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