Projective stratum

The first stratum is the projective one. It is the less structured one and has therefore the least number of invariants and the largest group of transformations associated with it. The group of projective transformations or collineations is the most general group of linear transformations.

As seen in the previous chapter a projective transformation of 3D space can be represented by a $4 \times 4$ invertible matrix

$${\bf T}_{P} \sim \left[ \begin{array}{cccc} p_{11} & p_{12} ... ..._{34} \\ p_{41} & p_{42} & p_{43} & p_{44} \end{array} \right]$$ (B21)

This transformation matrix is only defined up to a nonzero scale factor and has therefore 15 degrees of freedom.

Relations of incidence, collinearity and tangency are projectively invariant. The cross-ratio is an invariant property under projective transformations as well. It is defined as follows: Assume that the four points ${\tt M}_1, {\tt M}_2, {\tt M}_3$ and ${\tt M}_4$ are collinear. Then they can be expressed as ${\tt M}_i={\tt M}+\lambda_i {\tt M}'$ (assume none is coincident with ${\tt M}'$). The cross-ratio is defined as

$$\{ {\tt M}_1, {\tt M}_2 ; {\tt M}_3 , {\tt M}_4 \} = \frac{... ... : \frac{\lambda_2-\lambda_3}{\lambda_2-\lambda_4} \enspace .$$ (B22)

The cross-ratio is not depending on the choice of the reference points ${\tt M}$ and ${\tt M}'$ and is invariant under the group of projective transformations of ${\cal P}^3$. A similar cross-ratio invariant can be derived for four lines intersecting in a point or four planes intersecting in a common line.

The cross-ratio can in fact be seen as the coordinate of a fourth point in the basis of the first three, since three points form a basis for the projective line ${\cal P}^1$. Similarly, two invariants could be obtained for five coplanar points; and, three invariants for six points, all in general position.