Polygons and iteratively regularizing affine transformations

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• 作者：Otto Roeschel
• 关键词：Affine Iterations ; Affine Regularization ; Regular n ; gons
• 刊名：Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：58
• 期：1
• 页码：69-79
• 全文大小：742KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Algebra; Geometry; Algebraic Geometry; Convex and Discrete Geometry;
• 出版者：Springer Berlin Heidelberg
• ISSN：2191-0383
• 卷排序：58

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 \(k \longrightarrow \infty \).