## 2 Orbit Trap Method
Moreover, it reduces to the Picard-Mann iteration [24] if \(\alpha_{k}\ =\ 1\) and \(\beta_{k}\
eq 0\) for all \(k\in\mathbb{N}\). To use (11) in the orbit trap method, we need to specify a metric space and the mapping \(T\). We take the metric space \((\mathbb{R}^{2},d)\), where \(d\) is any metric in \(\mathbb{R}^{2}\), and as the mapping, we take \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) given by the formulas introduced in Section 2. In (11), we use sequences of parameters \(\alpha_{k}\) and \(\beta_{k}\), but for simplicity we will use constant sequences, i.e., \(\alpha_{k}\ =\ \alpha\) and \(\beta_{k}\ =\ \beta\). The conditions about the range of the parameters in (11) were introduced in fixed point theory to obtain convergence to a fixed point of the mapping. In the orbit trap method, we are not interested in finding fixed points. Instead, we are interested in generating interesting artistic patterns, so we can omit these conditions and take any real values. We can even go further and take \(\alpha\) and \(\beta\) as two-dimensional points from \(\mathbb{R}^{2}\), but in such a case, we need to define multiplication in order to be able to perform calculations in the S-iteration. Thus, we define the multiplication in a similar way as for complex numbers, i.e.,

\[ \begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix} \cdot \begin{bmatrix}x_{2}\\ y_{2}\end{bmatrix} = \begin{bmatrix}x_{1}x_{2}-y_{1}y_{2}\\ x_{1}y_{2}+y_{1}x_{1}\end{bmatrix} . \tag{12}\]

The last thing that we need to consider--if we want to use S-iteration in the orbit trap method--is the formula for \(p_{k+1}\). In the orbit trap method, we generate patterns with wallpaper symmetry, so in each form of the \(f\) function, we calculate the final result modulo \(2\pi\). Therefore, in the formula for \(p_{k+1}\) in (11), we need to do the calculations modulo \(2\pi\). The modified S-iteration that we will use in the orbit trap method has the following form:

\[ \begin{split} p_{k+1}&=([1,0]^{T}-\alpha)\cdot f(p_{k} )+\alpha\cdot f(u_{k})\mod\ \ \ [2\pi,2\pi]^{T},\\ u_{k}&=([1,0]^{T}-\beta)\cdot p_{k}+\beta\cdot f(p_{k}), \end{split}  \tag{13}\]

where \(p_{0}\in\mathbb{R}^{2}\) is a starting point; \(\alpha,\beta\in\mathbb{R}^{2}\) are parameters; and \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\) is the mapping introduced in Section 2. Let us notice that when the parameters \(\alpha\) and \(\beta\) are of the following form: \(\alpha\ =\ [x_{\alpha},0]^{T}\), \(\beta\ =\ [x_{\beta},0]^{T}\), where \(x_{\alpha},x_{\beta}\in\mathbb{R}\), then (13) reduces to the S-iteration with scalar parameters \(\alpha\ =\ x_{\alpha},\beta\ =\ x_{\beta}\). Therefore, for \(\alpha\ =\ [1,0]^{T}\) and \(\beta\ =\ [0,0]^{T}\), iteration (13) reduces to the standard Picard iteration. Looking again at Algorithm 2 and the orbit traps presented in Figure 1, we see that the circles forming the orbit traps are defined in a metric space in which we use the Euclidean metric. When introducing the S-iteration to the orbit trap method, we assumed that we take the metric space \((\mathbb{R}^{2},d)\), where \(d\) is any metric in \(\mathbb{R}^{2}\). Thus, the circles will have different shapes for various metrics. In the rest of the paper, we will use three metrics defined in \(\mathbb{R}^{2}\)[25]:

* \(l_{p}\)-metric, where \(p\in[1,\infty)\) \[d_{p}([x_{1},y_{1}]^{T},[x_{2},y_{2}]^{T})\ =\ (|x_{1}-x_{2}|^{p}+|y_{1}-y_{2}|^{p})^{\frac{1}{p}},\] (14)
* Chebyshev metric (or \(l_{\infty}\)-metric) \[d_{\infty}([x_{1},y_{1}]^{T},[x_{2},y_{2}]^{T})\ =\ \max\{|x_{1}-x_{2}|,|y_{1}-y_{2}|\},\] (15)
* Rickman's rug metric \[d_{\gamma}([x_{1},y_{1}]^{T},[x_{2},y_{2}]^{T})\ =\ |x_{1}-x_{2}|+|y_{1}-y_{2}|^{\gamma},\] (16) where \(\gamma\in(0,1)\). Examples of circles with centre \([0,0]^{T}\) and radius \(1\) obtained with metrics (14)-(16) are presented in Figure 2.

Besides the use of the S-iteration and various metrics for orbit traps, we also introduce three new orbit traps:

1. \(c_{0}=[0,3.5]^{T}\), \(c_{1}=[0,-3.5]^{T}\), \(r_{0}=r_{1}=3.5\),
2.