Structures Associated with a Plane

There are a number of structures naturally associated with a plane. This section lists some functions for accessing or creating them.

VectorSpace(P): Plane -> ModTupFld

The vector space underlying the classical plane \(P\).

Field(P): Plane -> FldFin

The field over which the classical plane \(P\) is defined.

IncidenceMatrix(P): Plane -> AlgMatElt

The incidence matrix of the plane \(P\).

Dual(P): Plane -> Plane, PlanePtSet, PlaneLnSet

The dual of the projective plane \(P\).

Example: Vector Space and Field of a Plane (ex-6a749f)

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The functions in this section are illustrated by the following example.

> A := FiniteAffinePlane(4);
> VectorSpace(A);
Full Vector space of degree 2 over GF(2^2)
> Field(A);
Finite field of size 2^2
>
> P := FiniteProjectivePlane< 7 | {1, 3, 5}, {1, 2, 7}, {1, 4, 6}, {2, 3, 6},
>                            {2, 4, 5}, {3, 4, 7}, {5, 6, 7} >;
> IP := IncidenceMatrix(P);
> IP;
[1 1 1 0 0 0 0]
[0 1 0 1 1 0 0]
[1 0 0 1 0 1 0]
[0 0 1 0 1 1 0]
[1 0 0 0 1 0 1]
[0 0 1 1 0 0 1]
[0 1 0 0 0 1 1]
> D := Dual(P);
> D;
Projective Plane of order 2
> IncidenceMatrix(D) eq Transpose(IP);
true

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