Section 7.2 Stereographic Projection
¶The relationship between the sphere and the Klein disk is given by stereographic projection, which is illustrated in Figure 7.2.1. A point \(P\) on the sphere of radius \(r\) is mapped to the point \(X\) in the horizontal plane by projecting \(P\) along the line connecting it to the south pole, \(S\text{.}\) Using similar triangles and some trigonometric identities, the relationship between the spherical angle \(\theta\) and the polar radius \(R\) can be determined.
Under stereographic projection, points on the equator map to themselves, points above the equator map to the interior of the disk of radius \(r\text{,}\) and points below the equator map to the exterior of the disk. Where does the south pole go? To infinity, thought of as a single point.
It can be shown that stereographic projection is a conformal mapping, which means that it preserves angles, so that circles map to circles. The northern hemisphere of the sphere maps to the Klein disk under stereographic projection! Well, almost; we haven't yet accounted for the wraparound features of the Klein disk, under which antipodal points on the equator are identified.
We can examine the effect of this mapping by looking at how lines are projected. Lines on the sphere are the intersections of planes through the origin with the sphere, as shown in Figure 7.2.2, which also shows the stereographic projection of this line into the disk. Although not a proof, the resulting elliptic line in the disk does appear to be a line in the Klein disk.