Stopping power

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,

$\lim _{a\to 0}\phi ={\tfrac {1}{2}}\pi \quad {\text{ and }}\quad \lim _{a\to \infty }\phi =0.$

These five equivalent expressions relate φ and a:

$\sin \phi ={\frac {1}{\cosh a}}$

$\tan({\tfrac {1}{2}}\phi )=\exp(-a)$

$\tan \phi ={\frac {1}{\sinh a}}$

$\cos \phi =\tanh a$

$\phi ={\tfrac {1}{2}}\pi -\operatorname {gd} (a)$

where gd is the Gudermannian function.

Demonstration

The angle of parallelism, φ, formulated as: (a) The angle between the x-axis and the line running from x, the center of Q, to y, the y-intercept of Q, and (b) The angle from the tangent of Q at y to the y-axis

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

$x^{2}+y^{2}=(1-x)^{2},$

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 = e^a. Then the relation between φ and a can be deduced from the triangle {(x, 0), (0, 0), (0, y)}, for example:

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

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

• Marvin J. Greenberg (1974) Euclidean and Non-Euclidean Geometries, pp. 211–3, W.H. Freeman & Company.
• Robin Hartshorne (1997) Companion to Euclid pp. 319, 325, American Mathematical Society, ISBN 0821807978.
• Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press, Oxford (See pages 113 to 118).