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, reversing inclusion and preserving incidence. Correlations are also called reciprocities or reciprocal transformations.

In two dimensions

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. The composition of two correlations that share the same pencil is a perspectivity.

In three dimensions

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 higher dimensions

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, the dimension of the image of W under φ is, where n is the dimension of the vector space V used to produce the projective space P.

Existence of correlations

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.

Special types of correlations

Polarity

If a correlation φ is an involution then it is called a polarity. Polarities of projective spaces lead to polar spaces, which are defined by taking the collection of all subspace which are contained in their image under the polarity.

Natural correlation

There is a natural correlation induced between a projective space P and its dual P by the natural pairing between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as
Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P to itself. In this way, every nondegenerate semilinear map induces a correlation of a projective space to itself.