Proofs of Theorems A and B

Some projective transformations

To simplify our proofs, we often find it convenient to use a plane parallel to the z-axis.
This involves no loss of generality. If T is a map with domain Π, and P is an orthogonal
3x3 matrix, then the map PTP-1 has the same geometrical effect on the plane P(Π) as
the map T has on the plane Π.

Moving Axes
If A and A' are sets of rectangular axes on a plane Π, not through O, then there is
a projective transformation mapping Π with axes A to Π with axes A'.

Proof
As in the preamble, we may assume that Π is the plane z=λ, with λ ≠ 0.
Let t denote the euclidean transformation of Π = R2 which maps A' to A.
Then t(x) = Qx+b, where Q is orthogonal 2x2, and b a 2d vector.
Let T be the partitioned matrix on the right, where c=(1/λ)b, o=(0,0).
Simple calculation shows that T maps Π to Π,and has the same effect on
Π (as a plane) as Q.

Note that the map may also be seen as the perspectivity from Π with axes A,
to Π', the same plane, but with axes T-1(A), where TεE(2) is as in the proof.

| Q  c |
T = ||
| o  1 |

Parallel Projection
Suppose that Π and Π' are parallel planes, neither through the origin. Then
the projection from Π' to Π along their common normal corresponds to a
projective transformation.

Proof
As before, we may suppose that Π is z = λ. Then Π', being parallel to Π, has
equation z = λ'. Since O lies on neither plane, λ, λ' ≠ 0.
We will first assume that the chosen axes lie over the x- and y-axes in R3.
Then we see that the map is given by the matrix T=diag(1,1,λ'/λ) since this
maps (x,y,λ') to (x,y,λ), the projection on Π.
After the previous result, we can use a second matrix to bring the image of
any axes on Π' into line with chosen axes on Π.

Part 2 - Proof of Theorem A

Theorems A and B