DEVELOPMENT OF FRACTAL PATTERNS FOR JACQUARD FABRICS BASED ON THE IFS METHOD

РАЗРАБОТКА ФРАКТАЛЬНЫХ УЗОРОВ ДЛЯ ЖАККАРДОВЫХ ТКАНЕЙ ПО МЕТОДУ IFS

Bekmurodova M.Sh.

28.12.2024 54

12(129)

10. Информатика, вычислительная техника и управление

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Bekmurodova M.Sh. DEVELOPMENT OF FRACTAL PATTERNS FOR JACQUARD FABRICS BASED ON THE IFS METHOD // Universum: технические науки : электрон. научн. журн. 2024. 12(129). URL: https://7universum.com/ru/tech/archive/item/18956 (дата обращения: 14.03.2025).

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ABSTRACT

This article describes the characteristics of Jacquard fabrics and their production technology. The importance and effectiveness of using fractal patterns in the digital processing of Jacquard fabrics are analyzed. Equations for generating fractal patterns based on the affine transformation method of IFS (Iteration Function System) are developed. Using these equations, an algorithm was created to generate fractal patterns with a complex geometric shape. Fractal patterns with n iterations were generated using a software environment.

АННОТАЦИЯ

Данная статья описывает характеристики жаккардовых тканей и технологию их производства. Анализируется значимость и эффективность использования фрактальных узоров при цифровой обработке жаккардовых тканей. Разработаны уравнения для генерации фрактальных узоров на основе метода аффинного преобразования IFS (итерационная функциональная система). С использованием этих уравнений был создан алгоритм для генерации фрактальных узоров со сложной геометрической формой. В программной среде были сгенерированы фрактальные узоры с n итерациями.

Keywords: Fractal pattern, Jacquard fabric, IFS (Iteration Function System), affine transformation, complex geometric shape.

Ключевые слова: Фрактальный узор, жаккардовая ткань, IFS (Iteration Function System), аффинное преобразование, сложная геометрическая форма.

Introduction

Fractal patterns, when used in jacquard fabrics, not only enhance their aesthetic appeal but also enrich the fabric’s structure and quality. Jacquard technology ensures high flexibility in creating these patterns, providing designers with the opportunity to develop new and innovative styles.

A jacquard fabric is a textile material with a distinctive structure, where complex patterns are created using threads of different colors. These fabrics are produced using a special Jacquard loom. The French inventor Joseph-Marie Jacquard revolutionized the textile industry by inventing the Jacquard loom in 1804, which enabled the creation of patterned fabrics and brought about revolutionary changes in the textile industry [1].

Figure 1. Jacquard fabrics

The design of patterns in Jacquard woven fabrics has been widely studied by researchers. In recent years, it is evident that many pattern shapes are being used in the design of woven garments [2]. In her research, Cai Yanyan explored the geometry of fractal patterns in garment design and its application [3]. This article systematically examines the method of creating fractal images and applies fractal patterns in garment design, presenting the specific effects of fractal patterns. Based on fractal theory, the article covers the basic principles of fractal theory in garment design, the algorithm for fractal graphic design, and the application of design research in creating garment patterns based on previous studies. The design patterns were obtained by drawing or rendering fractal graphics using software like Photoshop and Corel Draw [4].

Researcher Yang Zhi analyzed the application of fractal patterns in garment design, the importance of digital jet printing on clothing, and the use of fractals in ready-made garments [5]. Long Xiaotian and Luo Weiyan discussed the design and application of fractal graphic art in clothing materials. This study presents the aesthetic values of fractal graphic art and contemporary visual arts. It highlights the application of fractal graphic art in fabrics and its feasibility [6].

Methods

One of the methods for generating fractal structures is the Iteration Function System (IFS), developed in the mid-1980s. The Iteration Function System (IFS) is one of the most widely used methods for generating fractal shapes and is a set of contracting affine transformations [7].

The mathematical aspects of the IFS method were developed by John Hutchinson. The method gained popularity through the research of Michael Barnsley, a mathematician at the Georgia Institute of Technology [8]. It serves as a theoretical foundation for studying many problems in mathematics, and this approach provides good results for generalizing findings.

The process of constructing fractals, that is, the transition from one set to another, is an iterative process. At each step, the set obtained from the previous step is transformed. This process was proposed by Russian scientists L.M. Pererva and V.V. Yudin through the following algorithm [8]:

(1)

where, - transformations, i - the number of steps to be performed, - the initial compact set.

At each step of the iteration process, the same type of transformations are performed. At each step, the different changes are described as follows:

(2)

The above formula (2) is called Hutchinson transformations. Hutchinson transformations allow the construction of fractals of various structures by choosing elementary transformations [8,9].

It should be noted that if a predetermined initial compact set is chosen, the result obtained from the -transformations is completely independent. Moreover, it produces fractals that are not related to any transformation T.

Using two affine transformations via Hutchinson transformations, it can be generated as follows:

(3)

where,

will be formed. If four affine transformations of the form are used:

we get. Through this affine transformation, we can construct a fractal called a "crystal" (Figure 2).

Figure 2. The initial nine iterations of affine transformation of a unit square using the IFS method to generate a crystal fractal

Let’s consider the algorithm for generating the Koch snowflake fractal pattern using the affine transformation method of IFS. To construct the Koch snowflake, we replace each side of an equilateral triangle with fourline segments. Using affine transformations through the Hutchinson contraction, we generate the following:

If we convert the measurement to , i.e. 30°, we get the following:

Each of the four lines is one-third the length of the original side. The central base forms an equilateral triangle, making an angle of 60° with the base line. If we apply it to all three sides of the original triangle, we get the following result in n iterations (Figure 3):


n=0	n=1	n=2	n=3

Figure 3. Construction of the Koh snowflake fractal pattern in n iterations

Results and discussion

Algorithm for solving the problem. An algorithm for constructing fractals consisting of pentagons is presented in [10]. The algorithm for constructing a fractal pattern based on complex geometric shape is as follows:

First, we create a complex geometric shape: we draw polygons and triangles inside the circle, respectively.

Here, we take the center of the circle as . In the recursion process, the new circles are divided into k pieces. The centers of the new circles for each piece, C_i, are calculated using the following formula:

where r_n - is the radius of the circle and decreases at each iteration as follows:

Here, is the scaling factor. In most cases, is chosen.

In the programming environment it is written as follows:

def draw_circle(ax, center, radius, color='black', linewidth=0.5):

""" Function to draw a circle with a center and radius"""

circle = plt.Circle(center, radius, edgecolor=color, facecolor='none', linewidth=linewidth)

ax.add_artist(circle)

def draw_lines(ax, points, color='black', linewidth=0.5):

""" Function to draw lines between given points"""

for i, start in enumerate(points):

for end in points[i + 1:]:

ax.plot([start[0], end[0]], [start[1], end[1]], color=color, linewidth=linewidth)

def recursive_fractal(ax, center, radius, depth=3, colors=None, current_depth=0):

""" Recursive fractal structure drawing function"""

if depth == 0: return

We recursively transform the resulting complex shape using IFS affine transformations, and in n iterations we obtain different patterns. In this case, the vertices of the triangle inside the circle form the centers of small circles in the next iteration. In the 3rd and 4th iterations, a complex fractal pattern with a symmetrical base is formed.


n=0	n=1	n=2

n=3	n=3

Figure 5. Examples of fractal patterns built in n iterations of a geometric shape with a hexagonal base

Conclusions

In conclusion, the use of fractal patterns in jacquard fabrics not only increases the aesthetic and design aspects, but also increases the efficiency of the production process. In this regard, taking into account the types and characteristics of jacquard fabrics, equations for generating fractal patterns were constructed based on the IFS affine transformation method. Using these equations, an algorithm for generating fractal patterns based on a hexagonal geometric shape was developed. Fractal patterns with n iterations were generated using the Python programming environment, which supports graphics.

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