Section 7.4 Single Elliptic Geometry
¶The incidence postulates were our very first postulates, so they seem to be fundamental. Is there a way to implement the elliptic parallel postulate without changing the incidence postulates?
The answer is yes, and leads us back to the Klein disk.
Great circles on the sphere intersect in two antipodal points. How can we avoid counting to two? By defining the points in our model to be pairs of Euclidean points! There are actually two equivalent ways to implement this idea.
If you simply identify each pair of antipodal points on the sphere, you get, well, something that looks like half of a sphere. This model is called the real projective plane; the points can also be thought of as representing lines through the origin in three-dimensional Euclidean space. We will return to this model in Sections 9.1–9.2.
Equivalently, you can project this three-dimensional construction into the horizontal plane using stereographic projection. But that's exactly how we constructed the Klein disk! Points in the northern hemisphere, representing pairs of points on the sphere, map to points in the interior of the disk. But on the equator, we must still identify antipodal points.
The underlying geometry in this case is called single elliptic geometry, since the incidence postulates still hold. The ruler postulates must also be modified, as the lines are still copies of Euclidean circles. Distance is still measured using arclength, but the circumference of these circles is now half of what it was before, given the due to the identification of antipodal points. You only need to go half way around to get back to where you started!
Have any other postulates changed in single elliptic geometry. Yes! The plane separation postulate says that any line divides a plane in half; you can't get from one side to the other without crossing the line. This property still holds on the sphere; you can't get from east to west without crossing the great circle that divides them. But that is not the case in the Klein disk, as all you have to do is go away from the line, and jump to the other side at the boundary.
The failure of the plane separation postulate in single elliptic goemetry actually means that the Klein disk is not orientable; you can not consistently choose a "right-handed" orientation. Imagine sliding across the boundary to the left with your hand pointing north. Since antipodal points are identified, your hand winds up pointing south!