The attractor-wrapping approach to approximating convex hulls of 2D affine IFS attractors

Martyn, Tomasz

doi:10.1016/j.cag.2008.08.003

Cited by 6 publications

(7 citation statements)

References 12 publications

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“…In terms of computer graphics (virtual representation) of fractals, convex-hull creation has gained a great deal of attention. For example, Martyn [20] has developed an algorithm for approximating the rectangular convex hulls of 2D affine IFS fractals, ensuring a given accuracy. Martyn [19] has developed an approach to realistic real-time rendering of 3-D IFS fractals that requires both approximations of convex hulls of IFS fractal subsets and estimations of normals on a fractal surface.…”

Section: Related Workmentioning

confidence: 99%

A Hybrid Approach of Dynamic Programming and Genetic Algorithm for Multi-criteria Optimization on Sustainable Architecture design

Chang¹,

Shih²

2014

Computer-Aided Design and Applications

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This study deals with the design for manufacturing (DFM) of fractals created by a random walk called iterated function system (IFS). In particular, the DFM of an IFS-created fractal called Barnsley's fern-leaf is considered. The IFS dedicated for creating virtual models of a fern-leaf uses a set of four strictlycontracting affine mappings in the onto manner. The interactions among these mappings are studied in detail in order to identify some data structures. Based on the identified data structures, a DFM procedure is proposed. In the proposed DFM procedure, three out of the four mappings are employed in both the onto and one-to-one manner. The proposed DFM procedure is applied to the redesign of the shape (fern-leaf). Physical models of the redesigned fern-leaf are manufactured using both additive and subtractive manufacturing technologies (3-D printing and milling). The factors affecting accuracy of the physical models are also described. Although this study is limited to the shape of the fern-leaf, other IFS-created shapes can be redesigned using the proposed DFM procedure. Nevertheless, this study sheds some light on our understanding of how to develop more accurate physical models of IFS-created fractals.

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Section: Related Workmentioning

confidence: 99%

A Hybrid Approach of Dynamic Programming and Genetic Algorithm for Multi-criteria Optimization on Sustainable Architecture design

Chang¹,

Shih²

2014

Computer-Aided Design and Applications

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“…More recently, Duda [5] and Martyn [1] have presented methods to calculate the approximate convex hull of the affine IFS attractor. The two methods are similar.…”

Section: Related Workmentioning

confidence: 99%

“…We will describe the basic idea only in the following; details are available in [1]. We then explain how to compute the convex hull more efficiently by taking advantage of certain properties of affine transformations.…”

Section: Calculating the Approximate Convex Hullmentioning

confidence: 99%

“…The article is organized in the following way: we start by recalling the basic concepts of an IFS and notations in section 3. In section 4, we examine each step of Martyn's approach [1] in order to generalize it to 3D and to optimize it. We show how to simplify the approximation of the convex hull, i.e., to reduce the number of points without losing accuracy.…”

Section: Introductionmentioning

confidence: 99%

“…In order to compute an approximate convex hull by Martyn's approach in the same conditions, we implemented it as described in [1]. Table 1 demonstrates the execution time of our algorithm compared to Martyn's for 2D IFS attractors.…”

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Approximate convex hull of affine iterated function system attractors

Mishkinis

Gentil

Lanquetin

et al. 2012

Chaos, Solitons & Fractals

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In this paper, we present an algorithm to construct an approximate convex hull of the attractors of an affine iterated function system (IFS). We construct a sequence of convex hull approximations for any required precision using the self-similarity property of the attractor in order to optimize calculations. Due to the affine properties of IFS transformations, the number of points considered in the construction is reduced. The time complexity of our algorithm is a linear function of the number of iterations and the number of points in the output convex hull. The number of iterations and the execution time increases logarithmically with increasing accuracy. In addition, we introduce a method to simplify the approximation of the convex hull without loss of accuracy.

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On the localization of Hutchinson–Barnsley fractals

Anghelina,

Miculescu

2023

Chaos, Solitons & Fractals

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The attractor-wrapping approach to approximating convex hulls of 2D affine IFS attractors

Cited by 6 publications

References 12 publications

A Hybrid Approach of Dynamic Programming and Genetic Algorithm for Multi-criteria Optimization on Sustainable Architecture design

A Hybrid Approach of Dynamic Programming and Genetic Algorithm for Multi-criteria Optimization on Sustainable Architecture design

Approximate convex hull of affine iterated function system attractors

On the localization of Hutchinson–Barnsley fractals

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