Duality (projective geometry)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. 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.

A projective plane C may be defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. These sets can be used to define a plane dual structure.

Interchange the role of "points" and "lines" in

to obtain the dual structure

where I^∗ is the inverse relation of I. C^∗ is also a projective plane, called the dual plane of C.

If C and C^∗ are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any field (or, more generally, for every division ring(skewfield) isomorphic to its dual) K are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.

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