### Projective mappings of lines and conics in

lines:
Let be a set of points on a line and consider the action of a projective transformation on the the points. Since the points lie on the line we have

One can easily verify that

Thus the points all lie on the line . Hence, if points are transformed as , lines are transformed as .
conics:
Note that a conic is represented (homogeneously) as

Under a point transformation the conic becomes

which is the quadratic form of with . This gives the transformation rule for a conic.

Subhashis Banerjee 2008-01-20