Subsections

• Introduction
• Creating Point-Sets and Line-Sets
• Constructing Points and Lines

An affine or projective plane created by Magma consists of three objects: the point-set V, the line-set L and the plane P itself. The point-set and line-set of a plane are enriched sets and consequently constitute types. Note the use of a hyphen to distinguish between ordinary sets of points and lines and these type of "sets". As in the constructors given above, the point-set and line-set are returned as the second and third arguments, respectively, of all functions which create planes. Alternatively, a pair of functions are provided to extract the point-set and line-set of a plane P. The main purpose of having point-sets and line-sets as types is to provide a convenient mechanism for referring to the points and lines of a plane. Here, 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. The category names for point-sets and line-sets are PlanePtSet and PlaneLnSet respectively.

PointSet(P) : Plane -> PlanePtSet

Given a plane P, return the point-set V of P.

Given a plane P, return the line-set L of P.

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.

The category names for points and lines of planes in Magma are PlanePt and PlaneLn respectively. Points and lines can be created in the following ways.

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.

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

Given the point-set V of a plane P, return the point of P corresponding to the element x of the indexed set used to create P. (In the case of classical planes, x must be a vector.)

Rep(V) : PlanePtSet -> PlanePt

Given the point-set V of a plane P, return a representive point of P.

Given the point-set V of a plane P, return a random point of P.

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, and 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, SeqEnum -> PlaneLn

Given the line-set L of a plane P and a sequence of collinear points of P, return the line containing the points of S.

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

Rep(L) : PlaneLnSet -> PlaneLn

Given the line-set L of a plane P, return a representive line of P.

Given the line-set L of a plane P, return a random line of P.

Rep(l) : PlaneLn -> PlanePt

Given a line l of the plane P, return a representive 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.