In chapter two, we have seen that a projective transformation of the
projective line is completely determined by the images of three distinct
points. The higher dimensional analogue of this result makes uses of the
notion of a *projective frame.*

The proof depends on the following simple exercise.

In order to keep things simple, we shall prove this theorem only in the case . The reader is asked to extend the proof to arbitrary dimensions. We perform a series of reductions until we are left with a problem in affine geometry.

From now on, we will consider the projective plane as the plane plus the line at infinity (see figure 1). We will also indentify the plane with . Equivalently, we shall use the chart and think of as the line at infinity.

- The restriction of to is a bijection of .
- As a bijection of , sends straight lines to straight lines and parallel lines to parallel lines.
- sends the -axis, the -axis, and the diagonal unto themselves.
- fixes the origin of and the point

We now define by the equation . It follows immediately from the previous exercises that , and .

Figure 6.

Figure 7.

As a consequence of these exercises, we have that is an automorphism of and, therefore, equal to the identity. If we define by the equation , then the same arguments prove that is the identity.

Now for the final touch: