page_title: Duality (projective geometry)
section_title: Algebraic formulation
hierarchical_section_title: Duality (projective geometry) / Poles and polars / Reciprocation in the Euclidean plane / Algebraic formulation
context_page_description: 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.
context_section_description: We shall describe this polarity algebraically by following the above construction in the case that C is the unit circle (i.e., r = 1) centered at the origin.
  An affine point P, other than the origin, with Cartesian coordinates (a, b) has as its inverse in the unit circle the point Q with coordinates,
  The line passing through Q that is perpendicular to the line OP has equation ax + by = 1.
  Switching to homogeneous coordinates using the embedding (a, b) ↦ (a, b, 1), the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:
  such that
  Or, using the alternate notation, π((x, y, z)P) = (x, y, −z)L. The matrix of the associated sesquilinear form (with respect to the standard basis) is:
  The absolute points of this polarity are given by the solutions of:
  where Pᵀ= (x, y, z). Note that restricted to the Euclidean plane (that is, set z = 1) this is just the unit circle, the circle of inversion.
caption: Three pairs of dual points and lines: one red pair, one yellow pair, and one blue pair.
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