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 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.)

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.

Lines through a given point P and asymptotic to line R

1 Properties

1.1 Relation to Euclidean geometry

1.2 Lines

1.2.1 Non-intersecting / parallel lines

1.3 Circles and disks

1.4 Hypercycles and horocycles

1.5 Triangles

1.6 Regular apeirogon

1.7 Tessellations

2 Standardized Gaussian curvature

2.1 Cartesian-like coordinate systems

2.2 Distance

3 History

3.1 19th-century developments

3.2 Philosophical consequences

3.3 Geometry of the universe (spatial dimensions only)

3.4 Geometry of the universe (special relativity)

4 Physical realizations of the hyperbolic plane

5 Models of the hyperbolic plane

5.1 The Beltrami–Klein model

5.2 The Poincaré disk model

5.3 The Poincaré half-plane model

5.4 The hyperboloid model

5.5 The hemisphere model

5.6 The Gans model

5.7 The band model

5.8 Connection between the models

6 Isometries of the hyperbolic plane

7 Hyperbolic geometry in art

8 Higher dimensions

9 Homogeneous structure

10 See also