Projective 3-space

Projective 3D space is the projective space ${\cal P}^3$. A point of ${\cal P}^3$ is represented by a 4-vector ${\tt M}=[X \, Y\, Z\, W]^\top$. In ${\cal P}^3$ the dual entity of a point is a plane, which is also represented by a 4-vector. A point ${\tt M}$ is located on a plane ${\tt\Pi }$ if and only if

$${\tt\Pi}^\top {\tt M} = 0 \enspace .$$ (B4)

A line can be given by the linear combination of two points $\lambda_1 {\tt M}_1 + \lambda_2 {\tt M}_2$ or by the intersection of two planes ${\tt\Pi}_1 \cap {\tt\Pi}_2$.