Elliptic Geometry
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." In order to achieve a consistent system, however, the basic axioms of neutral geometry must be partially modified. Most notably, the axioms of betweenness are no longer sufficient (essentially because betweenness on a great circle makes no sense, namely if A and B are on a circle and C is between them, then the relative position of C is not uniquely specified), and so must be replaced with the axioms of subsets.
Elliptic geometry is sometimes also called Riemannian geometry. It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is >180 degrees.
See also
Axiom of Subsets, Elliptic Space, Euclidean Geometry, Hyperbolic Geometry, Non-Euclidean GeometryExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Elliptic Geometry." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticGeometry.html