The twopoint theorem for affine parabolas
If P and Q are points on an affine parabola P, then there is a unique affine transformation
which map P to P_{0} and P to O(0,0), Q to U(1,1).
Proof
By Theorem AC2, there is an affine transformation t
which maps P to P_{0}
Suppose that t maps P to P', Q to Q'. As P, Q are on P, P',Q' are on H_{0}.
We can map P_{0} to P_{0} by any element of E_{P}(2).
As P', Q' are on P_{0}, they have
coordinates P'(u^{2},u) and Q'(v^{2},v) for some u,v.
The general element r of E_{P}(2) has the form r(x,y) =
(e^{2}x+2efy+f^{2},ey+f).
This maps P' to ((eu+f)^{2},eu+f) and Q' to ((ev+f)^{2},ev+f). To achieve the required
images, we need eu+f=0 and ev+f=1. Solving : e = 1/(vu), f = eu = u/(vu).
Then u = rot has the required effect on P, P and Q.
If v has the same effect as u then w = vou^{1} maps
P_{0} to P_{0}, and fixes O and U.
Using the above general form for elements of E_{P}(2), w =
(e^{2}x+2efy+f^{2},ey+f).
To fix O and U, we require f = 0, and e = 1, so w = e
and hence v = u.

