Correlation (projective geometry)

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This article is about correlation in projective geometry. For other uses, see Correlation (disambiguation).

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

In two dimensions[edit]

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

In three dimensions[edit]

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[edit]

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[edit]

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[edit]

Polarity[edit]

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[edit]

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 = {vV | ⟨w, v⟩ = 0, ∀wW}.[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 VV induces a correlation of a projective space to itself.

References[edit]

  1. ^ H. S. M. Coxeter (1974) Projective Geometry, second edition, page 57, University of Toronto Press ISBN 0-8020-2104-2
  2. ^ J. G. Semple and G. T. Kneebone (1952) Algebraic Projective Geometry, p 360, Clarendon Press
  3. ^ Paul B. Yale (1968, 1988. 2004) Geometry and Symmetry, chapter 6.9 Correlations and semi-bilinear forms, Dover Publications ISBN 0-486-43835-X
  4. ^ 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