Polygons and iteratively regularizing affine transformations

We start with a generic planar n-gon Q0 with veritices qj , 0 (j= 0 , ⋯ , n- 1) and fixed reals u, v, w∈ R with u+ v+ w= 1. We iteratively define n-gons Qk of generation k∈ N with vertices qj , k (j= 0 , ⋯ , n- 1) via qj,k:=uqj,k-1+vqj+1,k-1+wqj+2,k-1. We are able to show that this affine iteration process for general input data generally regularizes the polygons in the following sense: There is a series of affine mappings βk such that the sums Δ k of the squared distances between the vertices of βk(Qk) and the respective vertices of a given regular prototype polygon P form a null series for k⟶ ∞. © 2016, The Author(s).