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In geometry, a polar space is an incidence structure of points and lines which satisfies the following axioms:

There are at least three points on any line.
No line is collinear with every other point.
Find a nice way to say "finite Witt index". Remember that you want to include the infinite case.
Let p be a point not on a line l. Then p is collinear with either one point on l or every point on l.

A polar space of rank 2 is a generalized quadrangle. Polar spaces with a finite number of points and lines are also studied as combinatorial objects, and are a geometric interpretation of the classical groups.

History

Construction

Projective spaces

The main article doesn't cover this stuff. Eep.

Focus on finite? Nah, go all out, I say. Should I broaden it to anti-automorphisms and division rings, then?

Formed vector spaces

Alternating bilinear Hermitian (up to a scalar) Symmetric bilinear

Quadratics.

Constructing polar spaces

References

Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019

Category:families of sets Category:Projective geometry

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