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Duality Principle

Duality Principle. For any projective result established using points and hyperplanes, a symmetrical result holds in which the roles of hyperplanes and points are interchanged: points become planes, the points in a plane become the planes through a point, etc.

The duality principle was firstly noted independently  by Joseph Gergonne  and  Jean-Victor Poncelet in the twenties of the 19th century. They noted the principle of duality in the form that substituting for line, lie on for pass through, collinear for concurrent, intersection for join, or vice versa, in a theorem or definition of the projective plane geometry results in another theorem or valid definition, thus getting dual form of the initial one. In the projective geometry of three dimensional space, duality principle holds between points and planes.

To establish the duality principle, it is sufficient to verify that the axioms imply their own duals which are either dual of an axiom or is derivable from the remaining axioms.

The duality principle is valid only in projective spaces.

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