The incidence structure and the cross-ratio are fundamental invariants under projective transformations.

Under the projective transformations, the incidence structure and the cross-ratio are preserved.

The cross-ratio is invariant under the projective transformations of the line.

It is essentially unique, up to projective transformation (the twisted cubic, therefore).

This normal form thus classifies real non-singular quadrics up to a projective transformation.

Two twisted N-gons are equivalent if a projective transformation carries one to the other.

This explains the statement that the pentagram map commutes with projective transformations.

If two n-gons are related by a projective transformation, they get the same coordinates.

The scene composed of these world points is within a projective transformation of the true scene.

They went on to investigate W-curves, curves invariant under a group of projective transformations.