Absolute points have a surprising but important application: they can
be used to determine the angle between two lines. To see how this works,
let us assume that
we have two lines
and
which
intersect the ideal line at two points, say
and
.
Then, the cross ratio between these two points and the
two absolute points
and
yields the directed
angle
from the second line to the first:

which is known as the

To gain some intuition on why this formula is true, let us consider a simple
example. Suppose we have two lines

in the affine plane. It is clear that these two lines can be represented as two vectors and in the Euclidean plane. The directed angle between the two lines is the directed angle between the two vectors and is given by:

Now in the projective plane these lines are represented as [

Converting the complex numbers from rectangular to polar coordinates yields:

from which it follows that

which is the desired result.