Polygons and iteratively regularizing affine transformations

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Abstract

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:=u qj,k−1+v qj+1,k−1+w qj+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 Δk 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⟶∞.
Original languageEnglish
Pages (from-to)69
Number of pages79
JournalBeiträge zur Algebra und Geometrie
Volume58
Issue number1
DOIs
Publication statusPublished - Mar 2017

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    Fields of Expertise

    • Information, Communication & Computing

    Cite this

    Polygons and iteratively regularizing affine transformations. / Röschel, Otto.

    In: Beiträge zur Algebra und Geometrie, Vol. 58, No. 1, 03.2017, p. 69.

    Research output: Contribution to journalArticleResearchpeer-review

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    keywords = "Affine Iterations; Affine Regularization; Regular n-gons",
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    AB - 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:=u qj,k−1+v qj+1,k−1+w qj+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 Δk 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⟶∞.

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