# 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: *φ*^{2}(*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

- H. S. M. Coxeter (1974)
*Projective Geometry*, second edition, page 57, University of Toronto Press ISBN 0-8020-2104-2 - J. G. Semple and G. T. Kneebone (1952)
*Algebraic Projective Geometry*, p 360, Clarendon Press - Paul B. Yale (1968, 1988. 2004)
*Geometry and Symmetry*, chapter 6.9 Correlations and semi-bilinear forms, Dover Publications ISBN 0-486-43835-X - Irving Kaplansky (1974) [1969],
*Linear Algebra and Geometry*(2nd ed.), p. 104

- 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