The essential difference between the metric geometries is the nature of parallel lines. 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. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either relaxing the metric requirement, or replacing the parallel postulate with an alternative. In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.
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