Say we have $n$-gons $P$ and $Q$. Is there any necessary condition for $Q = f(P)$, for some linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^2$?


Sorry if this is too elementary / general.



Linear transformation takes a polygon to another one.
Jesse Douglas studied linear transformations of polygons on the complex plane in 1930s. He proved, in particular, that a transformation $z\_i{}'=\sum\_{i=1}^na\_{ij}z\_j$ (all numbers are complex) will transform a polygon $\pi=(z\_1,\cdots,z\_n)$ into a polygon $\pi'=(z\_1{}',\cdots,z\_n{}')$ if, and only if, the matrix $a\_{ij}$ is cyclic, that is, if, and only if, $a\_{ij}=\alpha\_{j-i}$, $\alpha\_{j-i}=\alpha\_k$ if $k\equiv j-1\ (\text{mod}\,n)$. (See his article ["On linear polygon transformations"](http://www.ams.org/journals/bull/1940-46-06/S0002-9904-1940-07259-3/S0002-9904-1940-07259-3.pdf), Bull. Amer. Math. Soc. 46, (1940) pp. 551 - 560.)


