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Correlation (projective geometry)
In projective geometry, a correlation is a transformation of a ''d''-dimensional projective space that transforms objects of dimension ''k'' into objects of dimension , preserving incidence. Correlations are also called reciprocities or reciprocal transformations.
==In two dimensions==
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.〔H. S. M. Coxeter (1974) ''Projective Geometry'', second edition, page 57, University of Toronto Press ISBN 0-8020-2104-2〕
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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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