Section 7.5 The Polar Property
¶In neutral geoemetry, two distinct lines perpendicular to a given line must be parallel, but in elliptic geometry they must intersect. Where?
On the sphere, the answer is geometrically obvious: at the poles. That is, imagining the given line to be the equator, all lines perpendicular to the given line become lines of longitude, which all intersect at both the north and south pole. Thus, any line on the sphere determines two special points called its poles. All lines from the line to either of its poles intersect the line at right angles, and the resulting line segments have the same length.
Turning this argument around, given any point on the sphere, we can imagine it to be the north pole. So there is a unique great circle, corresponding to the equator, which is called the polar of the given point. All lines from a point to its polar intersect the polar at right angles, and the resulting line segments have the same length.
Since the Klein disk is obtained from the sphere by stereographic projection, these properties must also hold there, except that the two poles are identified into a single point. These properties of poles and polars are illustrated in Figure 7.5.1.