Find an isomorphism between an open subset of $\mathbb{A}^1_\mathbb{C}$ and $V(x^2+y^2-1)\subset\mathbb{A}^2_\mathbb{C}$.
As an excercise, I am asked to find an isomorphism as in the question. This may be a very trivial example but i did not come across it, and have never really solved such excercises.
Obviously $V(x^2+y^2-1)\subset\mathbb{A}^2_\mathbb{C}$ is the unit circle in $\mathbb{C}^2$ as an affine variety. Moreover I know that the nonempty open sets of the Zariski topology on $\mathbb{A}^1_\mathbb{C}$ are $\mathbb{A}^1_\mathbb{C}$ itself and the cofinite sets. Hence I am trying to find a morphism between $\mathbb{S}^1$ and such sets. The fact is that I do not know many projections, apart from the stereographic one which I think won't do in this case since it excludes the "point at infinity"
Morever I know that distinguished open subsets of a variety $X$ in $\mathbb{A}^n_\mathbb{C}$ are always isomorphic to a "canonical" hyperbola-like variety $Y$ in $\mathbb{A}^{n+1}_\mathbb{C}$, but cannot figure out how make the isomorphism work with $Y=\mathbb{S}^1$.
Thanks for hints or help.
Hint: $x^2+y^2 = (x+iy)(x-iy) = z\cdot z^{-1}$