The unit sphere

Because the coordinates are unaffected by scalar multiplication, $\ensuremath{{\cal P}^2}$is two-dimensional, even though its points contain three coordinates. In fact, it is topologically equivalent to a sphere. Each point $\ensuremath{{\bf p}} = (X,Y,W)$, represented as a ``line'' in ray space, can be projected onto the unit sphere to obtain the point $\frac{1}{\sqrt{X^2+Y^2+W^2}}(X,Y,W)$ (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 $\ensuremath{{\bf u}}$. The ideal line is the great circle around the horizontal midsection of the sphere, and the ideal points lie on this circle.

  

Figure 4: The unit sphere.