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: sub (ex-6a749f)#
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