projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, a correlation is a transformation of a ''d''-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

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 In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

, 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

pencil A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage and keeps it from marking the user's hand. Pencils create marks by physical abrasion, leaving a trail of ...

s, pencils into ranges, ompletequadrangles into ompletequadrilaterals, and so on. 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 Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography * Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

of two correlations that share the same pencil is a

perspectivity In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point. Graphics The science of graphical perspective uses perspectivities to make realistic images in p ...

In three dimensions

In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook: :If ''κ'' is such a correlation, every point ''P'' is transformed by it into a plane , 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 In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

. 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'').Paul B. Yale (1968, 1988. 2004) ''Geometry and Symmetry'', chapter 6.9 Correlations and semi-bilinear forms,

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...

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 , where ''n'' is the dimension of the

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

''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 Involution may refer to: Mathematics * Involution (mathematics), a function that is its own inverse * Involution algebra, a *-algebra: a type of algebraic structure * Involute, a construction in the differential geometry of curves * Exponentiati ...

(that is, two applications of the correlation equals the identity: 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 In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W of a vector space V equipped with a bilinear form B is the set W^\perp of all vectors in V that are orthogonal to every vector in W. I ...

''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(''V'') to itself. In this way, every nondegenerate semilinear map induces a correlation of a projective space to itself.

References

* * {{DEFAULTSORT:Correlation (Projective Geometry) Projective geometry Functions and mappings