Angle of parallelism

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

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

[edit] 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) = 2e^a/ (e^2a − 1) = 1/sinh a.

[edit] Lobachevsky originator

The following presentation in 1826 by Nicolai Lobachevsky is from the 1891 translation by G. B. Halsted:

The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism) which we will here designate by Π(p) for AD = p

see second appendix of Non-Euclidean Geometry by Roberto Bonola, Dover edition.