Projective transformation a parabola to a circle

Question

Take the parabola $x^2 - y = 0$ in the cartesian plane. I'm not entirely sure about this, but we can express this using homogenous coordinates as $X^2 - Y = 0$ (the $W$ coefficient is $0$?) With the knowledge that each conic is equivalent under projective transformation, how do I find such a projective transformation that will transform this parabola to a circle?

Answer 1

If you call the homogeneous coordinates of ${\mathbb P}^2$ by $(X:Y:W)$ you must homogenize $x^2-y$ to $X^2 - Y W$ (set $x=X/W$ and $y=Y/W$ and take the numerator of the resulting rational expression). Now substitute $Y= Y_1-W_1$ and $W=Y_1+W_1$ and $X=X_1$). Then $X^2-Y W$ goes into $X_1^2 - (Y_1^2 - W_1^2) = X_1^2 + W_1^2 - Y_1^2$. This is $x_1^2 + w_1^2 = 1$ homogenized.