Correlation (projective geometry)

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In projective geometry, a correlation is a transformation of a d-dimensional projective space that transforms objects of dimension k into objects of dimension d − k −1, preserving incidence. Correlations are also called reciprocities or reciprocal transformations.

In two dimensions[edit]

For example, 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 V* is an inclusion reversing permutation of the proper subspaces of V*.[3]

He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any subspace W*, the dimension of the image of W* under φ is (n − 1) − dim W* where n is the dimension of the vector space used to produce the projective space.

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]

If a correlation σ is involutory (that is, two applications of the correlation equals the identity: σ²(P)=P for all points P) then it is called a polarity.

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