Duality (projective geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
For broader coverage of this topic, see Duality (mathematics).Duality as a mapping[edit]
Plane dualities[edit]
A plane duality is a map from a projective plane C = (P, L, I) to its dual plane C∗ = (L, P, I∗) (see § Principle of duality above) which preserves incidence. That is, a plane duality σ will map points to lines and lines to points (Pσ = L and Lσ = P) in such a way that if a point Q is on a line m (denoted by Q I m) then Q I m ⇔ mσ I∗Qσ. A plane duality which is an isomorphism is called a correlation.[6] The existence of a correlation means that the projective plane C is self-dual.
The projective plane C in this definition need not be a Desarguesian plane. However, if it is, that is, C = PG(2, K) with K a division ring (skewfield), then a duality, as defined below for general projective spaces, gives a plane duality on C that satisfies the above definition.
In general projective spaces[edit]
A duality δ of a projective space is a permutation of the subspaces of PG(n, K) (also denoted by KPn) with K a field (or more generally a skewfield (division ring)) that reverses inclusion,[7] that is:
History[edit]
The principle of duality is due to Joseph Diaz Gergonne (1771−1859) a champion of the then emerging field of Analytic geometry and founder and editor of the first journal devoted entirely to mathematics, Annales de mathématiques pures et appliquées. Gergonne and Charles Julien Brianchon (1785−1864) developed the concept of plane duality. Gergonne coined the terms "duality" and "polar" (but "pole" is due to F.-J. Servois) and adopted the style of writing dual statements side by side in his journal.
Jean-Victor Poncelet (1788−1867) author of the first text on projective geometry, Traité des propriétés projectives des figures, was a synthetic geometer who systematically developed the theory of poles and polars with respect to a conic. Poncelet maintained that the principle of duality was a consequence of the theory of poles and polars.
Julius Plücker (1801−1868) is credited with extending the concept of duality to three and higher dimensional projective spaces.
Poncelet and Gergonne started out as earnest but friendly rivals presenting their different points of view and techniques in papers appearing in Annales de Gergonne. Antagonism grew over the issue of priority in claiming the principle of duality as their own. A young Plücker was caught up in this feud when a paper he had submitted to Gergonne was so heavily edited by the time it was published that Poncelet was misled into believing that Plücker had plagiarized him. The vitriolic attack by Poncelet was countered by Plücker with the support of Gergonne and ultimately the onus was placed on Gergonne.[27] Of this feud, Pierre Samuel[28] has quipped that since both men were in the French army and Poncelet was a general while Gergonne a mere captain, Poncelet's view prevailed, at least among their French contemporaries.