We have seen that a perspectivity p from Π to Π' with vertex O gives us a partial map of
points on Π, to points on Π'. If Π and Π' are non-parallel, then we have to omit a line from
each plane. The map extends naturally to a bijection between the extended planes.
Note.
From now on we regard any plane in R3 as equiped with rectangular axes, with
the same scale on all planes
With given axes on Π and Π', we have a transformation of the
extended plane R2. To show
the dependence on the choice of Π and Π' with their axes, we denote this by p(Π,Π').
Now suppose that tεP(2). Then t([x]) = [Ax], where A is an invertible 3x3 matrix.
Let Φ be a plane in R3, not through O. We obtain a partial map from Φ to itself as follows.
For a point Q on Φ with position vector q, let q' = Aq.
Then [q'] defines a point Q' on Φ,
if the ray cuts Φ, and an ideal point of Φ otherwise. Thus we have a partial map from Φ
to itself which extends to a transformation of the extended plane. Since we assume that
we have axes on Φ, we have a transformation of the extended plane R2. We shall
denote
this by t(Φ).
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).
This may be restated as "each perspectivity is a projective transformation".
Theorem B
A projective transformation may be expressed as a composite of
at most three perspectivities.
This may be restated as "the projective group is generated by perspectivities".
The proofs are rather complicated, and do not contribute much to the understanding of
either the group or the geometry. We give the proofs for completeness, but they could
well be omitted at first reading.
Proofs
- Some projective transformations
- The proof of Theorem A
- Some preliminary results
- The proof of Theorem B
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