the one-point theorem for affine ellipses

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).

By Theorem AC2, there is an affine transformation t which maps E to E0
Suppose that t maps P to P'. As P is on E, P' is on E0.
We can map E0 to E0 by any element of E0(2). There are two elements which
map P' to X, namely the rotation ρ about O through angle XOP', and the
reflection r in the bisector of angle XOP'. Then ρot and rot each have the
required property - mapping E to E0, and P to X.

If s maps E to E0, and P to X, then so(rot)-1 maps E0 to E0, and X to X,
so the composite is the identity or r0 reflection in the x-axis. It is then easy
to see that s = rot or r0orot = ρot.

affine symmetry page