Engineering images designed by fractal subdivision scheme

This paper is concerned with the modeling of engineering images by the fractal properties of 6-point binary interpolating scheme. Association between the fractal behavior of the limit curve/surface and the parameter is obtained. The relationship between the subdivision parameter and the fractal dimension of the limit fractal curve of subdivision fractal is also presented. Numerical examples and visual demonstrations show that 6-point scheme is good choice for the generation of fractals for the modeling of fractal antennas, bearings, garari’s and rock etc.

Nowadays, many techniques to generate fractals have been devised, such as IFS (iterated function systems) method Barnsley and Demko (1985), L-system method Prusinkiewicz and Lindenmayer (1990) and few others. Recently it has been shown that subdivision technique is not only an important tool for the fast generation of smooth engineering objects, but also an efficient tool for the fast generation of fractals. Zheng et al. (2007a, b) analyzed fractal properties of 4-point binary and three point ternary interpolatory subdivision schemes. Wang et al. (2011) discussed the fractal properties of the generalized chaikin corner-cutting subdivision scheme with two tension parameters. They gave the fractal range of scheme on the basis of the discussion of limit points on the limit curve. Li et al. (2013) designed the fractal curves by using the normal vector based subdivision scheme. Sarfraz et al. (2015) designed some engineering images by using rational spline interpolation. In this paper, we explore the properties of Weissman (1989) fractal subdivision scheme in different areas including engineering images. We conclude that 6-point scheme of Weissman can create engineering images for curves and surfaces with true fractal allotting and can provide some ways of shape control.
The paper is organized as follows. In "Fractal properties of the scheme" section, fractal range of 6-point subdivision scheme is being discussed. In "Numerical examples and demonstrations" section, some numerical examples are presented to confirm the correctness and effectiveness of the engineering images in the form of curve and surface. Finally, we give some concluding remarks in "Conclusions" section.

Fig. 1 Fractal antenna
According to interpolatory property p k 0 ≡ p 0 0 , k ≥ 0. Suppose p m i and p m j are two fixed control points after m subdivision steps, ∀ m ∈ Z, m ≥ 0. The role of parameter µ is required to be evaluated on the sum of all small edges among the two points after another k iterations. First we discuss and analyze the effect of µ among the two initial control points p 0 0 and p 0 1 . For i = −2 in the odd rule p k+1 2i+1 of scheme (1), we have Substituting p k+1 −4 = p k −2 and p k+1 −2 = p k −1 in (2), we get Putting i = −1, 0 in odd rule p k+1 2i+1 of scheme (1) by using p k+1 −2 = p k −1 and p k+1 2 = p k 1 , we have and Here, we define two edge vectors between the points p 0 0 and p 0 1 after k steps defined by v k = p k 1 − p k 0 and R k = p k 2 − p k 1 . Since p k i = i 2 k so we have p k 2 = 2 2 k = 1 2 k−1 = p k−1 1 and we can write as Since by dyadic parametrization p k+1 can be written as

This implies
So we have Equation (4) is the second order linear difference equation, such that The characteristic equation is (1), we get

This implies
Substitute k + 1 = 0 and i = −1 in (1), we have This implies Subtracting (7) from (6), we have The solution of Eq. (8)  Since v k = p k 1 − p k 0 and W k+1 = p k 0 − p k −1 . Then (9) From Theorem 1, we know that subdivision fractal can be gotten by keeping the corresponding subdivision parameter µ within the interval −1+ 16 . Therefore by Theorem 4 the fractal dimension of the 6-point subdivision fractal as a limit will be no more than d = 2 − α ∼ = 2.5430. Similarly, one can compute the fractal dimension of subdivision fractal for the interval − 3 16 < µ < −6+3 √ 2

Numerical examples and demonstrations
The proposed work is used to construct the engineering structures such as fractal antennas, bearings and garari's etc. Figure 2a-d present the fractal antennas generated after third, seventh, tenth and thirteenth subdivision levels at µ = 0.1718. The fractal dimension of these fractals is 2.4066. The initial sample of another fractal antenna is shown in Fig. 3a. Figure 3b, c show the fractal antennas generated by the scheme (1) at µ = −0.099. Figure 4a shows the initial sample for a bearing and Fig. 4b, c show the actual bearing generated at parametric values µ = 0 and µ = 3 256 respectively. The initial mesh for rock surface is shown in Fig. 5a. Figure 5b, c show the rock surfaces at third level with µ = −0.1 and µ = −0.05 respectively. Figure 6 presents the structure of garari type shapes.
In the case of given initial control points, shapes and dimensions of the fractals can be adjusted and controlled by adjusting the parameter µ. Hence the obtained results in "Fractal properties of the scheme" section can be used to generate fractal in a fast and efficient way.

Conclusions
In this article, we have reorganized the engineering images by fractal subdivision scheme. We have identified two different parametric intervals to generate different types of engineering models. The relationship between the subdivision parameter and the a b c d Fig. 2 Fractals: Dotted lines with initial control points show the initial control polygon (a, b, c, d) whereas solid lines show the fractal antennas at third, seventh, tenth and thirteenth level with µ = 0.1718 respectively c b a Fig. 3 Smooth curves: a shows the initial structure of bearing and b, c shows the bearing at third level with µ = 0 and µ = 3 256 respectively fractal dimension of the limit fractal curve of the 6-point binary interpolatory subdivision fractal is also presented. It is concluded that 6-point subdivision scheme is an efficient tool for the fast generation of self similar fractals useful in fractal antennas. It is also an appropriate technique for the designing of bearings and garari's etc.