In hyperbolic geometry, the angle of parallelism at a point is the angle at the non-right-angle vertex of a right hyperbolic triangle whose two other sides are asymptotically parallel, and it depends only on the length of the perpendicular segment between the point and the line it is being compared to. As that segment length approaches zero the angle of parallelism approaches a right angle, and as the length grows without bound the angle approaches zero; five equivalent trigonometric expressions relate the angle of parallelism to the segment length using the hyperbolic sine, cosine, tangent and related functions, and it can also be written in terms of the Gudermannian function.
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1. Angle of Parallelism (Wikipedia)
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In hyperbolic geometry, angle of parallelism Π ( a ) {\displaystyle \Pi (a)} is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides.
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The angle of parallelism was developed in 1840 in the German publication "Geometrische Untersuchungen zur Theory der Parallellinien" by Nikolai Lobachevsky.
In Branch: Geometry, Lead sentence
In hyperbolic geometry, angle of parallelism Π ( a ) is the angle at the non-right angle vertex of a right hyperbolic triangle hav
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