Angle of parallelism

Since two sides are asymptotically parallel, There are five equivalent expressions that relate

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.

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: In the Poincaré half-plane model of the hyperbolic plane (see Hyperbolic motions), one can establish the relation of Φ to a with Euclidean geometry.

Thus, the radius squared of Q is hence The metric of the Poincaré half-plane model of hyperbolic geometry parametrizes distance on the ray {(0, y) : y > 0 } with logarithmic measure.

Angle of parallelism in hyperbolic geometry
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.
This diagram, with yellow ideal triangle , is similar to one found in a book by Smogorzhevsky. [ 4 ]