hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, the angle of parallelism

\Pi(a)

, is the

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

at the non-right angle vertex of a right

hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''. Just as in the Euclidean case, three po ...

having two 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

\Pi(a)

be the least angle such that the line drawn through the point does not intersect the given line. Since two sides are asymptotically parallel, :

\lim_ \Pi(a) = \tfrac\pi\quad\text\quad\lim_ \Pi(a) =  0.

There are five equivalent expressions that relate ''

\Pi(a)

'' and ''a'': :

\sin\Pi(a) = \operatorname a  = \frac =\frac \ ,

\cos\Pi(a) = \tanh a = \frac    \ ,

\tan\Pi(a) = \operatorname a = \frac = \frac  \  ,

\tan \left( \tfrac\Pi(a) \right) = e^,

\Pi(a) = \tfrac\pi - \operatorname(a),

where sinh, cosh, tanh, sech and csch are

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...

s and gd is the

Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...

Construction

János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathem