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 
. For curvature 
, the complete simply connected model in dimension 
is hyperbolic
space 
.
Its two-dimensional case is the hyperbolic plane 
(Mann 2017, p. 5).
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 
. For curvature 
with 
, the Gauss-Bonnet
formula gives its area as
 |
(1)
|
where 
,
,
and 
are its interior angles in radians.
The smaller the angle sum, the larger the area
(Anderson 1999; Seidel 2023, p. 262). 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. For fixed curvature,
the standard models of a given dimension are related
by isometries and describe the same intrinsic geometry
(Anderson 1999).
Daina Taimina's crocheted models illustrate the hyperbolic plane (Wertheim et al. 2004-2005).
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
 |  |  |
(2)
|
 |  |  |
(3)
|
The special linear group acts by
 |
(4)
|
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
 |
(5)
|
(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/NumberedEquation4.svg) |
(6)
|
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,
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
Hilbert extended the definition to general bounded sets in a Euclidean space.
