Non-Euclidean geometry

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, non-Euclidean geometry consists of two geometries based on <a href="/facts/Axiom/mLjP14Mc">axioms</a> closely related to those that specify <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>. As Euclidean geometry lies at the intersection of <a href="/facts/Metric_geometry/gnBBRXtc">metric geometry</a> and <a href="/facts/Affine_geometry/yUGgTx2n">affine geometry</a>, non-Euclidean geometry arises by either replacing the <a href="/facts/Parallel_postulate/xhSxUX4u">parallel postulate</a> with an alternative, or relaxing the metric requirement. In the former case, one obtains <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a> and <a href="/facts/Elliptic_geometry/oMnFAoVb">elliptic geometry</a>, 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.
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Non-Euclidean geometry open-in-new

Non-Euclidean geometry