Planes and lines in ${\cal P}^3$

The duality that exist between points and lines in ${\cal P}^2$ exist between points and planes in ${\cal P}^3$ . Thus a plane is defined as a 4-tuple $(u_1,u_2,u_3,u_4)$ and the equation of this plane is given as

$$\sum_{i=1}^4 u_ix_i = 0$$

Analogous to the line at infinity ( ${\bf l_\infty}$ ) in ${\cal P}^2$ we have the plane at infinity ( ${\bf\pi_\infty}$ ) in ${\cal P}^3$ whose representation is $(0,0,0,1)^T$ .