PlanetMath: polarity

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (511)
Corrections (9)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

polarity

(Definition)

Definition 1

Given finite dimensional vector spaces and , a duality of the projective geometry to is an order-reversing bijection $f:PG(V)\rightarrow PG(W)$ . If then we can refer to as a correlation.
A correlation of order 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 to the hyperplanes of 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 (order-preserving map) $f:PG(V)\rightarrow PG(V^*)$ .

Through the use of the fundamental theorem of projective geometry, dualities and polarities can be identified with non-degenerate sesquilinear forms. (See Polarities and forms.)

"polarity" is owned by Algeboy.

(view preamble)

Other names:

order reversing

Also defines:

polarity, duality, correlation, pole, polar

Keywords:

projective geometry, projective point, hyperplane, order preserving, order reversing

Log in to rate this entry.
(view current ratings)

Cross-references: sesquilinear forms, non-degenerate, fundamental theorem of projective geometry, order-preserving map, projectivity, subspace, codimension, intersect, hyperplanes, points, collinear, map, composition, operation, group, collineations, order, bijection, projective geometry, vector spaces, finite dimensional
There are 59 references to this entry.

This is version 9 of polarity, born on 2006-06-09, modified 2006-06-21.
Object id is 7981, canonical name is Polarity2.
Accessed 6693 times total.

Classification:

AMS MSC:	51A05 (Geometry :: Linear incidence geometry :: General theory and projective geometries)
	51A10 (Geometry :: Linear incidence geometry :: Homomorphism, automorphism and dualities)

Pending Errata and Addenda

None.[ View all 5 ]

Discussion

forum policy

No messages.

Interact