Bolyai and Lobachevsky were the two mathematicians in addition to gauss finally rejected the parallel postulate creating non-euclidean geometry.
Non-Euclidean geometry in mathematics is made up of two geometries that are founded on axioms that are connected to those that define Euclidean geometry. Since affine geometry and metric geometry meet to form Euclidean geometry, non-Euclidean geometry results from either replacing the parallel postulate with a different one or loosening the metric requirement.
The traditional non-Euclidean geometries, hyperbolic geometry and elliptic geometry, are obtained in the first scenario. There exist affine planes associated with the planar algebras when the metric constraint is loosened, and these kinematic geometries, also known as non-Euclidean geometries, result from these.
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