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
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: