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affine symmetries of the hyperbola

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

For the web, it is easier to
give an affine transformation
in the linear form
t(x,y)=(ax+by+c,dx+ey+f)
This is equivalent to the
usual vector form
t(x) = Ax+b, with
A having rows (a,b), (c,d),
and b^T = (e,f).

affine symmetry page

The affine symmetries of H₀. Theorem AS2 The affine symmetry group of H₀ is E_H(2) = {t : t(x,y) = (kx,y/k) or t(x,y) = (ky,x/k), k ≠ 0}. proof 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	For the web, it is easier to give an affine transformation in the linear form t(x,y)=(ax+by+c,dx+ey+f) This is equivalent to the usual vector form t(x) = Ax+b, with A having rows (a,b), (c,d), and b^T = (e,f).