Editing Angle of parallelism

{{short description|An angle in certain right triangles in the hyperbolic plane}}
[[File:Angleofparallelism.svg|thumb|Angle of parallelism in hyperbolic geometry]]

In [[hyperbolic geometry]], '''angle of parallelism ''' <math> \Pi(a) </math> is the [[angle]] at the non-right angle vertex of a right [[hyperbolic triangle]] having two [[limiting parallel|asymptotic parallel]] sides. The angle depends on the segment length ''a'' between the right angle and the vertex of the angle of parallelism. 

Given a point not on a line, drop a perpendicular to the line from the point. Let ''a'' be the length of this perpendicular segment, and <math> \Pi(a) </math> be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel,

: <math> \lim_{a\to 0} \Pi(a) = \tfrac{1}{2}\pi\quad\text{ and }\quad\lim_{a\to\infty} \Pi(a) =  0.  </math>

There are five equivalent expressions that relate '' <math> \Pi(a)</math>'' and ''a'':

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: <math> \sin\Pi(a) = \operatorname{sech} a  = \frac{1}{\cosh a} =\frac{2}{e^a + e^{-a}} \ , </math>

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: <math> \cos\Pi(a) = \tanh a = \frac {e^a - e^{-a}} {e^a + e^{-a}}  \ , </math>

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: <math> \tan\Pi(a) = \operatorname{csch} a = \frac{1}{\sinh a} = \frac {2}{e^a - e^{-a}} \  , </math>

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: <math> \tan \left( \tfrac{1}{2}\Pi(a) \right) = e^{-a}, </math>

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: <math> \Pi(a) = \tfrac{1}{2}\pi - \operatorname{gd}(a), </math>

where sinh, cosh, tanh, sech and csch are [[hyperbolic function]]s and gd is the [[Gudermannian function]].

==Construction==
[[János Bolyai]] discovered a construction which gives the asymptotic parallel ''s'' to a line ''r'' passing through a point ''A'' not on ''r''.<ref>"Non-Euclidean Geometry" by Roberto Bonola, page 104, Dover Publications.</ref> Drop a perpendicular from ''A'' onto ''B'' on ''r''. Choose any point ''C'' on ''r'' different from ''B''. Erect a perpendicular ''t'' to ''r'' at ''C''. Drop a perpendicular from ''A'' onto ''D'' on ''t''. Then length ''DA'' is longer than ''CB'', but shorter than ''CA''. Draw a circle around ''C'' with radius equal to ''DA''. It will intersect the segment ''AB'' at a point ''E''. Then the angle ''BEC'' is independent of the length ''BC'', depending only on ''AB''; it is the angle of parallelism. Construct ''s'' through ''A'' at angle ''BEC'' from ''AB''.
:<math> \sin BEC = \frac{ \sinh {BC} }{ \sinh {CE} } = \frac{ \sinh {BC} }{ \sinh {DA} } = \frac{ \sinh {BC} }{ \sin {ACD} \sinh {CA} } = \frac{ \sinh {BC} }{ \cos {ACB} \sinh {CA} } = \frac{ \sinh {BC} \tanh {CA} }{ \tanh {CB} \sinh {CA} } = \frac{ \cosh {BC} }{ \cosh {CA} } = \frac{ \cosh {BC} }{ \cosh {CB} \cosh {AB} } = \frac{ 1 }{ \cosh {AB} } \,.</math>

See [[Hyperbolic triangle#Trigonometry of right triangles|Trigonometry of right triangles]] for the formulas used here.

==History==
{{Anchor|Lobachevsky originator}} <!-- Lobachevsky's formula links here -->

The '''angle of parallelism''' was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien"  by [[Nikolai Lobachevsky]].

This publication became widely known in English after the Texas professor [[G. B. Halsted]] produced a translation in 1891. (''Geometrical Researches on the Theory of Parallels'')

The following passages define this pivotal concept in hyperbolic geometry:
:''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''.<ref name=Lob>[[Nikolai Lobachevsky]] (1840) [[G. B. Halsted]] translator (1891) [https://archive.org/details/geometricalresea00loba Geometrical Researches on the Theory of Parallels]</ref>{{rp|13}}<ref>{{cite book|last1=Bonola|first1=Roberto|title=Non-Euclidean geometry : a critical and historical study of its developments|url=https://archive.org/details/noneuclideangeom0000bono|url-access=registration|date=1955|publisher=Dover|location=New York, NY|isbn=0-486-60027-0|edition=Unabridged and unaltered republ. of the 1. English translation 1912.}}</ref>

==Demonstration==

[[Image:Angle of parallelism half plane model.svg|thumb|400px|right|The angle of parallelism, ''&Phi;'', 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.<br>This diagram, with yellow [[ideal triangle]], is similar to one found in a book by Smogorzhevsky.<ref>A.S. Smogorzhevsky (1982) ''Lobachevskian Geometry'', §12 Basic formulas of hyperbolic geometry, figure 37, page 60, [[Mir Publishers]], Moscow</ref>]] 

In the [[Poincaré half-plane model]] of the hyperbolic plane (see [[Hyperbolic motion]]s), one can establish the relation of ''&Phi;'' 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 ''&Phi;'' with ''Q''. The angle at the center of ''Q'' subtended by the radius to (0,&nbsp;''y'') is also ''&Phi;'' 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'',&nbsp;0), ''x'' < 0, so its radius is 1&nbsp;&minus;&nbsp;''x''. Thus, the radius squared of ''Q'' is

: <math> x^2 + y^2 = (1 - x)^2, </math>

hence

: <math> x = \tfrac{1}{2}(1 - y^2). </math>

The [[Metric (mathematics)|metric]] of the [[Poincaré half-plane model]] of hyperbolic geometry parametrizes distance on the ray {(0,&nbsp;''y'') : ''y'' > 0 } with [[logarithmic measure]]. Let the hyperbolic distance from (0,&nbsp;''y'') to (0,&nbsp;1) be ''a'', so: log&nbsp;''y'' &minus; log 1 = ''a'', so ''y'' = ''e<sup>a</sup>'' where [[e (mathematical constant)|''e'']] is the base of the [[natural logarithm]]. Then 
the relation between ''&Phi;'' and ''a'' can be deduced from the triangle {(''x'',&nbsp;0),&nbsp;(0,&nbsp;0),&nbsp;(0,&nbsp;''y'')}, for example:

: <math> \tan\phi = \frac{y}{-x} = \frac{2y}{y^2 - 1} = \frac{2e^a}{e^{2a} - 1} = \frac{1}{\sinh a}. </math>

==References==
{{Reflist}}

* [[Marvin Greenberg|Marvin J. Greenberg]] (1974) ''Euclidean and Non-Euclidean Geometries'', pp.&nbsp;211&ndash;3, [[W.H. Freeman & Company]].
* [[Robin Hartshorne]] (1997) ''Companion to Euclid'' pp.&nbsp;319,&nbsp;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).
* [[Béla Kerékjártó]] (1966) ''Les Fondements de la Géométry'', Tome Deux, §97.6 Angle de parallélisme de la géométry hyperbolique, pp. 411,2, Akademiai Kiado, Budapest.

[[Category:Hyperbolic geometry]]
[[Category:Functions and mappings]]
[[Category:Angle]]