Correlation (projective geometry)

In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension d − k − 1, reversing inclusion and preserving incidence. Correlations are also called reciprocities or reciprocal transformations.

In the real projective plane, points and lines are dual to each other. As expressed by Coxeter,

Given a line m and P a point not on m, an elementary correlation is obtained as follows: for every Q on m form the line PQ. The inverse correlation starts with the pencil on P: for any line q in this pencil take the point m ∩ q. The composition of two correlations that share the same pencil is a perspectivity.

In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook:

Three-dimensional correlations also transform lines into lines, so they may be considered to be collineations of the two spaces.

In general n-dimensional projective space, a correlation takes a point to a hyperplane. This context was described by Paul Yale:

He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any projective subspace W of P(V), the dimension of the image of W under φ is (n − 1) − dim W, where n is the dimension of the vector space V used to produce the projective space P(V).

Correlations can exist only if the space is self-dual. For dimensions 3 and higher, self-duality is easy to test: A coordinatizing skewfield exists and self-duality fails if and only if the skewfield is not isomorphic to its opposite.

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