Editing Hyperbolic geometry (section)

{{More citations needed|date=June 2023}}
{{Short description|Type of non-Euclidean geometry}}
{{Other uses|Hyperbolic (disambiguation)}}
[[File:Hyperbolic.svg|frame|right|Lines through a given point ''P'' and asymptotic to line ''R'']]
{{General geometry |branches}}
[[File:Hyperbolic triangle.svg|thumb|250px|right|A triangle immersed in a saddle-shape plane (a [[hyperbolic paraboloid]]), along with two diverging ultra-parallel lines]]

In [[mathematics]], '''hyperbolic geometry''' (also called '''Lobachevskian geometry''' or '''[[János Bolyai|Bolyai]]–[[Nikolai Lobachevsky|Lobachevskian]] geometry''') is a [[non-Euclidean geometry]]. The [[parallel postulate]] of [[Euclidean geometry]] is replaced with:
:For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''.
(Compare the above with [[Playfair's axiom]], the modern version of [[Euclid]]'s [[parallel postulate]].)

The '''hyperbolic plane''' is a [[plane (mathematics)|plane]] where every point is a [[saddle point]]. 
Hyperbolic plane [[geometry]] is also the geometry of [[pseudosphere|pseudospherical surfaces]], surfaces with a constant negative [[Gaussian curvature]]. [[Saddle surface]]s have negative Gaussian curvature in at least some regions, where they [[local property|locally]] resemble the hyperbolic plane.

The [[hyperboloid model]] of hyperbolic geometry provides a representation of [[event (relativity)|event]]s one temporal unit into the future in [[Minkowski space]], the basis of [[special relativity]]. Each of these events corresponds to a [[rapidity]] in some direction.

When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; [[Felix Klein]] finally gave the subject the name '''hyperbolic geometry''' to include it in the now rarely used sequence [[elliptic geometry]] ([[spherical geometry]]), parabolic geometry ([[Euclidean geometry]]), and hyperbolic geometry.
In the [[Post-Soviet states|former Soviet Union]], it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer [[Nikolai Lobachevsky]].