polarity
Definition 1.
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Given finite dimensional vector spaces and , a duality of the projective geometry to is an order-reversing bijection . If then we can refer to as a correlation.
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A correlation of order is called a polarity.
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The set of correlations and collineations form a group denoted 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) .
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 (http://planetmath.org/PolaritiesAndForms).)
Title | polarity |
Canonical name | Polarity |
Date of creation | 2013-03-22 15:57:58 |
Last modified on | 2013-03-22 15:57:58 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 12 |
Author | Algeboy (12884) |
Entry type | Definition |
Classification | msc 51A10 |
Classification | msc 51A05 |
Synonym | order reversing |
Related topic | SesquilinearFormsOverGeneralFields |
Related topic | PolaritiesAndForms |
Defines | polarity |
Defines | duality |
Defines | correlation |
Defines | pole |
Defines | polar |