Editing Projective plane (section)

==Correlations==
{{main|correlation (projective geometry)}}
A '''duality''' is a map from a projective plane {{nowrap|1=''C'' = (''P'', ''L'', ''I'')}} to its dual plane {{nowrap|1=''C''* = (''L'', ''P'', ''I''*)}} (see [[#Plane duality|above]]) which preserves incidence. That is, a duality ''σ'' will map points to lines and lines to points ({{nowrap|1=''P''<sup>''σ''</sup> = ''L''}} and {{nowrap|1=''L''<sup>''σ''</sup> = ''P''}}) in such a way that if a point ''Q'' is on a line ''m'' (denoted by {{nowrap|''Q'' ''I'' ''m''}}) then {{nowrap|''Q''<sup>''σ''</sup> ''I''* ''m''<sup>''σ''</sup> ⇔ ''m''<sup>''σ''</sup> ''I'' ''Q''<sup>''σ''</sup>}}.  A duality which is an isomorphism is called a '''correlation'''.{{sfnp|Dembowski|1968|p=151}} If a correlation exists then the projective plane ''C'' is self-dual.

In the special case that the projective plane is of the [[Projective space|PG(2,&nbsp;''K'')]] type, with ''K'' a division ring, a duality is called a '''reciprocity'''.{{sfnp|Casse|2006|p=94}} These planes are always self-dual. By the [[fundamental theorem of projective geometry]] a reciprocity is the composition of an [[automorphic function]] of ''K'' and a [[homography]]. If the automorphism involved is the identity, then the reciprocity is called a '''projective correlation'''.

A correlation of order two (an [[Involution (mathematics)|involution]]) is called a '''polarity'''. If a correlation ''φ'' is not a polarity then ''φ''<sup>2</sup> is a nontrivial collineation.