Non-Euclidean geometry that rejects Euclid's fifth postulate (the parallel postulate) and modifies his second postulate. It is also known as Riemannian geometry, after Bernhard Riemann.
It asserts that no line passing through a point not on a given line is parallel to that line. It also states that while any straight line of finite length can be extended indefinitely, all straight lines are the same length. Though many of elliptic geometry's theorems are identical to those of Euclidean geometry, others differ (e.g., the angles in a triangle add up to more than 180°). It can most easily be pictured as geometry done on the surface of a sphere where all lines are great circles.
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