Editing Angle of parallelism (section)

==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.