PROJECTIVE GEOMETRY COURSE

§ 18: *Correlations*

*Definition :* A correlation γ of *P*^{2} is a bijection from the set of points and lines of *P*^{2} onto itself that maps every point to a line and every line to a point, whilst

i) *X* on *l* ↔ γ(*X*) through γ(*l*);

ii) γ preserves cross ratio.

*Remark:* Every correlation induces projectivities from pencils of points to pencils of lines and from pencils of lines to pencils of points.

*Definition :* The *natural* correlation π_{o} is determined by:

(π_{o}(*X*) = *l* and π_{o}(*l*) = *X*) ↔ (*X* and *l* have the same projective coordinates).

(See *O37*.)

*Fundamental theorem (FT) :* Given a complete quadrangle *ABCD* and a complete quadrilateral *pqrs*, there is exactly one correlation γ of *P*^{2} with
γ(*A*) = *p*, γ(*B*) = *q*, γ(*C*) = *r*, γ(*D*) = *s*.

*Proof:*

i) unicity:

Suppose γ_{1} and γ_{2} meet the requirements. Then γ_{1}^{-1}γ_{2} is a projective transformation with fixed points *A*, *B*,
*C*, *D*, so it is identity. Then γ_{1} = γ_{2}.

ii) existence:

Let φ be the projective transformation defined by φ(*A*) = π_{o}(*p*), φ(*B*) = π_{o}(*q*), φ(*C*) = π_{o}(*r*),
φ(*D*) = π_{o}(*s*).

Then γ := π_{o}φ is a correlation as required.

*Problems :*

*O71* Study again the natural correlation in the model of *O6*. Why do we call *X* the pole of π(*X*), and π(*X*) the polar line of *X*?

*O72* Prove we can write every projective transformation of *P*^{2} as the product of two correlations.

*O73* Prove that a correlation of *P*^{2} always preserves all harmonic positions, without using requirement ii) of the definition.

*O74* Consider the complete quadrangle *ABCD* and the complete quadrilateral *pqrs*.

Let γ be the correlation with γ(*A*) = *p*, γ(*B*) = *q*, γ(*C*) = *r*, γ(*D*) = *s*.

Let *G* := *AB*. *CD*, *H* := *AD*. *BC*, *u* := γ(*G*), *v* := γ(*H*).

Draw the quadrilateral *pqrs*, and construct *u* and *v*.

Let *l* be a line that intersects *AD* in *X* and *AB* in *Y*, where (*A*, *D*; *X*, *H*) = (*A*, *B*; *Y*, *G*) = -1.

Construct γ((*l*).