Augmented affine plane

To complete our geometrical tour of , let us project the unit sphere onto the plane W=1. Each point (X,Y,W) on the sphere is thus mapped to the point $(\frac{X}{W},\frac{Y}{W},1)$ which lies at the intersection of the W=1 plane with the ``line'' representing the point. Similarly, lines are mapped to the intersection of the W=1 plane with the ``plane'' representing the line. Ideal points and the ideal line are projected, respectively, to points at infinity and the line at infinity, as shown in figure 5. Thus we have returned to a representation in which points are points and lines are lines. A concise definition of the projective plane can now be given:

Definition 1   The projective plane, , is the affine plane augmented by a single ideal line and a set of ideal points, one for each direction, where the ideal line and ideal points are not distinguishable from regular lines and points.

The affine plane contains the same points as the Euclidean plane. The only difference is that the former also allows for nonuniform scaling and shear.

  

Figure 5: The affine plane plus the ideal line and ideal points.