Duality (projective geometry)

Duality (projective geometry)

In <a href="/en/Geometry" title="Geometry" class="wl">geometry</a>, a striking feature of <a href="/en/Projective_plane" title="Projective plane" class="wl">projective planes</a> is the <a href="/en/Symmetry" title="Symmetry" class="wl">symmetry</a> of the roles played by <a href="/en/Point_(geometry)" title="Point (geometry)" class="wl">points</a> and <a href="/en/Line_(geometry)" title="Line (geometry)" class="wl">lines</a> in the definitions and theorems, and (<a href="/en/Plane_(geometry)" title="Plane (geometry)" class="mw-redirect wl">plane</a>) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language (<a href="#Principle_of_duality" target="_blank">§&nbsp;Principle of duality</a>) and the other a more functional approach through special <a href="/en/Map_(mathematics)" title="Map (mathematics)" class="wl">mappings</a>. These are completely equivalent and either treatment has as its starting point the <a href="/en/Axiomatic" title="Axiomatic" class="mw-redirect wl">axiomatic</a> 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 <a href="/en/Projective_geometry" title="Projective geometry" class="wl">projective geometry</a>.

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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 duality is the formalization of this concept. There are two approaches to the subject of duality, one through language 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. 