Subsections

• Elementary Operations
• Deconstruction Functions
• Other Point and Line Functions

True if the points p and q are equal, otherwise false.

True if the points p and q are not equal, otherwise false.

True if the lines l and m are equal, otherwise false.

True if the lines l and m are not equal, otherwise false.

True if point p lies on the line l, otherwise false.

True if point p does not lie on the line l, otherwise false.

Given a subset S of the point set of the plane P and a line l of P, return true if the subset S of points lies on the line l, otherwise false.

Given a subset S of the point set of the plane P and a line l of P, return true if the subset S of points does not lie on the line l, otherwise false.

The unique point common to the lines l and m.

Rep(l) : PlaneLn -> PlanePt

Given a line l of the plane P, return a representative point of P which is incident with l.

Given a line l of the plane P, return a random point of P which is incident with l.

Given a point p from the point-set V of a plane P, return the index of p, i.e. the integer i such that p is V.i.

Index(P, l) : PlaneLn -> RngIntElt

Given a line l, return the index of l in the plane P, i.e. the integer i such that l is L.i (where L is the line-set of P).

The i-th coordinate of the point p, which must be from a classical plane. If p is from a projective plane, then i must satisfy 1 <= i <= 3; if p is from an affine plane, then i must satisfy 1 <= i <= 2.

l[i] : PlaneLn, RngIntElt -> FldFinElt

The i-th coordinate of the line l, which must be from a classical plane. The integer i must satisfy 1 <= i <= 3. Recall that in a classical plane <a:b:c> (where a, b, c in K) represents the line given by the equation ax + by + cz = 0 in a projective plane or ax + by + c = 0 in an affine plane.

Given a point p = (a:b:c) from a classical projective plane P (or p = (a, b) from a classical affine plane P), return the sequence [a, b, c] (or [a, b] in the affine case) of coordinates of p.

Coordinates(P, l) : Plane, PlaneLn -> [ FldFinElt ]

Given a line l = <a:b:c> from a classical plane P (projective or affine), return the sequence [a, b, c] of coordinates of l.

Eltseq(p) : PlanePt -> [ FldFinElt ]

Given a point p = (a:b:c) from a classical projective plane P (or p = (a, b) from a classical affine plane P), return the sequence [a, b, c] (or [a, b] in the affine case) of coordinates of p.

ElementToSequence(l) : PlaneLn -> [ FldFinElt ]

Eltseq(l) : PlaneLn -> [ FldFinElt ]

Given a line l = <a:b:c> from a classical plane P (projective or affine), return the sequence [a, b, c] of coordinates of l.

The set of points contained in the line l.

> K<w> := GF(4);
> P, V, L := ProjectivePlane(K);

> l := L![1, 0, 1];
> l;
< 1 : 0 : 1 >

> Set(l);
{  ( 0 : 1 : 0 ), ( 1 : w^2 : 1 ), ( 1 : 0 : 1 ),
   ( 1 : w : 1 ), ( 1 : 1 : 1) }

> Coordinates(P, l);
[ 1, 0, 1 ]
> l[1];
1

> Index(P, l);
8
> l eq L.8;
true

> V![1, 0, 1] in l;
true

> S := {V.1, V.2};
> S;
{  ( 1 : 0 : 0 ), ( 0 : 1 : 0 ) }
> S subset l;
false

> l2 := L!S;
> l2;
< 0 : 0 : 1 >
> S subset l2;
true

> p := l meet l2;
> p;
( 0 : 1 : 0 )
> p[3];
0

True if the set S of points of the plane P are collinear, otherwise false. If the points are collinear, the line which they define is also returned.

True if the set R of lines of the plane P are concurrent, otherwise false. If the lines are concurrent, their common point is returned as a second value.

True if the set S of points of a plane P contains a quadrangle.

The pencil of lines passing through the point p in the plane P.

The slope of the line l of a classical affine plane P.

True if the line l is parallel to the line m in the affine plane P.

The parallel class containing the line l of an affine plane P.

The partition into parallel classes of the lines of the affine plane P.

> A, V, L := AffinePlane(3);

> l := L![2, 1];
> l;
< 1 : 1 : 2 >
> Slope(l);
2

> ParallelClass(l);
{ 
    < 1 : 1 : 0 >,
    < 1 : 1 : 1 >,
    < 1 : 1 : 2 >
}
> [Slope(m): m in ParallelClass(l)];
[ 2, 2, 2 ]

> p := Rep(l);
> p;
( 1, 0 )
> Pencil(A, p);
{ 
    < 1 : 0 : 2 >,
    < 1 : 1 : 2 >,
    < 1 : 2 : 2 >,
    < 0 : 1 : 0 >
}