A non-Euclidean geometry, also called Lobachevsky-Bolyai-Gauss geometry, having constant sectional curvature 
. This geometry
satisfies all of Euclid's postulates except
the parallel postulate, which is modified to
read: For any infinite straight line 
and any point 
not on it, there are many other infinitely extending
straight lines that pass through 
and which do not intersect 
.
In hyperbolic geometry, the sum of angles of a triangle is less than 
,
and triangles with the same angles have the same areas.
Furthermore, not all triangles have the same angle
sum (cf. the AAA theorem for triangles
in Euclidean two-space). There are no similar triangles in hyperbolic geometry. The
best-known example of a hyperbolic space are spheres in
Lorentzian four-space. The Poincaré
hyperbolic disk is a hyperbolic two-space. Hyperbolic geometry is well understood
in two dimensions, but not in three dimensions.
Geometric models of hyperbolic geometry include the Klein-Beltrami model, which consists of an open disk in the Euclidean
plane whose open chords correspond to hyperbolic lines. A two-dimensional model is
the Poincaré hyperbolic disk. 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)
|
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(5)
|
Hilbert extended the definition to general bounded sets in a Euclidean space.
