Theorem AC2
 Each parabola is affine congruent to P_{0} : y^{2} = x.
 Each ellipse or circle is affine congruent to E_{0} :
x^{2} + y^{2} = 0.
 Each hyperbola is affine congruent to H_{0} :
xy^{ } = 1.
Proof
(1) Suppose that P is a parabola. We already know that there is a similarity s
mapping P to P_{0}. As S(2) is a subgroup of A(2), s is an affine transformation,
so the parabolas are affine congruent.
(2) Suppose that E is an ellipse or circle. From euclidean geometry, we know
that there is a euclidean (and hence affine) transfromation s such that s(E) has
equation x^{2}/a^{2}+y^{2}/b^{2} = 1, with a = b in the case of a circle.
Now consider the transfromation r(x,y) = (x/a,y/b)  this has matrix
diag(1/a,1/b),
and hence is affine. It clearly maps the locus s(E) to the locus with equation
x^{2}+y^{2} = 1, i.e. to E_{0}. Thus the affine transfromation
ros maps E to E_{0}, so the loci
are affine congruent.
(3) Suppose that H is a hyperbola. Much as in (2), there is an affine transformation
t mapping H to the hyperbola E* : x^{2}y^{2} = 1. Now let
u be the affine transformation
u(x,y) = (x+y,xy)  it has matrix with rows (1,1) and (1,1). Then u(E*) has the
equation xy = 1, i.e. is H_{0}. Thus the affine transformation uot
sends H to H_{0}, so the
hyperbolas are affine congruent.

