- Autor
- TitelPolygons and iteratively regularizing affine transformations
- Datei
- DOI10.1007/s13366-016-0313-7
- Persistent Identifier
- Erschienen inBeiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
- Band58
- Erscheinungsjahr2017
- Heft1
- Seiten69-79
- ISSN2191-0383
- Download Statistik1422
- Peer ReviewJa
- AbstractWe 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 \(k \longrightarrow \infty\).
[742.09 KB]