Correlation (projective geometry)
   HOME

TheInfoList



OR:

In
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, ...
, a correlation is a transformation of a ''d''-dimensional projective space that maps subspaces of dimension ''k'' to subspaces of dimension , reversing
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
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 is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, 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 In the Hebrew Bible and in the Old Testament, the word ranges has two very different meanings. Leviticus In Leviticus 11:35, ranges probably means a cooking furnace for two or more pots, as the Hebrew word here is in the dual number; or perhaps ...
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 ...
s, pencils into ranges, quadrangles into quadrilaterals, 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 v ...
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 three dimensions

In a 3-dimensional projective space a correlation maps a point to a
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
. 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
collineation In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thu ...
s 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'').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 whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''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: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
(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 space In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these ax ...
s, 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''⊥ of all vectors in ''V'' that are orthogonal to every ...
''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