**Correlations**

A **duality** is a map from a projective plane *C* = (*P*, *L*, I) to its dual plane *C** = (*L*, *P*, I*) (see above) which preserves incidence. That is, a duality σ will map points to lines and lines to points (*P*σ = *L* and *L*σ = *P*) in such a way that if a point *Q* is on a line *m* (denoted by *Q* I *m*) then *Q*σ I* *m*σ ⇔ *m*σ I *Q*σ. A duality which is an isomorphism is called a **correlation**. If a correlation exists then the projective plane *C* is self-dual.

In the special case that the projective plane is of the PG(2,*K*) type, with *K* a division ring, a duality is called a **reciprocity**. These planes are always self-dual. By the fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of *K* and a homography. If the automorphism involved is the identity, then the reciprocity is called a **projective correlation**.

A correlation of order two (an involution) is called a **polarity**. If a correlation φ is not a polarity then φ2 is a nontrivial collineation.

Read more about this topic: Projective Plane

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