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| 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 [[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 ''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,
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| : <math> \lim_{a\to 0}\phi = \tfrac{1}{2}\pi\quad\text{ and }\quad\lim_{a\to\infty} \phi = 0. </math>
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| These five equivalent expressions relate ''φ'' and ''a'':
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| : <math> \sin\phi = \frac{1}{\cosh a} </math>
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| <!-- extra blank line between two lines of "displayed" [[TeX]] for legibility -->
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| : <math> \tan(\tfrac{1}{2}\phi) = \exp(-a) </math> | |
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| <!-- extra blank line between two lines of "displayed" [[TeX]] for legibility -->
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| : <math> \tan\phi = \frac{1}{\sinh a} </math>
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| <!-- extra blank line between two lines of "displayed" [[TeX]] for legibility -->
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| : <math> \cos\phi = \tanh a </math> | |
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| <!-- extra blank line between two lines of "displayed" [[TeX]] for legibility -->
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| : <math> \phi = \tfrac{1}{2}\pi - \operatorname{gd}(a) </math>
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| where gd is the [[Gudermannian function]].
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| ==Demonstration==
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| [[Image:Angle of parallelism half plane model.svg|thumb|400px|right|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]]
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| In the '''half-plane model''' of the hyperbolic plane (see [[hyperbolic motion]]s) 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
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| : <math> x^2 + y^2 = (1 - x)^2, </math>
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| hence
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| : <math> x = \tfrac{1}{2}(1 - y^2). </math>
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| The [[Metric (mathematics)|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<sup>''a''</sup>. Then
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| the relation between ''φ'' and ''a'' can be deduced from the triangle {(''x'', 0), (0, 0), (0, ''y'')}, for example:
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| : <math> \tan\phi = \frac{y}{-x} = \frac{2y}{y^2 - 1} = \frac{2e^a}{e^{2a} - 1} = \frac{1}{\sinh a}. </math>
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| ==Lobachevsky originator==
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| The following presentation in 1826 by [[Nicolai Lobachevsky]] is from the 1891 translation by [[G. B. Halsted]]:
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| :''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''
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| :: see second appendix of ''Non-Euclidean Geometry'' by Roberto Bonola, [[Dover Publications|Dover edition]].
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| ==References==
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| * Marvin J. Greenberg (1974) ''Euclidean and Non-Euclidean Geometries'', pp. 211–3, [[W.H. Freeman & Company]].
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| * Robin Hartshorne (1997) ''Companion to Euclid'' pp. 319, 325, [[American Mathematical Society]], ISBN 0821807978.
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| * Jeremy Gray (1989) ''Ideas of Space: Euclidean, Non-Euclidean, and Relativistic'', 2nd edition, [[Clarendon Press]], Oxford (See pages 113 to 118).
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| [[Category:Hyperbolic geometry]]
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| [[Category:Functions and mappings]]
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| [[Category:Angle]]
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Wilber Berryhill is the name his mothers and fathers gave him and he totally digs that name. I am currently a travel agent. Her family lives in Ohio but her spouse wants them to transfer. Playing badminton is a thing that he is completely addicted to.
Take a look at my blog post: psychic readers (linked website www.socialairforce.com)