Application of Fractal Graphics in Packaging and Decorating (II)

2. Julia set

The Julia set is actually a subset of the Mandelbort set. It corresponds to every point in and outside the Mandelbort set, so the Julia set is countless. The Julia set drawing method is exactly the same as the Mandelbort set and is implemented using the same iteration formula. The difference is that the Mandelbort set is fixed to zero, the change c, the result of the iterative calculation is examined, and the position of point c is colored according to the result, thereby drawing a graph. The Julia set is fixed c, change, also according to the result of the iteration to the point location and draw the graphics. Therefore, according to different c values, different patterns can be obtained. The values ​​of c in the figure below are: c = -1.0+0.05 i, c = -0.5+0.55 i, c = 0.25+0.52 i, c = 0.66 i, and c = -i.

3. L system

The L system is a method developed by the American biologist Aristid Lindenmayer to study plant morphology and growth. At the beginning, it only focused on the topological structure of the plant, that is, the adjacent relationship between the trunk and the side branch of the plant. Later, the geometric explanation was added to the description. The process forms the so-called L system.

The L system is a formal language. Its axioms and productions are described by strings. To associate the L system with graphics, it is necessary to assign a specific meaning to each letter in the L system. Imagine a turtle crawling on a plane. Its state is described by three values, (x, y, and ∝), where x and y are the rectangular coordinates of where the turtle is, and ∝ indicates the orientation of the turtle's head. The length of the step d and the angle of rotation in the direction of twist §, the following is the interpretation of the meaning of the symbol:

F: move one step forward, step d, turtle's state is (x`, y `, ∝`), where X`=X+dcos∝, Y`=Y+dcos∝

Draw a straight line from (x, y) to (X`, Y`);

+: Turn left corner §. The next state of the turtle is (x, y, ∝ + §), which specifies that the forward angle is counterclockwise and the negative angle is clockwise;

-: Turn to the right. The next state of the turtle is (x, y, ∝-§).

[ : Push the current state of the turtle's crawl onto the stack. The information includes the location and direction of the turtle;

] : Pops a state from the stack as the turtle's current state, but does not draw a line.

According to the above character definition, the branch structure of the plant can be described in a simple manner. The following figure is generated by the string P:FF - [- F + F + F ] + [+ F - F - F ]. It is not difficult to see that by defining different strings, the growth direction and growth type of the plant can be controlled; by varying the number of iterations, the degree of plant growth can be controlled. Plants drawn in this way can often achieve more realistic results. The L system's algorithm is simply a simple implementation of the graph. If you want to achieve perfect results in the packaging design of fractal graphics, you can also use the palette and its animation technology to form a natural color of the scene and achieve the change of scenery in spring, summer, autumn and winter seasons. The two trees in the following figure use palette changes to represent the differences in time between summer and autumn.

Similar to the Mandelbrot set and the Julia set, the L system can also change parameters to obtain different plant forms. Such as: define different strings, control the type of plant growth; define different number of iterations, control the height and density of plant growth; define different deflection angles, and control the plant growth direction and inclination. The L system branch structure diagram of different forms in the following figure, where (a), (b) is generated from a single character string, and (c) is generated from five character strings.

Therefore, while the common CAD software uses copying, cutting, deletion, synthesis, and art processing as the main methods, drawing graphics using fractal theory not only provides a new design method for packaging designers, but also enables designers to have The greater imagination and space to be used can more directly reflect the designer’s creative intentions. In addition, the application of various unpredictable, colorful and beautiful fractal patterns drawn to the packaging and decoration design not only has good practicality, but also has high ornamental value. It is believed that the fractal drawing is in the packaging design. There will be very good prospects for development.