KöMaL 2017 A711

For which pairs $(m,n)$ does there exist an injective function $f\colon \mathbb{R}^2\to\mathbb{R}^2$ under which the image of every regular $m$-gon is a regular $n$-gon. (Note that $m,n\ge 3$, and that by a regular $N$-gon we mean the union of the boundary segments, not the closed polygonal region.)