the one-point theorem for affine hyperbolas

The one-point theorem for affine hyperbolas
If P is a point on an affine hyperbola H, then there are exactly two affine transformations
which map H to H0 and P to U(1,1).

By Theorem AC2, there is an affine transformation t which maps H to H0
Suppose that t maps P to P'. As P is on H, P' is on H0.
We can map H0 to H0 by any element of EH(2). As P' is on H0, it has coordinates
(t,1/t) for some t. The elements of EH(2) which map (t,1/t) to (1,1) are precisely
r(x,y) = (x/t,yt), and s(x,y) = (ty,x/t). Then rot and sot are of the required form.

Now, the elements of EH(2) fixing (1,1) are e and v(x,y) = (y,x).

If u has the required effect, then uo(rot)-1 maps H0 to H0 and U to U.
Thus u = rot or u = vorot = sot.

affine symmetry page