Polygons and iteratively regularizing affine transformations (Q515720)

The author defines a certain iteration process by which a given planar \(n\)-gon with vertices \(q_{j,0}\) (with \(j=0,\ldots, n-1\)) is transformed, given three fixed real numbers \(u\), \(v,\) and \(w\) with \(u+v+w=1\), in stages, to polygons with vertices \(q_{j,k}\) defined by \(q_{j,k}=uq_{j, k-1}+ vq_{j+1, k-1}+wq_{j+2,k-1}\) (addition being modulo \(n\)) at stage \(k\). This affine iteration process is shown to have a regularizing effect on the original polygon, in the sense that it brings it closer to a regular \(n\)-gon as \(k\rightarrow \infty\).