Let
*m* = (*x*, *y*, *t*)^{t} be the homogeneous coordinates of a point in
the first image and
*e* = (*u*, *v*, *w*) be the coordinates of the
epipole of the second camera in the first image. The
epipolar line through *m* and *e* is represented by the vector
(*c.f.* section
2.2.1). The mapping
is linear and can be
represented by a
rank 2 matrix *C*:

The mapping of epipolar lines

The constraints on *A* are encapsulated by the correspondence of 3
distinct epipolar lines. The first two correspondences each provide
two constraints, because a line in the plane has 2 dof. The third line
must pass through the intersection of the first two, so only provides
one further constraint. The correspondence of any further epipolar
line is then determined, for example by its cross ratio with the three
initial lines. Since *A* has eight degrees of freedom and we only have
five constraints, it is not fully determined. Nevertheless, the matrix
*F* = *A C* *is* fully determined. Using (5.1) we
get

*F* defines a bilinear constraint between the coordinates of corresponding
image points. If *m*' is the point in the second image corresponding
to *m*, it must lie on the epipolar line *l*'=*Fm*, and hence
(*c.f.* 5.1). The epipolar
constraint can therefore be written: