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}. proof 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). proof

affine symmetry page