In
mathematics,
non-Euclidean geometry consists of two geometries based on
axioms closely related to those specifying
Euclidean geometry. As Euclidean geometry lies at the intersection of
metric geometry and
affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the
parallel postulate is replaced with an alternative one. In the latter case one obtains
hyperbolic geometry and
elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry.