A correlation is a duality

Duality (projective geometry)

A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...

 (collineation

Collineation

In projective geometry, a collineation is a one-to-one and onto map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. All projective linear transformations induce a collineation...

 from a projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

 onto its dual space, taking points to hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

s (and vice versa) and preserving incidence

Incidence (geometry)

In geometry, the relations of incidence are those such as 'lies on' between points and lines , and 'intersects' . That is, they are the binary relations describing how subsets meet...

) from a projective space to itself. In the case of projective planes

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

 correlations can only exist if the plane is self-dual.

If a correlation σ is involutory (that is, two applications of the correlation equals the identity: σ²(P)=P for all points P) then it is called a polarity.