Elliptic geometry

Elliptic geometry is a non-Euclidean geometry with no parallel lines. In its standard spherical model, points are antipodal pairs on a sphere and lines are great circles.

Spherical geometry: Geometry on a sphere, where points lie on the surface and lines are great circles. Unlike elliptic geometry, its spherical model keeps antipodal points distinct.
Euclidean geometry: Geometry based on Euclid's postulates, including the parallel postulate. Its parallel postulate is replaced by elliptic geometry's claim that no parallels exist.
Euclid: An ancient Greek mathematician whose treatise Elements organized geometry axiomatically. His fifth postulate became the target of attempts to establish Euclidean geometry's necessity.
Hyperbolic geometry: A non-Euclidean geometry in which infinitely many parallels pass through a point outside a line. It shares the rejection of Euclid's postulate but has negative curvature and many parallels.
Great circle: A circle on a sphere whose plane passes through the sphere's center. Great circles represent lines in the spherical model of elliptic geometry.
Parallel postulate: Euclid's postulate stating that through a point outside a line exactly one parallel can be drawn. Elliptic geometry rejects this condition by allowing no parallel through the point.
Giovanni Girolamo Saccheri: An Italian Jesuit mathematician who investigated alternatives to Euclid's parallel postulate. His work examined hypotheses that anticipate non-Euclidean geometry.
Euclidean plane: The flat two-dimensional space of ordinary Euclidean geometry. Its zero curvature and unique parallels differ from elliptic geometry's positive curvature.
Antipodal point: A point diametrically opposite another point on a sphere. Elliptic geometry identifies each point with its antipode.
Geodesic: A locally shortest path in a curved space, generalizing a straight line. Great-circle lines are geodesics in the spherical model.

Linked from 11 pages