Why a hiperbola is birational isomorphic to a parabola?

Question

Let $C$ be the parabola given by equation $y=x^2$ and $D$ be the hiperbola given by equation $x^2-y^2+1 = 0$. Show that the hiperbola is birational isomorphic to parabola. It is useful to consider the parametric equations?

Answer 1

Consider the completion in projective plane, $C$ is defined as $yz=x^2$ and $D$ is defined as $(x-y)(x+y)=z^2$. They are the same!