Editing Involution (mathematics) (section)

=== Projective geometry ===
An involution is a [[projectivity]]  of period 2, that is, a projectivity that interchanges pairs of points.<ref name=AGP>A.G. Pickford (1909) [https://archive.org/details/elementaryprojec00pickrich/page/n5 Elementary Projective Geometry], [[Cambridge University Press]] via [[Internet Archive]]</ref>{{rp|24}}
* Any projectivity that interchanges two points is an involution.
* The three pairs of opposite sides of a [[complete quadrangle]] meet any line (not through a vertex) in three pairs of an involution. This theorem has been called [[Desargues]]'s Involution Theorem.<ref>[[Judith V. Field|J. V. Field]] and J. J. Gray (1987) ''The Geometrical Work of Girard Desargues'', (New York: Springer), p. 54</ref> Its origins can be seen in Lemma IV of the lemmas to the ''Porisms'' of Euclid in Volume VII of the ''Collection'' of [[Pappus of Alexandria]].<ref>Ivor Thomas (editor) (1980) ''Selections Illustrating the History of Greek Mathematics'', Volume II, number 362 in the [[Loeb Classical Library]] (Cambridge and London: Harvard and Heinemann), pp. 610&ndash;3</ref>
* If an involution has one [[fixed point (mathematics)|fixed point]], it has another, and consists of the correspondence between [[projective harmonic conjugate|harmonic conjugates]] with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called '''double points'''.<ref name=AGP/>{{rp|53}}

Another type of involution occurring in projective geometry is a '''polarity''' that is a [[correlation (projective geometry)|correlation]] of period 2.<ref>[[H. S. M. Coxeter]] (1969) ''Introduction to Geometry'', pp. 244–8, [[John Wiley & Sons]]</ref>