Section 7.3 Double Elliptic Geometry
¶We start with the sphere. Lines are great circles, and all great circles intersect. But they intersect in two antipodal points, not one, so we must change the incidence postulates accordingly. Furthermore, antipodal points, such as the north and south pole, do not determine a unique line, as infinitely many lines of longitude connect them. However, any two non-antipodal points do determine a unique great circle.
Since great circles are bounded, we must also change the ruler postulates. Instead of all lines being copies of the number line, all lines must be a copy of a Euclidean circle, with distance being measured using arclength along the great circles.
With these changes to the incidence and ruler postulates, we indeed obtain a consistent set of postulates that are satisfied by the sphere. The underlying geometry is called double elliptic geometry due to the doubling of points in the revised incidence postulates.