klein view page

affine symmetries of the hyperbola

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