- Angle of parallelism
In
hyperbolic geometry , the angle of parallelism Φ is theangle at one vertex of a righthyperbolic 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"→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 motion s) one can establish the relation of Φ to "a" withEuclidean 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" = (1–"y"2)/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 Φ = "y"/(−"x") = 2"y"/ ("y"2 − 1) = 2"e""a"/ (e2"a" − 1) = 1/sinh "a".
Lobachevsky originator
The following presentation in 1826 by
Nicolai Lobachevsky is from the 1891 translation byG. 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. 211-13, W. H. Freeman & Co.
* Robin Hartshorne (1997) "Companion to Euclid" p.319,p.325, AMS, [ISBN 0821807978] .
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