You are here
Homepolarity
Primary tabs
polarity
Definition 1.

Given finite dimensional vector spaces $V$ and $W$, a duality of the projective geometry $PG(V)$ to $PG(W)$ is an orderreversing bijection $f:PG(V)\rightarrow PG(W)$. If $W=V$ then we can refer to $f$ as a correlation.

A correlation of order $2$ is called a polarity.

The set of correlations and collineations $f:PG(V)\rightarrow PG(V)$ form a group denoted $P\Gamma L^{*}(V)$ with the operation of composition.
Remark 2.
Dualities are determined by where they map collinear triples. Given a map define on the points of $PG(V)$ to the hyperplanes of $PG(W)$ which maps collinear triples to triples of hyperplanes which intersect in a codimension 2 subspace, this specifies a unique duality.
Remark 3.
A polarity/duality necessarily interchanges points with hyperplanes. In this context points are called “poles” and hyperplanes “polars.”
An alternative definition of a duality is a projectivity (orderpreserving map) $f:PG(V)\rightarrow PG(V^{*})$.
Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with nondegenerate sesquilinear forms. (See Polarities and forms.)
Mathematics Subject Classification
51A10 no label found51A05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections