Projective mappings of lines and conics in ${\cal P}^2$

lines:

Let ${\bf x}_i$ be a set of points on a line ${\bf l}$ and consider the action of a $3 \times 3$ projective transformation ${\bf H}$ on the the points. Since the points lie on the line we have

$${\bf l}^T{\bf x}_i = 0$$

One can easily verify that

$${\bf l}^T{\bf H}^{-1}{\bf H}{\bf x}_i = 0$$

Thus the points ${\bf H}{\bf x}_i$ all lie on the line ${\bf H}^{-T}{\bf l}$ . Hence, if points are transformed as ${\bf x'}_i = {\bf H}{\bf x}_i$ , lines are transformed as ${\bf l'} = {\bf H}^{-T}{\bf l}$ .

conics:

Note that a conic is represented (homogeneously) as

$${\bf x}^T{\bf C}{\bf x} = 0$$

Under a point transformation ${\bf x'} = {\bf H}{\bf x}$ the conic becomes

$${\bf x}^T{\bf C}{\bf x} = {\bf x'}^T [{\bf H}^{-1}]^T{\bf C}{\bf H}^{-1}{\bf x'} = {\bf x'}^T{\bf H}^{-T}{\bf C}{\bf H}^{-1}{\bf x'} = 0$$

which is the quadratic form of ${\bf x'}^T{\bf C'}{\bf x'}$ with ${\bf C'} = {\bf H}^{-T}{\bf C}{\bf H}^{-1}$ . This gives the transformation rule for a conic.