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

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

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

One can easily verify that

$\displaystyle {\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}$ .
Note that a conic is represented (homogeneously) as

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

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

$\displaystyle {\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.

Subhashis Banerjee 2008-01-20