Research and Application of Three-dimensional Fractal in Packaging Design Patterns

Today, CAD software for packaging graphic image processing, such as Photoshop, 3DMAX, AutoCAD, CorelDraw, etc., is already very mature and widely used. However, most of these softwares use the Euclidean geometry theory to describe the graphs based on the traditional computer drawing methods. The drawings are all geometric patterns with smooth surfaces and regular shapes. For example, cubes, cones, spheres, and so on. However, the scenes in the natural world are mostly rough and irregular, and their shapes cannot be accurately represented by equations. Naturally, they cannot be generated by computers. But on many occasions, it is necessary to simulate those natural scenes, such as film and television, game production, advertising creativity, and virtual reality technology, all of which are inseparable from the simulation of natural scenery. In the past, computer software was used to simulate natural scenery by using an interactive method or by using a digital camera, scanner, or other tools to input pictures into a computer before processing. Almost no pure mathematics is used to simulate the modeling, and the use of mathematical methods to simulate three-dimensional modeling is minimal.

Packaging designers often need to design more arbitrary graphics when designing packaging patterns. Sometimes they need some abstract, realistic three-dimensional graphics to satisfy consumers' aesthetic requirements. Functions are focused on graphics and image processing and structural design. Can't do this task well. The use of fractal theory to draw graphics not only provides package designers with a completely new design approach, but also allows designers to have more room for imagination and space, and can fully express the designer's ideas. Moreover, the fractal patterns are unpredictable and colorful and beautiful. Applying it to the packaging design not only has practicality, but also has a very good artistic value, and it can attract consumers and stimulate their desire for purchase.

Fractal is a main content of modern mathematics. Fractal geometry is a powerful tool to study and deal with irregular patterns in nature and engineering. Its application involves almost every field of natural science. There is no consistent definition of fractals, but K.Falconner's description of fractal F is often cited by people. The description is as follows:

1. F has a fine structure, that is to say it has complex details under any small scale;
2. F is irregular, and its whole and part cannot be described by traditional geometric language;
3. F usually has a self-similar form, and this self-similarity can be approximate or statistically significant;
4. In general, the fractal dimension under some definition of F is greater than its topological dimension.
5. In most cases, F is determined in a very simple way, possibly by an iterative process.

First, the study of three-dimensional fractal

In order to draw three-dimensional fractal maps, we must first obtain the basic types of three-dimensional fractal, and then according to these types of computer graphics, at the same time in order to get better visual effects, but also to the graphics for color, lighting and other aspects of treatment to reflect The unique properties of three-dimensional fractals. According to this basic idea, the design of three-dimensional fractal graphics is divided into the following parts:

1. The use of three-dimensional fractal theory to achieve a variety of fractal algorithms, in the computer to generate a variety of typical three-dimensional fractal graphics.

For example, three-dimensional Sierpinski mats, Menger sponges, various forms of plants, undulating mountains (Figure 1), floating clouds (Figure 2), and patterns that are not present in nature but are very singular and decorative.

2. Use appropriate color patterns to enrich the color of the graphics, making it more realistic to simulate natural scenery.

For colorful color images, different methods can be used to define values ​​for colors. The two most common color modes are RGB mode and CMYK mode. The RGB mode uses the R (red), G (green), and B (blue) color components to represent the color values ​​of digital image pixels. CMYK (cyan, product, yellow, black) patterns are mostly used for printing. Here, we use RGB color mode.

3. Produce a Chinese user interface with good results.

There are two ways to create applications under Windows. The first one is to use image development tools, such as OpenGL, which is a set of graphics standards, designed in strict accordance with the principles of computer graphics, in line with optical and visual principles, very suitable for three-dimensional graphics Development. Its characteristics are relatively simple, but not enough targeted, can only use other tools that have been prepared to develop, it is difficult to expand; the second is the use of excellent program development language, such as Visual C + +, C + + Builder or Visual Basic. Such a program development language has powerful functions, good flexibility and extensibility, and is suitable for writing a program that directly operates on the bottom of the system, so that we can perform targeted program writing according to the characteristics of packaging and decorating.

Based on the above analysis, we can use Visual C++ 6.0 as the development tool for basic graphics. At the same time, with the aid of the knowledge of three-dimensional fractal theory and computer graphics, various fractal algorithms are implemented in computer languages, and then graphics are rendered to complete three-dimensional Fractal software program code work and apply it to packaging CAD. The objects studied in this paper are mainly three-dimensional fractals. Due to the special nature of three-dimensional graphics, the following related processes should be implemented in the design:

First of all, in order to draw three-dimensional fractal graphics, we must first establish a mathematical model. The establishment of a three-dimensional model of an object requires the input of three-dimensional data to the computer. However, in practical applications, the input of three-dimensional data is a complex problem. It is necessary to establish a model by using curve and surface fitting methods. Therefore, according to the specific conditions of different three-dimensional fractal graphics, find suitable methods to establish their own mathematical models.

Second, we must implement the three-dimensional fractal algorithm on the computer. The model that is realistically displayed on the display device needs to perform coordinate transformation, hidden face elimination and shading processing on the original graphic data, and finally generate a realistic image to be displayed. For different three-dimensional fractal graphics, how to use the appropriate method to achieve on the computer, need to be selected after the specific analysis of the problem.

For example, when we plot a three-dimensional fractal mountain, we can use the random midpoint displacement method as follows:

Take a triangle in the horizontal plane as the initial element, find the midpoint of each edge in turn, and make these three points move randomly in the Y direction for a certain distance, then reduce the range of values ​​of the random number, and connect the generated three points. Make a triangle so that it repeats the above process as a second-level initial element. This is iterated n times until a satisfactory graph is generated. If the shape of the initial element is a polygon, it can be divided into several triangles and the above operation is performed for each triangle.

Finally, save the generated three-dimensional fractal graphics. In the Windows operating system, the two most common image file types are bitmap format and vector format. Bitmap formats are easier to implement and apply to any image and are therefore the most frequently used image representation methods.

The Windows bitmap is divided into a DDB (Device-Dependent Bitmap) bitmap and a DIB (Device-Independent Bitmap) bitmap. DIB bitmaps can display images inherent to bitmaps on different machines or systems. Compared to DDB bitmaps, DIB bitmaps are an external bitmap format, often storing bitmap files with BMP extensions. Because the materials used in packaging and decorating design can be stored as images in the hard disk of the computer through peripherals, DIB bitmap programming is used.

Second, the application of three-dimensional fractal

The three-dimensional fractal graphics are rich in levels and profound in meaning. They embody various types of chaos and order, can better simulate the objective world, and can only be produced by computers. Any brilliant painter is difficult to draw. With the improvement of people's living standards, people's requirements for the appearance of the product design and decoration are also increasing. Fantastic and unpredictable fractal maps can precisely meet people's needs.

In addition, computer-generated fractal images can be superimposed on the product's anti-counterfeit trademarks or logos, and can even be used as part of the logo to achieve a good anti-counterfeiting effect. At present, domestic and foreign anti-counterfeiting technologies are mainly limited to: special inks, paper, plate making and printing, laser holographic anti-counterfeiting, and the combination of printing and computer digital technology to produce anti-counterfeit trademarks is bound to be a trend. Fractal theory can use this exactly. Because the fractal pattern has uncertainty for the system itself, it is very sensitive to the parameters, and a small error will cause a huge difference in the results, which makes it extremely difficult to copy and copy it. A pixel difference will cause the complete graphics. Change, and the three-dimensional fractal image vivid, complex drawing, more difficult to copy, which provides a great use of space for packaging anti-counterfeiting.

In addition, three-dimensional fractal patterns can also be used as three-dimensional texture materials in three-dimensional packaging container renderings, materials and textures of packaging containers, and packaging material patterns. The effects are novel and vivid.

Because the theory, modeling, and programming of three-dimensional fractals are not familiar to the average designer, their use has been greatly limited. At present, there are not many achievements in directly applying computer three-dimensional fractals to the design industry. If the three-dimensional fractal computer generated by the parametric design, change a small number of parameters can be recursive levels of different similar graphics, so that it goes from the abstract theoretical research into practical design applications, can reduce the designer's preparation, debugging drawing program time It is helpful to promote the application of three-dimensional fractal technology in the field of packaging, with considerable practical value and broad prospects for development.

(Author/Xie Li Zhao Rongli)

Reprinted from: Guangdong Packaging Magazine