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Proofs of Theorems A and B Part 2

Proof of Theorem A

Theorem A
If p is the perspectivity from Π to Π' with vertex O, then there is a
projective transformation t with p(Π,Π') = t(Π).
Proof
Let R be the rotation about O which maps Π' to a plane Π" parallel to Π.
This is a linear transformation which acts rigidly on Π', so maps afigure on Π' to
an exact copy on Π". It maps the given axes on Π' to axes on Π".
Now project Π" onto Π along their common normal. Once again, this is rigid.
In particular, it maps the axes on Π" to axes on Π. By the Parallel Projection
result, this map P is given by a projective transformation.
Finally, we can map the axes produced on Π by the above transformations to
the given axes. By the Moving Axes result, this can be achieved by another
projective transformation T.
Combining these, we see that the perspectivity is described by ToPoR. This is
a projective transformation whose matrix is the product of those for T, P and R.

This shows that a perspectivity can be interpreted as a projective transformation.
It also gives an important step in the proof of Theorem B.
Suppose that R is a rotation about the origin, P is a projection between parallel
planes Π', Π along their common normal, and T is a euclidean transformation of
the plane Π (regarded as a projective transformation).
Let Φ be the plane R^-1(Π'), and choose axes A on Φ. Then B = (ToPoR)^-1(A) is
a set of axes on Π. The details of the above proof show that ToPoR corresponds
to the perspectivity from Π, with axes B to Φ with axes A.

Part 1 - Some projective transformations

Part 3 - Some preliminary results

Theorems A and B

Theorem A If p is the perspectivity from Π to Π' with vertex O, then there is a projective transformation t with p(Π,Π') = t(Π). Proof Let R be the rotation about O which maps Π' to a plane Π" parallel to Π. This is a linear transformation which acts rigidly on Π', so maps afigure on Π' to an exact copy on Π". It maps the given axes on Π' to axes on Π". Now project Π" onto Π along their common normal. Once again, this is rigid. In particular, it maps the axes on Π" to axes on Π. By the Parallel Projection result, this map P is given by a projective transformation. Finally, we can map the axes produced on Π by the above transformations to the given axes. By the Moving Axes result, this can be achieved by another projective transformation T. Combining these, we see that the perspectivity is described by ToPoR. This is a projective transformation whose matrix is the product of those for T, P and R.
This shows that a perspectivity can be interpreted as a projective transformation. It also gives an important step in the proof of Theorem B. Suppose that R is a rotation about the origin, P is a projection between parallel planes Π', Π along their common normal, and T is a euclidean transformation of the plane Π (regarded as a projective transformation). Let Φ be the plane R^-1(Π'), and choose axes A on Φ. Then B = (ToPoR)^-1(A) is a set of axes on Π. The details of the above proof show that ToPoR corresponds to the perspectivity from Π, with axes B to Φ with axes A.