## 2 Orbit Trap Method
**Institutional Review Board Statement**: Not applicable. **Informed Consent Statement**: Not applicable. **Data Availability Statement**: The data that support the findings of this study are available from the corresponding author, upon reasonable request. **Conflicts of Interest**: The authors declare that they have no conflict of interest. ## Appendix A Symmetry Conditions

In this appendix, we gathered the conditions that function \(f\) given by (1) must satisfy to generate a pattern with one of the symmetry types: p2, pm, pg, cm, cmm, pmm, pmg, pgg, p4, p4m, and p4g. The symmetry of the considered function \(f\) depends on the matrix:

\[P\ =\ \begin{bmatrix}P_{00}&\ldots&P_{04}\\ \vdots&\vdots&\vdots\\ P_{40}&\ldots&P_{44}\end{bmatrix} ,\]

where \(P_{00},P_{01},\ldots,P_{44}\in\mathbb{R}^{2}\). For simplicity, we present the conditions in the form of a matrix \(M\) (a mask) that has the same dimensions as \(P\). To obtain a matrix \(P^{\prime}\) that will define \(f\) with symmetry type given by the mask \(M\), we need to calculate \(P^{\prime}\ =\ P\circ M\), where \(\circ\) is element-wise product of matrices and vectors. Let us introduce the following notation:

\[\mathbf{0}= \begin{bmatrix}0\\ 0\end{bmatrix} ,\ \mathbf{1}= \begin{bmatrix}1\\ 1\end{bmatrix} ,\ \uparrow= \begin{bmatrix}1\\ 0\end{bmatrix} ,\ \downarrow= \begin{bmatrix}0\\ 1\end{bmatrix} .\]

The masks for the p2, pm, pg, cm, cmm, pmm, pmg, and pgg symmetries are the following [15]:

\[M_{p2}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{1}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{1}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{1}\\ \mathbf{1}&\mathbf{1}&\mathbf{1}&\mathbf{0}&\mathbf{0}\\ \mathbf{1}&\mathbf{1}&\mathbf{1}&\mathbf{0}&\mathbf{0}\\ \end{bmatrix} ,\ M_{pm}= \begin{bmatrix}\uparrow&\uparrow&\uparrow&\downarrow& \downarrow\\ \uparrow&\uparrow&\uparrow&\downarrow&\downarrow\\ \uparrow&\uparrow&\uparrow&\downarrow&\downarrow\\ \uparrow&\uparrow&\uparrow&\downarrow&\downarrow\\ \uparrow&\uparrow&\uparrow&\downarrow&\downarrow\end{bmatrix} ,\ M_{pg}= \begin{bmatrix}\uparrow&\uparrow&\uparrow&\downarrow&\downarrow\\ \downarrow&\downarrow&\downarrow&\uparrow&\uparrow\\ \uparrow&\uparrow&\downarrow&\downarrow\\ \downarrow&\downarrow&\uparrow&\uparrow&\downarrow\\ \uparrow&\uparrow&\downarrow&\downarrow\end{bmatrix} \]

\[M_{pm}= \begin{bmatrix}\uparrow&\mathbf{0}&\uparrow&\mathbf{0}&\downarrow\\ \mathbf{0}&\uparrow&\mathbf{0}&\downarrow\\ \uparrow&\mathbf{0}&\uparrow&\downarrow&\mathbf{0}\\ \uparrow&\mathbf{0}&\uparrow&\downarrow&\mathbf{0}\\ \uparrow&\mathbf{0}&\uparrow&\downarrow&\mathbf{0}\end{bmatrix} ,\ M_{pm}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\downarrow\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\downarrow\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\downarrow\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\downarrow\\ \uparrow&\uparrow&\uparrow&\mathbf{0}&\mathbf{0}\\ \uparrow&\uparrow&\uparrow&\mathbf{0}&\mathbf{0}\end{bmatrix} \]

In the case of the p4, p4m, and p4g symmetries, we cannot give the conditions in the form of masks because the coefficients of \(P\) must satisfy some symmetry conditions thatcannot be placed in the mask. Therefore, for these three symmetry types, we directly give the form of the matrix \(P\)[15]:

\[P_{p4}= \begin{bmatrix}\mathbf{0}&\mathbf{0}&\mathbf{0}&\begin{bmatrix}m_{00}\\ m_{01}\end{bmatrix}&\begin{bmatrix}m_{10}\\ m_{11}\\ m_{11}\end{bmatrix}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\begin{bmatrix}m_{20}\\ m_{21}\end{bmatrix}&\begin{bmatrix}m_{30}\\ m_{31}\end{bmatrix}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\begin{bmatrix}m_{40}\\ m_{41}\end{bmatrix}&\begin{bmatrix}m_{50}\\ m_{51}\end{bmatrix}\\ \begin{bmatrix}m_{01}\\ -m_{00}\end{bmatrix}&\begin{bmatrix}m_{21}\\ -m_{20}\end{bmatrix}&\begin{bmatrix}m_{41}\\ -m_{40}\end{bmatrix}&\mathbf{0}&\mathbf{0}\\ \begin{bmatrix}m_{11}\\ -m_{10}\end{bmatrix}&\begin{bmatrix}m_{31}\\ -m_{30}\end{bmatrix}&\begin{bmatrix}m_{51}\\ -m_{50}\end{bmatrix}&\mathbf{0}&\mathbf{0}\end{bmatrix} \]

\[P_{p4m}=