## 2 Orbit Trap Method
The obtained patterns are presented in Figure 5, and the parameters used to generate them were the following: colour map from Figure 4b, area \([-2\pi,2\pi]^{2},M=100,\) Picard iteration (i.e., \(\alpha=[1,0]^{T},\beta=[0,0]^{T}\)), Euclidean metric, function \(f\) given by (8) with p3m1 symmetry and \(G_{\mathbb{P}_{0}}=[0.343244,0.711418]^{T},G_{\mathbb{P}_{1}}=[-0.671499,-0.28995 1]^{T},G_{\mathbb{P}_{2}}=[-0.959405,-0.868905]^{T},H_{\mathbb{P}_{0}}=[-0.22 4996,0.0844803]^{T},\)\(H_{\mathbb{P}_{1}}=[-0.0571376,0.0250126]^{T},\) and \(H_{\mathbb{P}_{2}}=[0.0250892,0.553883]^{T}\). From the images in Figure 5, we see that only Figure 5a,c,g have the same symmetry type as \(f\), namely the p3m1 symmetry. All the other images have another type of symmetry, namely, the cm symmetry. When we look at the orbit traps used to obtain the patterns with p3m1 symmetry, then we notice that they have the same rotational symmetries and reflections as the p3m1 symmetry. Thus, to generate a pattern with a given symmetry type, the function \(f\) and the orbit traps should satisfy similar symmetry conditions. Moreover, from this example, we can observe one more interesting thing. When we look at the orbit traps in Figures 1c and 3b, we see that they are the same circles but rotated by 45\({}^{\circ}\). However, if we look at the patterns generated using them with the same other parameters (see Figure 5d,e), then we notice that the patterns are completely \begin{table}
\begin{tabular}{c c} \hline \hline
**Base Colour** & **Colour Ratio** \\ \hline  & **(a)** \\ \hline \((0,0,0)\) & 0.0 \\ \((255,0,0)\) & 0.7 \\ \((255,128,0)\) & 0.75 \\ \((0,0,0)\) & 0.8 \\ \((255,128,0)\) & 0.85 \\ \((255,255,255)\) & 0.9 \\ \((26,0,0)\) & 1.0 \\ \hline  & **(b)** \\ \hline \((0,0,0)\) & 0.0 \\ \((0,0,150)\) & 0.5 \\ \((161,202,241)\) & 0.7 \\ \((243,195,0)\) & 0.75 \\ \((0,0,0)\) & 0.8 \\ \((243,195,0)\) & 0.85 \\ \((255,255,255)\) & 0.9 \\ \((26,0,0)\) & 1.0 \\ \hline  & **(c)** \\ \hline \((0,0,0)\) & 0.0 \\ \((6,204,2)\) & 0.6 \\ \((0,0,0)\) & 0.8 \\ \((19,158,245)\) & 0.85 \\ \((200,200,200)\) & 0.9 \\ \((115,106,55)\) & 1.0 \\ \hline  & **(d)** \\ \hline \((0,0,0)\) & 0.0 \\ \((254,0,0)\) & 0.25 \\ \((0,255,0)\) & 0.3 \\ \((255,255,127)\) & 0.47 \\ \((25,0,0)\) & 0.6 \\ \((255,52,0)\) & 0.7 \\ \((0,128,255)\) & 0.85 \\ \((25,0,0)\) & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Base colours and colour ratios used to generate colour maps in Figure 4.

different. Thus, a modification of the orbit traps obtained by rotation could lead to a significant shape change of the resulting pattern. In the next example, we present the effect of changing the metric used in the definition of the orbit traps. To show this, we fix orbit traps and change the metric and its parameter, i.e., \(p\) for the \(l_{p}\) metric and \(\gamma\) for the Rickman's rug metric. In the example, we use orbit traps from Figure 3a, and the other parameters defining the pattern were the following: area \([-2\pi,2\pi]^{2}\), \(M\ =\ 100\), \(E\ =\ 16\), Picard iteration (\(\alpha\ =\ 1\), \(\beta\ =\ 0\) in the S-iteration), colour map from Figure 4c, and function \(f\) given by (1) with the cmm symmetry defined by the coefficients (we give only the non-zero elements of the matrix that defines (1)) \(P_{04}\ =\ [0,-0.890893]^{T}\), \(P_{13}\ =\ [0,0.594811]^{T}\), \(P_{24}\ =\ [0,0.166299]^{T}\), \(P_{31}\ =\ [2.33836,0]^{T}\), \(P_{40}\ =\ [-1.53665,0]^{T}\), and \(P_{42}\ =\ [0.149503,0]^{T}\). In Figure 6, we show patterns obtained with the \(l_{p}\) metric with the following values of \(p\): (a) 1.0, (b) 1.5, (c) 2.0, (d) 2.5, (e) 3.0, and (f) \(\infty\), whereas in Figure 7, we see patterns generated using Rickman's rug metric with the following values of \(\gamma\): (a) 0.9, (b) 0.7, (c) 0.5, and (d) 0.1. The image from Figure 6c was obtained with the Euclidean metric that was used in the original method presented in [17]. It will serve as a reference point.