Section 6.1 The Klein Disk
¶One model for elliptic geometry is the Klein Disk. 1 In this model, one again starts with a disk \(D\) in the Euclidean plane. 2
- The points of the Klein disk are all Euclidean points in the interior of \(D\text{,}\) together with all pairs of opposite points on the boundary of \(D\text{.}\) (Such pairs are called antipodal points.)
- The lines of the Klein disk are all (arcs of) Euclidean circles that meet the boundary circle at opposite points. Again, diameters of \(D\) are included as a special case, and can be thought of as arcs of Euclidean circles of infinite radius.
- The angles of the Klein disk are Euclidean angles in \(D\text{.}\) 3
Do not confuse the Klein Disk model of elliptic geometry with the Beltrami–Klein model of hyperbolic geometry.
Again, the disk \(D\) is often taken to have unit radius, but this assumption is not necessary.
Angles between curves are of course measured using their tangent lines.
You can explore constructions in the Klein disk using the new tools shown in Figure 6.1.1. Don't confuse these new tools (in the “tool” menu near the right) with their Euclidean analogs!
The GeoGebra applet in Figure 6.1.1 models the sphere using stereographic projection, with points inside the disk corresponding to the Northern Hemisphere, and points outside the disk corresponding to the Southern Hemisphere. The Klein disk itself contains only the Northern Hemisphere (and the equator), but has a “wraparound” feature that identifies antipodal points.