 1. Since "Q" is tangent to the unit semicircle centered at the origin, the two semicircles represent "parallel hyperbolic lines". The "y"-axis crosses both semicircles, making a right angle with the unit semicircle and a variable angle "φ" with "Q". The angle at the center of "Q" subtended by the radius to (0, "y") is also "φ" because the two angles have sides that are perpendicular, left side to left side, and right side to right side. The semicircle "Q" has its center at ("x", 0), "x" < 0, so its radius is 1 − "x". Thus, the radius squared of "Q" is
hence
The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, "y") : "y" > 0 } with logarithmic measure. Let log "y" = "a", so "y" = e"a" where e is the base of the natural logarithm. Then 
the relation between "φ" and "a" can be deduced from the triangle {("x", 0), (0, 0), (0, "y")}, for example: