All functions which create a plane return three values:

• the plane itself;
• the point-set of the plane;
• the line-set of the plane.

ProjectivePlane< V | X : parameters > : SetIndx, List -> ProjPl

    Check: BoolElt                      Default: true

Construct the projective plane P having as point set the indexed set V (or {@ 1, 2, ... . v @} if an integer v is given), and as line set L = { L_1, L_2, ..., L_b } given by the list X. The value of X must be either:

• A list of subsets of the set V.
• A sequence, set or indexed set of subsets of V.
• A list of lines of an existing plane.
• A sequence, set or indexed set of lines of an existing plane.
• A combination of the above.
• A v x b (0, 1)-matrix A, where A may be defined over any coefficient ring. The matrix A will be interpreted as the incidence matrix for the plane P.
• A set of codewords of a linear code with length v. The line set of P is taken to be the set of supports of the codewords.

The optional boolean argument Check indicates whether or not to check that the given data satisfies the projective plane axioms.

ProjectivePlane(W) : ModTupFld -> ProjPl

ProjectivePlane(F) : FldFin -> ProjPl

ProjectivePlane(q) : RngIntElt -> ProjPl

Given a 3-dimensional vector space W defined over the field F = GF(q), construct the classical projective plane defined by the one-dimensional and two-dimensional subspaces of W.

AffinePlane< V | X : parameters > : SetIndx, List -> AffPl

    Check: BoolElt                      Default: true

Construct the affine plane P having as point set the indexed set V (or {@ 1, 2, ... . v @} if an integer v is given), and as line set L = { L_1, L_2, ..., L_b } given by the list X. The value of X must be either:

• A list of subsets of the set V.
• A sequence, set or indexed set of subsets of V.
• A list of lines of an existing plane.
• A sequence, set or indexed set of lines of an existing plane.
• A combination of the above.
• A v x b (0, 1)-matrix A, where A may be defined over any coefficient ring. The matrix A will be interpreted as the incidence matrix for the plane P.
• A set of codewords of a linear code with length v. The line set of P is taken to be the set of supports of the codewords.

The optional boolean argument Check indicates whether or not to check that the given data satisfies the affine plane axioms.

AffinePlane(W) : ModFld -> AffPl

AffinePlane(F) : FldFin -> ProjPl

AffinePlane(q) : RngIntElt -> ProjPl

Given a 2-dimensional vector space W defined over the field F = GF(q), construct the classical affine plane defined by the cosets of the subspaces of W.

> P, V, L := ProjectivePlane(3);
> P;
Projective Plane PG(2, 3)
> V;
Point-set of Projective Plane PG(2, 3)
> L;
Line-set of Projective Plane PG(2, 3)

> A := AffinePlane< 4 | Setseq(Subsets({1, 2, 3, 4}, 2)) >;
> A: Maximal;
Affine Plane of order 2
Points: {@ 1, 2, 3, 4 @}
Lines:
    {1, 3},
    {1, 4},
    {2, 4},
    {2, 3},
    {1, 2},
    {3, 4}

> P, V, L := ProjectivePlane(16);
> time P2 := ProjectivePlane< Points(P) | {Set(l): l in L} : Check := true >;
Time: 10.769
> time P2 := ProjectivePlane< Points(P) | {Set(l): l in L} : Check := false >;
Time: 0.030