§ 18: Correlations

Definition : A correlation γ of P2 is a bijection from the set of points and lines of P2 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 πo is determined by:
o(X) = l and πo(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 P2 with γ(A) = p, γ(B) = q, γ(C) = r, γ(D) = s.


i) unicity:
Suppose γ1 and γ2 meet the requirements. Then γ1-1γ2 is a projective transformation with fixed points A, B, C, D, so it is identity. Then γ1 = γ2.

ii) existence:
Let φ be the projective transformation defined by φ(A) = πo(p), φ(B) = πo(q), φ(C) = πo(r), φ(D) = πo(s).
Then γ := πoφ 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 P2 as the product of two correlations.

O73 Prove that a correlation of P2 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).