the one-point theorem for affine parabolas

The two-point 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 P0 and P to O(0,0), Q to U(1,1).

By Theorem AC2, there is an affine transformation t which maps P to P0
Suppose that t maps P to P', Q to Q'. As P, Q are on P, P',Q' are on H0.
We can map P0 to P0 by any element of EP(2). As P', Q' are on P0, they have
coordinates P'(u2,u) and Q'(v2,v) for some u,v.

The general element r of EP(2) has the form r(x,y) = (e2x+2efy+f2,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/(v-u), f = -eu = -u/(v-u).
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 P0 to P0, and fixes O and U.

Using the above general form for elements of EP(2), w = (e2x+2efy+f2,ey+f).
To fix O and U, we require f = 0, and e = 1, so w = e and hence v = u.

affine symmetry page