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Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to a given line l through a point that is not on l. In hyperbolic geometric models, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometric models, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same). The pseudosphere has the appropriate curvature to model hyperbolic geometry.
See also

Projective geometry
Spherical geometry
Taxicab geometry
Riemannian geometry

References

Ian Stewart. Flatterland. Perseus Publishing; ISBN 0-7382-0675-X (softcover, 2001)
Marvin Jay Greenberg. Euclidean and non-Euclidean geometries: Development and history. Publisher: W H Freeman 1993. ISBN 0-7167-2446-4.

Undergraduate Texts in Mathematics

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Graduate Studies in Mathematics

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