affine symmetries of the hyperbola

The affine symmetries of P0.

Theorem AS3
The affine symmetry group of P0 is
EP(2) = {t : t(x,y) = (e2x+2efy+f2,ey+f), e ≠ 0}.


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 maps P to P0 and P to O(0,0), Q to U(1,1).


For the web, it is easier to
give an affine transformation
in the linear form


This is equivalent to the
usual vector form

t(x) = Ax+b, with

A having rows (a,b), (c,d),
and bT = (e,f).

affine symmetry page