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 two dimensions

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.^[1]

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 three dimensions

In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook:^[2]

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 higher dimensions

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).^[3]

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).

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 (that is, two applications of the correlation equals the identity: φ²(P) = P for all points P) 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(V) and its dual P(V^∗) 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 W^⊥ = {v ∈ V | ⟨w, v⟩ = 0, ∀w ∈ W}.^[4]

Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P(V) to itself. In this way, every nondegenerate semilinear map V → V^∗ induces a correlation of a projective space to itself.

References

• Robert J. Bumcroft (1969), Modern Projective Geometry, Holt, Rinehart, and Winston, Chapter 4.5 Correlations p. 90 
• Robert A. Rosenbaum (1963), Introduction to Projective Geometry and Modern Algebra, Addison-Wesley, p. 198