affine symmetries of the hyperbola

The affine symmetries of H0.

Theorem AS2
The affine symmetry group of H0 is
EH(2) = {t : t(x,y) = (kx,y/k) or t(x,y) = (ky,x/k), k ≠ 0}.


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).


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