Because the coordinates are unaffected by scalar multiplication,
is two-dimensional, even though its points contain three coordinates. In
fact, it is topologically equivalent to a sphere. Each point
,
represented as a ``line'' in ray space, can be projected onto the unit
sphere to obtain the point
(Notice that the denominator
is never zero, since the point (0,0,0) is not allowed). Thus, points
in the projective plane
can be visualized as points on the unit sphere, as shown in figure
4 (Since each ``line'' in ray space pierces the
sphere twice, both these intersections represent the same point;
that is, antipodal points are identified).
Similarly, the ``planes'' that represent lines in ray space intersect the
unit sphere along great circles, so lines are visualized as great
circles perpendicular to
.
The ideal line is the great circle around
the horizontal
midsection of the sphere, and the ideal points lie on this circle.