Knowra Projective duality Projective duality Projective duality is a correspondence in projective geometry that exchanges points and hyperplanes while preserving incidence relations. It turns geometric statements into dual statements by interchanging their roles.
Homogeneous coordinates : Coordinates defined up to multiplication by a nonzero scalar, representing points in projective space. They encode projective points and hyperplanes in forms that duality can exchange.
Projective plane : A geometry in which any two distinct points determine a unique line and any two lines meet at a unique point. In a projective plane, duality exchanges points and lines directly.
Desargues' theorem : A projective geometry theorem relating the intersections of corresponding sides of two triangles to the alignment of corresponding vertices. Its point-and-line formulation has a dual theorem obtained by exchanging those roles.
Affine geometry : The geometry of points, lines, and planes that preserves affine combinations but distinguishes parallel directions. Projective duality relies on projective incidence and does not preserve affine notions such as parallelism.
Incidence (geometry) : The relation of containment between geometric objects, such as a point lying on a line. Duality preserves incidence while reversing which kinds of objects participate.
Projective space : The space of one-dimensional subspaces of a vector space, equipped with projective incidence. Projective duality is defined within projective spaces and their incidence structure.
Pascal's theorem : A theorem stating that the three pairs of opposite sides of a hexagon inscribed in a conic meet in collinear points. Its dual, Brianchon’s theorem, follows by exchanging points and lines.
Euclidean geometry : The geometry of points and shapes equipped with distance, angle, and perpendicularity. Those metric properties are not what projective duality preserves.
Dual vector space : The vector space of linear functionals on a given vector space. Linear functionals represent hyperplanes, providing the algebraic counterpart to points.
Projective subspace : A subset of projective space arising from a vector subspace and containing all projective points represented by its vectors. Duality extends from points and hyperplanes to subspaces of complementary dimensions.
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