Linear transformation takes a polygon to another one. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:53:43Z http://mathoverflow.net/feeds/question/33303 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33303/linear-transformation-takes-a-polygon-to-another-one Linear transformation takes a polygon to another one. Adeel 2010-07-25T17:54:20Z 2012-12-30T13:25:27Z <p>Say we have $n$-gons $P$ and $Q$. Is there any necessary condition for $Q = f(P)$, for some linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^2$?</p> <p>Sorry if this is too elementary / general.</p> http://mathoverflow.net/questions/33303/linear-transformation-takes-a-polygon-to-another-one/33309#33309 Answer by Andrey Rekalo for Linear transformation takes a polygon to another one. Andrey Rekalo 2010-07-25T18:31:45Z 2010-07-25T18:31:45Z <p>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 <a href="http://www.ams.org/journals/bull/1940-46-06/S0002-9904-1940-07259-3/S0002-9904-1940-07259-3.pdf" rel="nofollow">"On linear polygon transformations"</a>, Bull. Amer. Math. Soc. 46, (1940) pp. 551 - 560.)</p> http://mathoverflow.net/questions/33303/linear-transformation-takes-a-polygon-to-another-one/117624#117624 Answer by woodbass for Linear transformation takes a polygon to another one. woodbass 2012-12-30T13:25:27Z 2012-12-30T13:25:27Z <p>By the way, is there any result on linear transformation of polyhedra in $\mathbb{R}^n$?</p>