What are elliptic and hyperbolic geometries? Why were they developed?

1.What are elliptic geometries

2.what are hyperbolic geometries?

3.why was elliptic geometries developed

4.why was hyperbolic geometries developed?

Step-by-step explanation:

1.Elliptic geometry is a non-Euclidean geometry with positive curvature which replaces the parallel postulate with the statement "through any point in the plane, there exist no lines parallel to a given line.

2 .Hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that rejects the validity of Euclid's fifth, the “parallel,” postulate. ... In hyperbolic geometry, through a point not on a given line there are at least two lines parallel to the given line.

3.Felix Klein (1849–1925) modified the model by identifying each pair of antipodal points as a single point, see the Modified Riemann Sphere. With this model, the axiom that any two points determine a unique line is satisfied. Often an elliptic geometry that satisfies this axiom is called a single elliptic geometry.

4.The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

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