Non-Euclidean Geometry

E412207

branch of mathematics geometry

Non-Euclidean Geometry is a branch of mathematics that studies geometrical systems in which Euclid’s parallel postulate does not hold, leading to alternative models of space such as hyperbolic and elliptic geometry.

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All labels observed (3)

Label	Occurrences
Lobachevskian geometry	2
Imaginary Geometry	1
Non-Euclidean Geometry canonical	1

Statements (48)

Predicate	Object
instanceOf	branch of mathematics ⓘ geometry ⓘ
basedOn	alternative parallel axioms ⓘ
contrastsWith	Euclidean geometry ⓘ
definedBy	rejection of Euclid's parallel postulate ⓘ
developedBy	Bernhard Riemann ⓘ Carl Friedrich Gauss ⓘ Eugenio Beltrami ⓘ Felix Klein ⓘ János Bolyai ⓘ Nikolai Lobachevsky ⓘ
fieldOfStudy	mathematics ⓘ
formalizedBy	axiomatic systems ⓘ
generalizes	Euclidean geometry to curved spaces ⓘ
hasApplicationIn	computer graphics ⓘ cosmology ⓘ general relativity ⓘ navigation on curved surfaces ⓘ theoretical physics ⓘ topology ⓘ
hasKeyProperty	sum of angles in a triangle may differ from 180 degrees ⓘ
hasKeyResult	existence of consistent geometries with different parallel axioms ⓘ independence of Euclid's parallel postulate from other axioms ⓘ
hasModel	Beltrami–Klein model ⓘ Poincaré disk model ⓘ Poincaré upper half-plane model ⓘ surface form: Poincaré half-plane model sphere as a model of elliptic geometry ⓘ
hasSubfield	hyperbolic trigonometry ⓘ spherical trigonometry ⓘ
historicalRoot	19th-century mathematics ⓘ
includes	Non-Euclidean Geometry self-linksurface differs ⓘ surface form: Lobachevskian geometry Riemannian manifolds ⓘ surface form: Riemannian geometry elliptic geometry ⓘ hyperbolic geometry ⓘ projective models of geometry ⓘ
influenced	axiomatic method in mathematics ⓘ modern concept of space-time ⓘ
relatedTo	Riemannian manifolds ⓘ curved space ⓘ differential geometry ⓘ group theory via isometries ⓘ projective geometry ⓘ
studies	geometrical systems where Euclid's fifth postulate does not hold ⓘ
taughtIn	university mathematics curricula ⓘ
usesConcept	curvature ⓘ geodesic ⓘ metric space ⓘ models of geometry ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

H. S. M. Coxeter → notableWork → Non-Euclidean Geometry ⓘ

Nikolai Lobachevsky → notableWork → Non-Euclidean Geometry ⓘ

this entity surface form: Imaginary Geometry

Nikolai Lobachevsky → knownFor → Non-Euclidean Geometry ⓘ

this entity surface form: Lobachevskian geometry

Non-Euclidean Geometry → includes → Non-Euclidean Geometry self-linksurface differs ⓘ

subject surface form: Non-Euclidean geometry

this entity surface form: Lobachevskian geometry