What is "hyperbolic geometry"

n. a non-Euclidean geometry in which it is assumed that through any point there are two or more parallel lines that do not intersect a given line in the plane

In mathematics, hyperbolic geometry (also called Bolyai– Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. In hyperbolic 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 this with Playfair's axiom, the modern version of Euclid's parallel postulate)

Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.

A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.

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 Russia it is commonly called Lobachevskian geometry after one of its discoverers, the Russian geometer Nikolai Lobachevsky.

This page is mainly about the 2 dimensional or plane hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry.

Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.