文摘
We start with a generic planar n-gon \(Q_0\) with veritices \(q_{j,0}\) (\(j = 0, \dots , n-1\)) and fixed reals \(u, v, w \in \mathbb R\) with \(u+v+w = 1\). We iteratively define n-gons \(Q_k\) of generation \(k \in \mathbb N\) with vertices \(q_{j,k}\) (\(j = 0, \dots , n-1\)) via \(q_{j,k}:= u \, q_{j, k-1} + v \, q_{j+1, k-1} + w \, q_{j+2, k-1}\). 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 \(\beta _k\) such that the sums \(\Delta _k\) of the squared distances between the vertices of \(\beta _k(Q_k)\) and the respective vertices of a given regular prototype polygon P form a null series for \(k \longrightarrow \infty \).