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).
Proof
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.
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