Hyperbolic geometry is a non-Euclidean geometry of constant negative sectional curvature,
also called Lobachevsky-Bolyai-Gauss geometry. The curvature is often normalized
to 
;
after a rescaling by 
, it is 
. In the hyperbolic plane, the parallel
postulate is replaced by the statement that for a line 
and a point 
not on it, infinitely many lines through
do not intersect 
.
The angle sum of a hyperbolic triangle is less than 
.
Hyperbolic triangles with the same angles are congruent
and therefore have the same area, so there are no noncongruent similar triangles.
Two standard disk models are the Poincaré
hyperbolic disk and the Klein-Beltrami model.
Other standard models include the Poincaré half-plane model and hyperboloid
model. In the Poincaré half-plane model, the upper
half-plane represents the hyperbolic plane, with hyperbolic lines represented
by vertical Euclidean lines and semicircles centered on the real axis.
Relations among the sides and angles of hyperbolic triangles are given by the hyperbolic law of cosines, hyperbolic law of sines, and hyperbolic law of tangents (Anderson 1999, pp. 146-151).
In the Klein-Beltrami model, an open disk in the Euclidean plane represents the hyperbolic plane and its open chords
represent hyperbolic lines. Felix Klein constructed an analytic hyperbolic geometry
in 1870 in which a point is represented by a pair of real numbers 
with
 |
(1)
|
(i.e., points of an open disk in the complex plane) and the distance between two points is given by
![d(x,X)=acosh^(-1)[(1-x_1X_1-x_2X_2)/(sqrt(1-x_1^2-x_2^2)sqrt(1-X_1^2-X_2^2))].](/images/equations/HyperbolicGeometry/NumberedEquation2.svg) |
(2)
|
The geometry generated by this formula satisfies all of Euclid's postulates except the fifth. The metric of this geometry is given by the Cayley-Klein-Hilbert metric,
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Hilbert extended the definition to general bounded sets in a Euclidean space.
