Cross Ratios of Planes

A pencil of planes in $I\!\!P^{3}$ is a family of planes having a common line of intersection. The cross ratio of four planes $\Pi_{i}$of a pencil is the same as the cross ratio of the lines l_i of intersection of the planes with fifth, transversal plane (see fig. 3.3).

  

Figure 3.3: A pencil of planes

Once again, different transversal planes give the same cross ratio as the figures they give are projectively equivalent. The Möbius formula also extends to this case: let P, Q be any two distinct points on the axis of the plane pencil, and $A_i, i=1,\ldots, 4$ be points lying on each plane $\Pi_{i}$ (not on the axis), then

$$\{\Pi_{1}, \Pi_{2}; \Pi_{3}, \Pi_{4}\} = \frac {\mid P Q A_{... ...}\mid} {\mid P Q A_{1} A_{4}\mid\;\;\mid P Q A_{2} A_{3}\mid}$$

where ${\mid P Q A_i A_j\mid}$ stands for a $4\times 4$ determinant of 4-component column vectors.