== Alert == 

 This article is misleading. Consider the elliptic plane, the simplest example of elliptic geometry. There is no such thing as "spherical elliptic geometry", only spherical geometry. Two antipodal points on the sphere are identified to form the elliptic plane. The article real projective plane describes the actual "elliptic plane". See Coxeter, Introduction to Geometry (1969), pages 92 to 95. In conventional geometry, one presumes that two lines intersect in a single point or not at all. Great circles intersect in two points. The content of this article must be changed; perhaps I will.