Duality (projective geometry)

These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration.

The projective planes PG(2, K) for any field (or, more generally, for every division ring (skewfield) isomorphic to its dual) K are self-dual.

[1] These principles provide a good reason for preferring to use a "symmetric" term for the incidence relation.

[3] As the real projective plane, PG(2, R), is self-dual there are a number of pairs of well known results that are duals of each other.

A duality δ of a projective space is a permutation of the subspaces of PG(n, K) (also denoted by KPn) with K a field (or more generally a skewfield (division ring)) that reverses inclusion,[7] that is: Consequently, a duality interchanges objects of dimension r with objects of dimension n − 1 − r ( = codimension r + 1).

By the Fundamental theorem of projective geometry θ is induced by a semilinear map T: V → V∗ with associated isomorphism σ: K → Ko, which can be viewed as an antiautomorphism of K. In the classical literature, π would be called a reciprocity in general, and if σ = id it would be called a correlation (and K would necessarily be a field).

Some authors suppress the role of the natural isomorphism and call θ a duality.

[9] When this is done, a duality may be thought of as a collineation between a pair of specially related projective spaces and called a reciprocity.

Any duality of PG(n, K) for n > 1 is induced by a nondegenerate sesquilinear form on the underlying vector space (with a companion antiautomorphism) and conversely.

To simplify this discussion we shall assume that K is a field, but everything can be done in the same way when K is a skewfield as long as attention is paid to the fact that multiplication need not be a commutative operation.

Another way to put it is that the points of n-dimensional projective space are the 1-dimensional vector subspaces, which may be visualized as the lines through the origin in Kn+1.

Since K is a field, the dot product is symmetrical, meaning uH ⋅ xP = u0x0 + u1x1 + ... + unxn = x0u0 + x1u1 + ... + xnun = xH ⋅ uP.

Specifically, in the projective plane, PG(2, K), with K a field, we have the correlation given by: points in homogeneous coordinates (a, b, c) ↔ lines with equations ax + by + cz = 0.

In a projective space, PG(3, K), a correlation is given by: points in homogeneous coordinates (a, b, c, d) ↔ planes with equations ax + by + cz + dw = 0.

The correlation is given by: This correlation in the case of PG(2, R) can be described geometrically using the model of the real projective plane which is a "unit sphere with antipodes[11] identified", or equivalently, the model of lines and planes through the origin of the vector space R3.

[12] Polarities have been classified, a result of Birkhoff & von Neumann (1936) that has been reproven several times.

[12][13][14] Let V be a (left) vector space over the skewfield K and φ be a reflexive nondegenerate sesquilinear form on V with companion anti-automorphism σ.

[16] When composed with itself, the correlation φ(xP) = xH (in any dimension) produces the identity function, so it is a polarity.

On the other hand, if K = C the set of absolute points form a nondegenerate quadric (a conic in two-dimensional space).

In the orthogonal case, the absolute points lie on a conic if p is odd or form a line if p = 2.

The incidence preserving process, in which points and lines are transformed into their polars and poles with respect to C is called reciprocation.

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.

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).

The theory of poles and polars of a conic in a projective plane can be developed without the use of coordinates and other metric concepts.

The polar of P with respect to C is the side of the diagonal triangle opposite P.[24] The theory of projective harmonic conjugates of points on a line can also be used to define this relationship.

The principle of duality is due to Joseph Diaz Gergonne (1771−1859) a champion of the then emerging field of Analytic geometry and founder and editor of the first journal devoted entirely to mathematics, Annales de mathématiques pures et appliquées.

Gergonne and Charles Julien Brianchon (1785−1864) developed the concept of plane duality.

Jean-Victor Poncelet (1788−1867) author of the first text on projective geometry, Traité des propriétés projectives des figures, was a synthetic geometer who systematically developed the theory of poles and polars with respect to a conic.

Poncelet maintained that the principle of duality was a consequence of the theory of poles and polars.

Julius Plücker (1801−1868) is credited with extending the concept of duality to three and higher dimensional projective spaces.