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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 

You can explore constructions in the Klein disk using the new tools shown in Figure 6.1.1, or on the standalone page at handouts/Klein2.html 4 . Do not confuse these new tools (in the “disk” menu near the right) with their Euclidean analogs, even though their icons are the same!  5 

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.

Figure 6.1.1. A GeoGebra interface for the Klein disk. (You may need to scroll the toolbar to the right in order to gain access to the new menus. Be warned that some of the tools do not yet work properly for points on the unit circle, or for lines that are diameters.)
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.
handouts/Klein2.html
The new menu includes tools for drawing elliptic lines and line segments and measuring their length, an elliptic circle tool, an elliptic compass, and elliptic angle measurement. (The disk tool itself merely places points and should be avoided; use the Euclidean point tool instead.) These tools are labeled respectively with the (Euclidean!) icons below:
Use most of these tools as you would their Euclidean counterparts: to measure disance or draw lines or line segments, select two points; to measure angles, select three points, with the vertex second; to construct a circle, select the center point and a point on the circle. Unlike its Euclidean analog, however, the elliptic compass tool requires that the new center be selected first, followed by two points to set the distance.