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, A correlation is a point-to-line and a line-to-point transformation that preserves the relation of incidence in accordance with the principle of duality. Thus it transforms ranges into pencils, pencils into ranges, quadrangles into quadrilaterals, and so on. 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: If κ is such a correlation, every point P is transformed by it into a plane π′ = κP, and conversely, every point P arises from a unique plane π′ by the inverse transformation κ−1. 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: A correlation of the projective space P(V) is an inclusion-reversing permutation of the proper subspaces of P(V). 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.