## 2 Methodology
### Spiral Strip Design
\begin{pmatrix} \frac{b\cos(\phi^{\prime}+\varphi_{0})-(2\pi a+b\phi^{\prime})\sin(\phi^{ \prime}+\varphi_{0})}{2\pi\sqrt{a^{2}+\frac{ab\phi^{\prime}}{a}+\frac{2^{2}( \theta^{\prime}-1)}{4a^{2}}}}\\ \frac{b\sin(\phi^{\prime}+\varphi_{0})+(2\pi a+b\phi^{\prime})\cos(\phi^{ \prime}+\varphi_{0})}{2\pi\sqrt{a^{2}+\frac{ab\phi^{\prime}}{a}+\frac{2^{2}( \theta^{\prime}+1)}{4a^{2}}}}\end{pmatrix} \]  
The basis vector that points perpendicularly to the spiral curve (\(\mathbf{e}_{p}\)) can be expressed as  
\[\mathbf{e}_{p}=\frac{1}{\mathrm{d}l}\binom{\mathrm{d}y}{\mathrm{d}x}= \begin{pmatrix} \frac{b\sin(\phi^{\prime}+\varphi_{0})+(2\pi a+b\phi^{\prime})\cos(\phi^{ \prime}+\varphi_{0})}{2\pi\sqrt{a^{2}+\frac{ab\phi^{\prime}}{a}+\frac{2^{2}( \theta^{\prime}-1)}{4a^{2}}}}\\ -\frac{b\cos(\phi^{\prime}+\varphi_{0})-(2\pi a+b\phi^{\prime})\sin(\phi^{ \prime}+\varphi_{0})}{2\pi\sqrt{a^{2}+\frac{ab\phi^{\prime}}{a}+\frac{2^{2}( \theta^{\prime}-1)}{4a^{2}}}}\end{pmatrix} \]  
Figure 1b shows the resulting shape of individual segments that are arranged along the spiral. The width and length of the segments are denoted by \(w\) and \(l\). For simplicity, all segments shown in Figure 1b had the same width and length, but the width and length of each segment can also be different (this is demonstrated by some of the examples shown in the next section). The values of parameters \(a\) and \(b\) can be expressed with the use of segment width \(w\) as \(a=r_{0}+\frac{w}{2}\) and \(b=w+s\). \(r_{0}\) is the approximate distance of the first element from the center, while \(s\) represents the space (the width of the gap region) between the segments in two consequent loops (for details, please see the enlarged inset region in Figure 1b). We used vector \(\mathbf{m}=(x,y)\) to denote a point on the spiral, with \(x\) and \(y\) defined by Equations (2) and (3), respectively. For each \(\mathbf{m}\), we can define two additional vectors that point to two points on the opposite sides of the spiral:  
\[\mathbf{p} =\mathbf{m}+\frac{w}{2}\mathbf{e}_{p}, \tag{8}\] \[\mathbf{q} =\mathbf{m}-\frac{w}{2}\mathbf{e}_{p}. \tag{9}\]  
The locations of points \(\mathbf{m},\mathbf{p}\), and \(\mathbf{q}\) at the start of the spiral are shown in the enlarged inset region in Figure 1b (they are denoted as \(\mathbf{m}_{1},\mathbf{p}_{1}\) and \(\mathbf{q}_{1}\)). The distance between \(\mathbf{p}\) and \(\mathbf{q}\) is always equal to the width of the spiral segment, and the line that connects the two points is always perpendicular to the curve. Each spiral segment is defined as a multipoint polygon defined by a sequence of \(\mathbf{p}\) and \(\mathbf{q}\) points. The drawing procedure begins at the spiral start point (which corresponds to \(\varphi^{\prime}=0\)). First, the corresponding values of \(x,y\) and \(\mathbf{e}_{p}\) are calculated using Equations (2), (3), and (8) assuming \(\varphi^{\prime}=0\). With these, the locations of corresponding \(\mathbf{p}\) and \(\mathbf{q}\) points are calculated via Equations (8) and (9). Next, we move in small steps along the spiral curve. We denoted the length of each step as \(\Delta l\). The step size needs to be small enough; otherwise, the borders of the segments might not look smooth (e.g., at least 100 steps should be performed for each spiral loop). At each step, the new value of \(\varphi^{\prime}\) is calculated with the use of Equation (7) as  
\[\varphi^{\prime}_{\mathrm{new}}=\varphi^{\prime}_{\mathrm{old}}+\Delta l \left(a^{2}+\frac{ab\varphi^{\prime}_{\mathrm{old}}}{\pi}+\frac{b^{2}\left( \varphi^{\prime 2}_{\mathrm{old}}+1\right)}{4\pi^{2}}\right)^{-\frac{1}{2}}. \tag{10}\]  
At each step, a new set of \(\mathbf{p}\) and \(\mathbf{q}\) points are calculated in the same way as before, but using \(\varphi^{\prime}_{\mathrm{new}}\) instead of \(\varphi^{\prime}=0\). Once the length of the traveled path reaches the prescribed length of the segment (\(l\)), the list of all \(\mathbf{p}\) and \(\mathbf{q}\) points corresponding to this segment is used to draw a filled polygon with the prescribed color.