Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Lobachevskian geometry or BolyaiLobachevskian 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.
Lines through a given point P and asymptotic to line R
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), along with two diverging ultra-parallel lines

(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)

Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.

A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.

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 former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.

This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions.

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