Documentation

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 = PG₂(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 = AG₂(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 <a : b : c> (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 = AG₂(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 `Plane_points-lines (H150E2)`

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)

Create the third point of P:

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

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

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

Choose a random point of P:

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

Create the sixth line of P:

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

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

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

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 >

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

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

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

Get the third point of A: