Ray space

We have just seen that, in going from Euclidean to projective, a point in $\ensuremath{{\cal R}^2}$ becomes a set of points in $\ensuremath{{\cal R}^3}$which are related to each other by means of a nonzero scaling factor. Therefore, a point $\ensuremath{{\bf p}} = (X,Y,W)$ in $\ensuremath{{\cal P}^2}$ can be visualized as a ``line'' <sup>2</sup> in three-dimensional space passing through the origin and the point $\ensuremath{{\bf p}}$ (Technically speaking, the line does not include the origin). This three-dimensional space is known as the ray space (among other names) and is shown in Figure 3. Similarly, a line $\ensuremath{{\bf u}} = (a,b,c)$ in $\ensuremath{{\cal P}^2}$ can be visualized as a ``plane'' passing through the origin and perpendicular to $\ensuremath{{\bf u}}$. The ideal line is the horizontal W=0 ``plane'', and the ideal points are ``lines'' in this ``plane.''

  

Figure 3: Ray space.