Abstract
We start with a generic n-gon \(Q_0\) with vertices \(q_{j,0}\) (\(j = 0, \dots , n-1\)) in the d-dimensional Euclidean space \({\mathbb {E}}^d\). Additionally, \(m+1\) real numbers \(u_0, \ldots , u_m \in {\mathbb {R}} \, (m < n)\) with \(\sum _{\mu = 0}^m u_\mu = 1\) are given. From these initial data we iteratively define generations of n-gons \(Q_k\) in \({\mathbb {E}}^d\) for \(k \in {\mathbb {N}}\) with vertices \(q_{j,k}:= \sum _{\mu = 0}^m u_\mu \, q_{j+\mu , k-1}\). We can show that this affine iteration generally regularizes in an affine sense.
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Lang, J., Mick, S. & Röschel, O. Regularizing transformations of polygons. J. Geom. 108, 791–801 (2017). https://doi.org/10.1007/s00022-017-0373-3
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DOI: https://doi.org/10.1007/s00022-017-0373-3