Duality (projective Geometry)

Duality (projective 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 metamathematical concept. There are two approaches to the subject of duality, one through language (the Principle of Duality) and the other a more functional approach. 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 which is called a duality. In specific examples, 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.

Read more about Duality (projective Geometry):  Principle of Duality, Duality As A Mapping, Higher Dimensional Duality, Three Dimensions, Geometric Construction of A Reciprocity, Poles and Polars, Mapping The Sphere Onto The Plane, Duality Mapping Defined, Preservation of Incidence

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Duality (projective Geometry) - Preservation of Incidence
... The dualitymapping g is an isomorphism with respect to the incidence properties such as collinearity and concurrency) ... this property given a pair of lines L1 and L2 which intersect at a point P,then their dual points gL1 and gL2 define the unique line g1P ... points P1 and P2 through which passes line L,P1.P2 L,then what is the intersection of lines g1P1 and g1P2?If g1P1 g1P2 P then so that Given a ...