Affine transformation in random iterated function systems

2001 ◽  
Vol 22 (7) ◽  
pp. 820-826 ◽  
Author(s):  
Xiong Yong ◽  
Shi Ding-hua
Keyword(s):  
Affine Transformation ◽  
Iterated Function Systems ◽  
Iterated Function

scholarly journals IMPLEMENTASI TEOREMA COLLAGE UNTUK MENDESAIN SISTEM FUNGSI ITERASI (SFI): STUDI AWAL MENDESAIN SFI MOTIF SONGKET LOMBOK

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Afifurrahman Afifurrahman
Keyword(s):  
Programming Language ◽  
Fractal Geometry ◽  
Affine Transformation ◽  
Iterated Function Systems ◽  
Contractive Mapping ◽  
Self Similarity ◽  
Fractal Models ◽  
Iterated Function ◽  
Transformation Algorithm ◽  
Collage Theorem

Abstrak: Geometri fraktal merupakan cabang matematika yang memfokuskan kajiannya pada objek-objek fraktal. SFI yaitu teknik yang dapat digunakan untuk memodelkan objek fraktal. Tulisan ini memaparkan bagaimana mengaplikasikan teorema Collage untuk mendesain SFI suatu himpunan K  yang memiliki  sifat self-similarity. Mendesain SFI suatu himpunan K  berarti mencari sejumlah berhingga pemetaan kontraktif berupa transformasi affine: dengan  sedemikian sehingga  untuk n=1,2,…,N. Keenam parameter pada persamaan di atas disebut sebagai kode SFI. Penelitian ini bertujuan untuk merancang suatu algoritma berdasarkan ide dari teorema Collage dalam menentukan kode SFI yang akan digunakan untuk memvisualisasikan atraktor dari objek fraktal menggunakan bahasa pemrograman. Algoritma yang telah disusun selanjutnya diterapkan untuk memperoleh SFI motif songket Lombok dan diperoleh hasil sebagai berikut:dengan faktor kontraktivitas s = 0.70434.Kata Kunci: Teorema Collage; Sifat Self-Similarity; Transformasi Affine; Algoritma; SFI; Atraktor. Abstract: Fractal geometry is the branch of mathematics that focus its studies on fractals. Iterated Function Systems (IFS) acts as a technique to generate fractal models. This article presents how to implement the Collage Theorem to design IFS of  K  which hold self-similarity property. Designing IFS of K  means that finding the finite contractive mapping i.e. affine transformation:  where   such that  for n=1,2,…,N. The six parameters on the equation above are called IFS codes. The aim of the study is constructing the algorithm based on the Collage theorem to determine the IFS codes which are used to visualize the attractor of the fractal objects through programming language. The Algorithm is implemented to obtain the IFS of Songket’s texture of Lombok and the result is given below:with a contractivity factor s = 0.70434.Keywords: Collage Theorem; Self-Similarity; Affine Transformation; Algorithm; IFS; Attractor.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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