Duality (projective Geometry) - Principle of Duality

Principle of Duality

For more details on this topic, see Incidence structure#Dual structure.

If one defines a projective plane 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, then one may define a plane dual structure.

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

C=(P,L,I)

to obtain the dual structure

C* =(L,P,I*),

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 division ring K are 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.

In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words "point" and "line" and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of "Two points are on a unique line." is "Two lines meet at a unique point." Forming the plane dual of a statement is known as dualizing the statement.

If a statement is true in a projective plane C, then the plane dual of that statement must be true in the dual plane C*. This follows since dualizing each statement in the proof "in C" gives a statement of the proof "in C*."

The Principle of Plane Duality says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C.

The above concepts can be generalized to talk about space duality, where the terms "points" and "planes" are interchanged (and lines remain lines). This leads to the Principle of Space Duality. Further generalization is possible (see below).

These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation. Thus instead of saying "a point lies on a line" one should say "a point is incident with a line" since dualizing the latter only involves interchanging point and line ("a line is incident with a point").

Traditionally in projective geometry, the set of points on a line are considered to include the relation of projective harmonic conjugates. In this tradition the points on a line form a projective range, a concept dual to a pencil of lines on a point.