affine symmetries of the ellipse

The affine symmetries of E0.

The euclidean symmetry group is E0(2), the subgroup of E(2) consisting of all rotations
about the origin O, and all reflections in lines through O. In fact,

Theorem AS1
The affine symmetry group of E0 is E0(2) = {t : t(x) = Ax, with A orthogonal}.


This case is unusual since it turns out that the affine symmetry group is exactly the same
as the euclidean symmetry group.

The result allows us to prove a stronger form of Theorem AC2.

The one-point theorem for affine ellipses
If P is a point on an affine ellipse E, then there are exactly two
affine transformations which map E to E0 and P to X(1,0).


affine symmetry page