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The application of fractal graphics in packaging and decoration (Part 2)

2. Julia set

Julia set is actually a subset of Mandelbort set, which corresponds to each point inside and outside Mandelbort set, so Julia set is countless. Julia set drawing method is exactly the same as Mandelbort set, which is realized by the same iterative formula. The difference is that the Mandelbort set is fixed to zero, C is changed, the iterative calculation results are examined, and the position of point C is colored according to the results, so as to draw the graph. Julia set is a fixed C, which changes, and also draws the graph by coloring the point position according to the iteration result. Therefore, according to different C values, different shapes of graphics 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

l system is a description method for studying plant morphology and growth proposed by American biologist aristid Li, who bought an experimental machine ndenmayer. At first, it only focused on the topological structure of plants, that is, the adjacent relationship between the trunk and side branches of plants. Later, geometric interpretation was added to the description process to form the so-called L system

l system is a formal language. Its axioms and production are described by strings. If you want to connect L system with graphics, you need to give each letter in L system a specific meaning. Imagine a tortoise crawling on a plane, and its state is described by three values, marked with (x, y, ∝), where x, y are the rectangular coordinates of the tortoise's position, ∝ represents the orientation of the tortoise head, and then gives the crawling step D and the angle increment of the torsion direction §, the following is the explanation of the meaning of the sign:

f: move forward one step, the step is D, and the tortoise's state is (x`, y`, ∝ `), where x`=x+dcos ∝, Y`=y+dcos ∝

draw straight line segments from (x, y) to (x`, y`)

+: turn the § angle to the left, and the turtle's next state is (x, y, ∝ + §). It is specified that the positive angle is counterclockwise, and the negative angle is clockwise

－: turn the corner to the right, and the turtle's next state is (x, y, ∝ － §)

[: press the current crawling state of the tortoise into the stack, and the information includes the location and direction of the tortoise;

]: pop up a state from the stack as the current state of the tortoise, but do not draw a line

according to the above character definitions, the branching structure of plants can be described in a simple way. The following figure is generated by the string p:f f f - [- F + F + F] + [+ F - F - F]. It is not difficult to see that by defining different strings, the growth direction and type of plants can be controlled; The density of plant growth can be controlled by iterating and then by different times of electrodeposition methods. Plants drawn in this way can often achieve a more realistic effect. The algorithm of L system is only a simple implementation of graphics. If you want to make the fractal image achieve a perfect effect in packaging and decoration design, you can also use the palette and its animation technology to form a natural transition of scenery color, so as to realize the replacement of scenery in spring, summer, autumn and winter. The two trees in the figure below use palette changes to show the time difference between summer and autumn

similar to Mandelbrot set and Julia set, l system can also change parameters to obtain different plant forms. For example, define a string that does not contract with the main needle, and control the type of plant growth; Define different iteration times to control the height and density of plant growth; Define different deflection angles to control the growth direction and inclination of plants. As shown in the following figure, the branch structure of L system in different forms, in which, (a), (b) is generated by a single string, and (c) is generated by five strings

therefore, in common CAD software, copying, cutting, deletion, synthesis and artistic processing are the main factors, and hand roughness affects accuracy; The latter uses the higher cost of servo electromechanical and uses fractal theory to draw graphics, which not only provides a new design means for packaging designers, but also allows designers to have greater imagination and play space, and can more directly reflect the designer's creative intention. In addition, the application of various unpredictable and colorful beautiful fractal patterns to packaging decoration design not only has good practicability, but also has high ornamental value. It is believed that fractal images will have a good development prospect in packaging design

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