Let
be
four points on the projective line. (In this demonstration, we will only consider
finite points, although the cross ratio holds for infinite points as well.)
Define the distance
between two points *i* and *j* as
.
What we want to show is that the cross ratio

is preserved under projective projection of the points.

A point
is projected through a
transformation matrix *T* to
a new point
.
Therefore, the new coordinate
is defined by:

Then, the distance between two points

where . The ratio between two distances, one from a point

which is the original ratio , multiplied by a constant that is dependent only upon the coordinates

showing that the cross ratio is unaffected by projection.

Suppose that one of the points, say
,
is at infinity (i.e., its
second coordinate is zero).
Then, dividing by the second coordinate (which is what we normally do
to transform the point into the required form) yields
.
Substituting into the above formula yields:

since the terms with cancel each other. (Technically speaking, we must take the limit of the equation as