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Definitions

Elliptic geometry may be derived from spherical geometry by identifying antipodal points of the sphere to a single elliptic point. The elliptic lines correspond to great circles reduced by the identification of antipodal points. As any two great circles intersect, there are no parallel lines in elliptic geometry.

In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, all perpendiculars to a given line intersect at a single point called the absolute pole of that line.

Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.

The distance between a pair of points is proportional to the angle between their absolute polars.

As explained by H. S. M. Coxeter: