Let $K$ denote an arbitrary field. Let $X$ and $Y$ denote degree $2$ curves in affine $2$-space over $K$. Let $f : X \rightarrow Y$ denote a regular map. Is it true that $f$ must be an affine transformation? If not, is this at least true when $K$ is a perfect field?
Motivation. I'm trying to get a basic working intuition for regular maps.
No. The map $(x,y) \mapsto (x^{-1},y^{-1})$ is not affine, but is a regular automorphism of the affine conic $1 = xy$.
For intuition it is already good to look the case where the target is the affine line $\Bbb A_K^1$. A regular map $f : X \to K$ is a quotient of polynomials functions $f/g$ so that $Z(g) \cap X = \emptyset$.