Angle of parallelism

In hyperbolic geometry, the angle of parallelism φ, also known as Π(p), is the angle at one vertex of a right hyperbolic triangle that has two asymptotic parallel 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 asymptotic parallel,

\lim_{a\to 0}\varphi = \tfrac{1}{2}\pi\quad\text{ and }\quad\lim_{a\to\infty} \varphi = 0.

These five equivalent expressions relate φ and a:

\sin\varphi = \frac{1}{\cosh a}
\tan(\tfrac{1}{2}\varphi) = \exp(-a)
\tan\varphi = \frac{1}{\sinh a}
\cos\varphi = \tanh a
\varphi = \tfrac{1}{2}\pi - \operatorname{gd}(a)

where gd is the Gudermannian function.

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 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

x^2 %2B y^2 = (1 - x)^2,

hence

x = \tfrac{1}{2}(1 - y^2).

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

\tan\varphi = \frac{y}{-x} = \frac{2y}{y^2 - 1} = \frac{2e^a}{e^{2a} - 1} = \frac{1}{\sinh a}.

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.

References