PROJECTIVE GEOMETRY COURSE

*Chapter 3*: PROJECTIVE TRANSFORMATIONS OF THE PROJECTIVE PLANE

§ 15: *Collineations and projective mappings*

*Definition :* A collineation of *P*^{2} is a bijection φ from *P*^{2} onto itself that maps points to points and lines to lines, whilst
φ(*A*) lies on φ(*l*) if and only if *A* lies on *l*.

*Proposition :* Every collineation preserves the harmonic position of pairs of points (and likewise of pairs of lines).

*Proof :* Let {*A*,*B*} and {*C*,*D*} be pairs on a pencil of points *l* that separate harmonically.

Form a complete quadrangle *A**B**P**Q* with diagonal points *D* = *D*_{1} = *A**B*.*P**Q*,
*D*_{2} = *A**P*.*B**Q*, *D*_{3} = *A**Q*.*B**P*.

Then we must have *C* = *l*.*D _{2}*

If we denote by *X ' * the image point of *X* under the collineation, then it turns out that:

quadrangle *A**B**P**Q* maps to quadrangle *A ' **B ' **P ' **Q ' * with diagonal points
*D _{1} ' *,

So

*Definition :* A projective mapping from *P*^{2} onto *P*^{2} is a collineation of *P*^{2} that preserves cross ratio.

*Remark :* The projective mappings are exactly the continuous collineations.

*Fundamental theorem :* Let *A*, *B*, *C*, *D* freely situated points in *P*^{2}, and likewise
*A ' *, *B ' *, *C ' *, *D ' *. Then there exists exactly one projective transformation of *P*^{2} that maps *A* to *A ' *,
*B* to *B ' *, *C* to *C ' *, and *D* to *D ' *.

*Proof :*

1) *unicity* : Suppose φ and ψ meet the requirements. Then φψ^{-1} is a projective mapping with four freely situated fixed points, so identity (see *O59*).
Dus φ = ψ.

2) *existence* : Use projective coordinaten. Represent *X* in *P*^{2} by λ* x* with

Now let α, β, γ ∈ ℜ be determined by

Let ρ, σ, τ ∈ ℜ be determined by

Let

Then

Now see

*Remark :* From the proof under 2) above and *O61* we see that each projective transformation of *P*^{2} is induced by some regular linear
transformation of ℜ^{3} (and by each λ-tuple thereof , λ ≠ 0).

We used this in §6.

*Problem 59 :* Prove that every projective transformation with four freely situated fixed points is identity.

Hint: Start from the complete quadrangle *ABCD* of the fixed points, and use the propositions of §10.

*Problem 60 :* Let φ be a projective transformation of *P*^{2}. Let *L* be a point in *P*^{2}.

Prove that φ induces a projectivity from the pencil of lines *L* onto the pencil of lines *L '*, with *L '* = φ(*L*).

Provre also that φ is identity if φ leaves two pencils of lines linewise linvariant. Dualise.

*Problem 61 :* Let *P* be a regular linear transformation of ℜ^{3}.

Prove that *P* preserves the collinearity of any three points *A*, *B*, *C* on a line *l*, : *AB*/*AC* =
*P(A)P(B)*/*P(A)P(C)*.

Prove subsequently that the collineation φ, induced by *P* on *P*^{2}, preserves cross ratio.

*Problem 62 :* Consider four freely situated points *A*, *B*, *C*, *D*, and four freely situated points *A* ' , *B* ' , *C* ' , *D* ' .

Construct the image point of an arbitrary point *X* under the projective transformation that maps *A* to *A* ', *B* to *B* ', *C* to *C* ' , and
*D* to *D* ' .

(Hint: project *D* from every vertex of the triangle *ABC* onto the opposite side. Do the same with *D ' * and *A ' B ' C ' *. Now reduce the construction to
the projectivities induced the sides ofn *ABC* and use Steiner.)