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- Non-Euclidean_geometry abstract "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 set aside. 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.The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ (see the entries on hyperbolic geometry, elliptic geometry, and absolute geometry for more information).Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line:In Euclidean geometry the lines remain at a constant distance from each other even if extended to infinity, and are known as parallels.In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.In elliptic geometry the lines "curve toward" each other and intersect.".
- Non-Euclidean_geometry thumbnail Noneuclid.svg?width=300.
- Non-Euclidean_geometry wikiPageExternalLink 1183548588.
- Non-Euclidean_geometry wikiPageExternalLink Non-Euclidean_geometry.html.
- Non-Euclidean_geometry wikiPageExternalLink noneuclideangeom00bonorich.
- Non-Euclidean_geometry wikiPageExternalLink Non-Euclidean_geometries.
- Non-Euclidean_geometry wikiPageExternalLink 5koAax8ct.
- Non-Euclidean_geometry wikiPageID "58610".
- Non-Euclidean_geometry wikiPageRevisionID "605211347".
- Non-Euclidean_geometry hasPhotoCollection Non-Euclidean_geometry.
- Non-Euclidean_geometry id "4669".
- Non-Euclidean_geometry title "Non-euclidean geometry".
- Non-Euclidean_geometry subject Category:Non-Euclidean_geometry.
- Non-Euclidean_geometry comment "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 set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries.".
- Non-Euclidean_geometry label "Geometria nieeuklidesowa".
- Non-Euclidean_geometry label "Geometria non euclidea".
- Non-Euclidean_geometry label "Geometria não euclidiana".
- Non-Euclidean_geometry label "Geometría no euclidiana".
- Non-Euclidean_geometry label "Géométrie non euclidienne".
- Non-Euclidean_geometry label "Nichteuklidische Geometrie".
- Non-Euclidean_geometry label "Niet-euclidische meetkunde".
- Non-Euclidean_geometry label "Non-Euclidean geometry".
- Non-Euclidean_geometry label "Неевклидова геометрия".
- Non-Euclidean_geometry label "هندسة لاإقليدية".
- Non-Euclidean_geometry label "非ユークリッド幾何学".
- Non-Euclidean_geometry label "非欧几里得几何".
- Non-Euclidean_geometry sameAs Neeukleidovská_geometrie.
- Non-Euclidean_geometry sameAs Nichteuklidische_Geometrie.
- Non-Euclidean_geometry sameAs Μη_ευκλείδειες_γεωμετρίες.
- Non-Euclidean_geometry sameAs Geometría_no_euclidiana.
- Non-Euclidean_geometry sameAs Géométrie_non_euclidienne.
- Non-Euclidean_geometry sameAs Geometri_non-Euklides.
- Non-Euclidean_geometry sameAs Geometria_non_euclidea.
- Non-Euclidean_geometry sameAs 非ユークリッド幾何学.
- Non-Euclidean_geometry sameAs 비유클리드_기하학.
- Non-Euclidean_geometry sameAs Niet-euclidische_meetkunde.
- Non-Euclidean_geometry sameAs Geometria_nieeuklidesowa.
- Non-Euclidean_geometry sameAs Geometria_não_euclidiana.
- Non-Euclidean_geometry sameAs m.0g031.
- Non-Euclidean_geometry sameAs Q233858.
- Non-Euclidean_geometry sameAs Q233858.
- Non-Euclidean_geometry wasDerivedFrom Non-Euclidean_geometry?oldid=605211347.
- Non-Euclidean_geometry depiction Noneuclid.svg.
- Non-Euclidean_geometry isPrimaryTopicOf Non-Euclidean_geometry.