The Point-Set and Line-Set of a Plane

Introduction

An affine or projective plane in Magma consists of three objects: the plane \(P\) itself, the point–set \(V\) of \(P\), and the line–set \(L\) of \(P\).

Although called the point–set and line–set, \(V\) and \(L\) are not actual Magma sets. They simply act as the parent structures for the points and lines (respectively) of the plane \(P\), enabling easy creation of these objects via the ! and . operators.

The point–set \(V\) belongs to the Magma category PlanePtSet, and the line–set \(L\) to the category PlaneLnSet.

In this section, the functions used to create point–sets, line–sets and the points and lines themselves are described.

Creating Point-Sets and Line-Sets

As mentioned above, the point–set and line–set are returned as the second and third arguments of any function which creates a plane. They can also be created via the following two functions.

PointSet(P): Plane -> PlanePtSet

Given a plane \(P\), return the point–set \(V\) of \(P\).

LineSet(P): Plane -> PlaneLnSet

Given a plane \(P\), return the line–set \(L\) of \(P\).

Using the Point-Set and Line-Set to Create Points and Lines

For efficiency and clarity, the points and lines of a plane are given special types in Magma. The category names for points and lines are PlanePt and PlaneLn respectively. They can be created in the following ways.

V . i: PlanePtSet, RngIntElt -> PlanePt

Given the point–set \(V\) of a plane \(P\) and an integer \(i\), return the \(i\)-th point of \(P\).

V ! [a, b, c]: PlanePtSet, SeqEnum -> PlanePt

Given the point–set \(V\) of a classical projective plane \(P = PG_2(K)\), and elements \(a, b, c\) of the finite field \(K\), create the projective point \((a : b : c)\) in the plane \(P\).

V ! [a, b]: PlanePtSet, SeqEnum -> PlanePt

Given the point–set \(V\) of a classical affine plane \(P = AG_2(K)\), and elements \(a, b\) of the finite field \(K\), create the point \((a, b)\) in the plane \(P\).

V ! x: PlanePtSet, Elt -> PlanePt

Given the point–set \(V\) of a plane \(P\), return the point of \(P\) corresponding to the element \(x\), which should be coercible into the underlying point set for \(P\). (In the case of classical planes, \(x\) should be coercible to a vector.)

Representative(V): PlanePtSet -> PlanePt

Rep(V): PlanePtSet -> PlanePt

Given the point–set \(V\) of a plane \(P\), return a representative point of \(P\).

Random(V): PlanePtSet -> PlanePt

Given the point–set \(V\) of a plane \(P\), return a random point of \(P\).

L . i: PlanePtSet, RngIntElt -> PlanePt

Given the line–set \(L\) of a plane \(P\) and an integer \(i\), return the \(i\)-th line of \(P\).

L ! [a, b, c]: PlaneLnSet, SeqEnum -> PlaneLn

Given the line set \(L\) of a classical plane \(P\) defined over a finite field \(K\), and elements \(a, b, c\) of \(K\), create the line \(\left<a : b : c\right>\) (i.e. the line given by the equation \(ax + by + cz = 0\) if \(P\) is projective, or \(ax + by + c = 0\) if \(P\) is affine).

L ! [m, b]: PlaneLnSet, SeqEnum -> PlaneLn

Given the line set \(L\) of a classical affine plane \(P = AG_2(K)\), and elements \(m, b\) of the finite field \(K\), create the affine line \(y = mx + b\) in \(P\).

L ! S: PlaneLnSet, SetEnum -> PlaneLn

L ! S: PlaneLnSet, SeqEnum -> PlaneLn

Given the line–set \(L\) of a plane \(P\) and a set or sequence \(S\) of collinear points of \(P\), return the line containing the points of \(S\).

L ! l: PlaneLnSet, PlaneLn -> PlaneLn

Given the line–set \(L\) of a plane \(P\) and a line \(l\) of a (possibly) different plane (generally a subplane of \(P\)), return the line of \(P\) corresponding to \(l\).

Representative(L): PlaneLnSet -> PlaneLn

Rep(L): PlaneLnSet -> PlaneLn

Given the line–set \(L\) of a plane \(P\), return a representative line of \(P\).

Random(L): PlaneLnSet -> PlaneLn

Given the line–set \(L\) of a plane \(P\), return a random line of \(P\).

Example: Points Lines (ex-afb4c8)

The following example shows how points and lines of a plane can be created. First we study a classical projective plane.

> P, V, L := FiniteProjectivePlane(5);
> V;
Point-set of Projective Plane PG(2, 5)
> L;
Line-set of Projective Plane PG(2, 5)

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Create the third point of \(P\):

> V.3;
( 0 : 0 : 1 )

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Create the point \((1:2:3)\) of \(P\):

> V![1, 2, 3];
( 1 : 2 : 3 )

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Choose a random point of \(P\):

> Random(V);
( 1 : 0 : 0 )
> Random(V);
( 0 : 0 : 1 )

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Create the sixth line of \(P\):

> L.6;
< 1 : 1 : 3 >

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Create the line of \(P\) given by the equation 4x + 3y + 2z = 0:

> L![4, 3, 2];
< 1 : 2 : 3 >

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Create the line of \(P\) containing the points \((0:0:1)\) and \((0:1:0)\):

> L![ V | [0, 0, 1], [0, 1, 0] ];
< 1 : 0 : 0 >

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Get a representative from the line-set of \(P\), and a random line:

> Rep(L);
< 1 : 0 : 0 >
> Random(L);
< 1 : 2 : 4 >

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Now we look at a non-classical plane.

> V := {2, 4, 6, 8};
> A, P, L := FiniteAffinePlane< SetToIndexedSet(V) | Setseq(Subsets(V, 2)) >;
> A: Maximal;
Affine Plane of order 2
Points: {@ 2, 4, 6, 8 @}
Lines:
    {6, 8},
    {2, 6},
    {2, 8},
    {2, 4},
    {4, 6},
    {4, 8}
> P;
Point-set of Affine Plane of order 2
> L;
Line-set of Affine Plane of order 2

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Get the third point of \(A\):

> P.3;
6

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Create the point of \(A\) given by the integer 4:

> P!4;
4

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Get a representative from the point–set of \(A\):

> Rep(P);
2

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Get the third line of \(A\):

> L.3;
{2, 8}

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Create the line of \(A\) containing the integers 2 and 6:

> L![2, 6];
{2, 6}

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Choose a random line from \(A\):

> Random(L);
{6, 8}

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Retrieving the Plane from Points, Lines, Point-Sets and Line-Sets

The ParentPlane function allows you to access the plane to which a point, line, point–set or line–set belongs.

ParentPlane(V): PlanePtSet -> Plane, PlanePtSet, PlaneLnSet

The plane \(P\) for which \(V\) is the point–set.

ParentPlane(L): PlaneLnSet -> Plane, PlanePtSet, PlaneLnSet

The plane \(P\) for which \(L\) is the line–set.

ParentPlane(p): PlanePt -> Plane, PlanePtSet, PlaneLnSet

The plane \(P\) for which \(p\) is a point.

ParentPlane(l): PlaneLn -> Plane, PlanePtSet, PlaneLnSet

The plane \(P\) for which \(l\) is a line.