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The Connection between Projective and Affine Planes


The Connection between Projective and Affine Planes








There exist natural mathematical constructions to form a projective plane from
an affine plane and vice versa. The functions in the this section
provide a quick and easy way to do this in Magma.



AffinePlane(P, l) : ProjPl, PlaneLn -> AffPl, Map



The affine plane obtained by removing the line l from the
projective plane P, plus the embedding map from the affine plane to P.



ProjectiveEmbedding(P) : AffPl -> ProjPl, Map



The projective completion of the affine plane P, plus the embedding
map from P to the projective plane.





Example Plane_embedding (H57E7)

We begin with the classical affine plane A of order 3, and take the
projective embedding P of A. We then remove a randomly selected
line from P, and show that the affine plane produced by this action
is isomorphic to the original affine plane A.





> A := AffinePlane(3); 
> P := ProjectiveEmbedding(A); 
> P;
Projective Plane of order 3
> A2 := AffinePlane(P, Random(LineSet(P)));
> A2;
Affine Plane of order 3
> iso, map := IsIsomorphic(A, A2);
> is_iso, map := IsIsomorphic(A, A2);
> is_iso;
true
> map;
Mapping from: PlaneAff: A to PlaneAff: A2





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