Angle of parallelism - Definition

In hyperbolic geometry, the angle of parallelism Φ is the angle at one vertex of an right hyperbolic triangle that has two parallel sides.The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism Φ.Since two sides are parallel,

lim_a→0 Φ = π/2 and lim_a→∞ Φ = 0.

There are four equivalent expressions relating Φ and a:

sin Φ = 1/cosh a

tan(Φ/2) = exp(-a)

tan Φ = 1/sinh a

cos Φ = tanh a

Demonstration

In the 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 abscissa and through (0,y), y > 1, and (1,0).Since it is tangent to the unit semicircle centered at the origin, the two semicircles represent parallel hyperbolic lines.The ray {(0,y): y > 0 } 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 center at (x,0), x < 0, so the radius squared of Q is

x² + y² = (1-x)², hence x = (1 –y²)/2.

The metric of the half-plane model of hyperbolic geometry parametrizes distance on the ray {(0,y): y > 0 } with natural logarithm.Then log y = a, or y = e^a so that the relation between Φ and a can be deduced from the triangle {(x,0),(0,0),(0,y)}, for example

tan Φ = y/(-x) = 2y/(y²-1) = 2 e^a/(e^2a-1) = 1/sinh a .