geometry - 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 $kequiv j-1 (text{mod},n)$. (See his article "On linear polygon transformations", Bull. Amer. Math. Soc. 46, (1940) pp. 551 - 560.)