Duality (projective Geometry) - Higher Dimensional Duality

Higher Dimensional Duality

Duality in the projective plane is a special case of duality for projective spaces, transformations of PG(n,K) (also denoted by KPn) with K a field, that interchange objects of dimension r with objects of dimension n - 1 - r ( = codimension r + 1). That is, in a projective space of dimension n, the points (dimension 0) are made to correspond with hyperplanes (codimension 1), the lines joining two points (dimension 1) are made to correspond with the intersection of two hyperplanes (codimension 2), and so on.

The points of PG(n,K) can be taken to be the nonzero vectors in the (n + 1)-dimensional vector space over K, where we identify two vectors which differ by a scalar factor. Another way to put it is that the points of n-dimensional projective space are the lines through the origin in Kn + 1, which are 1-dimensional vector subspaces. Also the n- vector dimensional subspaces of Kn + 1 represent the (n − 1)- geometric dimensional hyperplanes of projective n-space over K.

A nonzero vector u = (u₀,u₁,...,u_n) in Kn + 1 also determines an (n - 1) - geometric dimensional subspace (hyperplane) H_u, by

H_u = {(x₀,x₁,...,x_n) : u₀x₀ + … + u_nx_n = 0 }.

When a vector u is used to define a hyperplane in this way it shall be denoted by u_H, while if it is designating a point we will use u_P. In terms of the usual dot product, H_u = {x_P : u_H • x_P = 0}. Since K is a field, the dot product is symmetrical, meaning u_H•x_P = u₀x₀ + u₁x₁ + ... + u_nx_n = x₀u₀ + x₁u₁ + ... + x_nu_n = x_H•u_P. A reciprocity can be given by u_P ↔ H_u between points and hyperplanes. This extends to a reciprocity between the line generated by two points and the intersection of two such hyperplanes, and so forth.

In the projective plane, PG(2,K), with K a field we have the reciprocity given by: points in homogeneous coordinates (a,b,c) ↔ lines with equations ax + by + cz = 0. In a corresponding projective space, PG(3,K), a reciprocity is given by: points in homogeneous coordinates (a,b,c,d) ↔ planes with equations ax + by + cz + dw = 0. This reciprocity would also map a line determined by two points (a₁,b₁,c₁,d₁) and (a₂,b₂,c₂,d₂) to the line which is the intersection of the two planes with equations a₁x + b₁y + c₁z + d₁w = 0 and a₂x + b₂y + c₂z + d₂w = 0.