I would like to show that
- The circle $X : x^2+y^2=1$ and the hyperbola $Y : u^2-v^2=1$ are not isomorphic over $\Bbb R$, i.e. there is no isomorphism $f:X(\Bbb C) \to Y(\Bbb C)$ having real coefficients.
- Their projective closure $X' : x^2+y^2=z^2$ and $Y': u^2-v^2=w^2$ are isomorphic over $\Bbb R$.
For part 1., I know that $f(x,y)=(x,iy)$ is an isomorphism, but it has non-real coefficients. I wasn't sure how to prove that any isomorphism must have non-real coefficients.
For part 2., I tried to prove that the rings $\Bbb R[x,y,z]/(x^2+y^2-z^2)$ and $\Bbb R[u,v,w]/(u^2-v^2-w^2)$ are isomorphic. I don't know whether this is true, but at least I was told that the projective closures are isomorphic over $\Bbb R$, as this seems to be confirmed by this answer. But I'm not sure if there is an isomorphism with homogeneous real polynomials.