Angle of parallelism
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,
- lima→0 Φ = π/2 and lima→∞ 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 = ea 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) = 2ea/(e2a − 1) = 1/sinh a.