Editing Projective geometry (section)

== Duality ==
{{further|Duality (projective geometry)}}

In 1825, [[Joseph Gergonne]] noted the principle of [[duality (projective geometry)|duality]] characterizing projective plane geometry: given any theorem or definition of that geometry, substituting ''point'' for ''line'', ''lie on'' for ''pass through'', ''collinear'' for ''concurrent'', ''intersection'' for ''join'', or vice versa, results in another theorem or valid definition, the "dual" of the first. Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping ''point'' and ''plane'', ''is contained by'' and ''contains''. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension ''R'' and dimension {{nowrap|''N'' − ''R'' − 1}}. For {{nowrap|1=''N'' = 2}}, this specializes to the most commonly known form of duality—that between points and lines.
The duality principle was also discovered independently by [[Jean-Victor Poncelet]].

To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).

In practice, the principle of duality allows us to set up a ''dual correspondence'' between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a [[conic]] curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical [[polyhedron]] in a concentric sphere to obtain the dual polyhedron.

Another example is [[Brianchon's theorem]], the dual of the already mentioned [[Pascal's theorem]], and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):
* '''Pascal:''' If all six vertices of a hexagon lie on a [[Conic section#In the real projective plane|conic]], then the intersections of its opposite sides ''(regarded as full lines, since in the projective plane there is no such thing as a "line segment")'' are three collinear points. The line joining them is then called the '''Pascal line''' of the hexagon.
* '''Brianchon:''' If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the '''Brianchon point''' of the hexagon.
: (If the conic degenerates into two straight lines, Pascal's becomes [[Pappus's hexagon theorem|Pappus's theorem]], which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)