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.
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. 


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, $a_{ij}=\alpha_{ji}$, $\alpha_{ji}=\alpha_k$ if $k\equiv j1\ (\text{mod}\,n)$. (See his article "On linear polygon transformations", Bull. Amer. Math. Soc. 46, (1940) pp. 551  560.) 


By the way, is there any result on linear transformation of polyhedra in $\mathbb{R}^n$? 

