Section 11.3 Hyperboloids
¶Just as stereographic projection (Section 7.2) can be used to project the sphere into the plane, there is an analogous projection, also called stereographic, that projects the unit hyperboloid into the plane, as shown in Figure 11.3.1.
As can be seen in Figure 11.3.1, the entire hyperboloid is projected into a disk in the \(xy\)-plane. In the elliptic case, the northern hemisphere is also projected to a disk, resulting in the Klein disk model (Section 6.1). In the hyperbolic case, the result is the Poincaré disk (Section 5.1)!
Just as lines on the sphere are the intersections of planes through the origin with the sphere, lines on the hyperboloid are the intersections of planes through the origin with the hyperboloid. This construction is illustrated in Figure 11.3.2, which also shows the stereographic projection of this line into the disk. Although not a proof, the resulting hyperbolic line in the disk does appear to be a line in the Poincaré disk.
This remarkable analogy between the constructions of the Klein and Poincaré disks (compare Figure 7.2.2 and Figure 11.3.2) concludes our tour of models of non-Euclidean geometry.