Hyperbolic geometry is a non-Euclidean geometry of constant negative sectional curvature,
also called Lobachevsky-Bolyai-Gauss geometry. The sectional
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.
In dimension 3, a useful matrix model identifies 3-dimensional hyperbolic space
with the positive definite 
Hermitian matrices
of determinant 1. Writing 
for the conjugate transpose,
this gives
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(1)
|
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(2)
|
The special linear group acts by
 |
(3)
|
The stabilizer of the identity matrix is the special unitary group, giving the displayed realization as a symmetric space (Helgason 1978).
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
 |
(4)
|
(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/NumberedEquation3.svg) |
(5)
|
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,
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
Hilbert extended the definition to general bounded sets in a Euclidean space.
