## 4 Cultural Heritage Preservation
### Point-Cloud Creation  
There are four main motifs as described above. The first three of them can be modeled by using the parametric-equation-based approach, whereas the other can be modeled by using the algorithmic approach.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Motif** & **Ainu Name** & **Description** \\ \hline
1 & Ayusi & A shape marked with thorn \\
2 & Moreu & A spiral shape \\
3 & Arus-moreu & A spiral shape with small thorns \\
4 & Sikike-nu-moreu & A spiral shape with corners \\
5 & Sik & An eye shape \\
6 & Utasa & A shape that intersects each other \\
7 & Uren-moreu & Two spiral shapes \\
8 & Ski-uren-moreu & Two spiral shapes with an eye shape \\
9 & Moreu-etok & A shape like a spirally plant \\
10 & Punkar & A shape like vine \\
11 & Apapo-piras (u) ke & A shape like flower \\
12 & Apapo-epuy & A shape like flower bud \\
13 & - & A shape like heart type \\
14 & - & A shape like fishing bell \\ \hline \hline \end{tabular}
\end{table}
Table 1: Different types of Hokkaido-based Ainu motifs [64].  
First consider the point-clouds for representing the motifs called Moreu, Sik, and Utasa using the parametric-equation-based approach. The equation for representing the motif called Moreu is described as follows. Let \(P_{i}=(P_{ix},P_{iy})\) be an arbitrary point on the motif called Moreu. The equation of \(P_{i}\) is defined as follows.  
\[P_{ix}=P_{0x}+a\theta_{i}\sin(\theta_{i}),\,P_{iy}=P_{0y}+b\theta_{i}\cos( \theta_{i}) \tag{1}\]  
In Equation (1), \(\theta_{i}=\theta_{0}+(i-1)\Delta\theta\), \(i=1,\,...,\,n\), \(a,\,b>0\), \(b\geq a\), and \(\Delta\theta>0\). Figure 9a shows the point-cloud of Moreu in accordance with Equation (1), where \(a=0.15\), \(b=0.2\), \(\theta_{0}=0^{\circ}\), \(\Delta\theta=10^{\circ}\), \(n=29\), \(P_{0x}=3\), and \(P_{0y}=3\).  
Similarly, the equation for representing the motif called Sik is described as follows. Let \(P_{i}=(P_{ix},P_{iy})\) be an arbitrary point on the motif called Sik. The equation of \(P_{i}\) is defined as follows.  
\[P_{ix}=P_{0x}+a\cos^{3}(\theta_{i}),\,P_{iy}=P_{0y}+b\sin^{3}(\theta_{i}), \tag{2}\]  
In Equation (2), \(\theta_{i}=\theta_{0}+(i-1)\Delta\theta\), \(i=1,\,...,\,n\), \(a,\,b>0\), \(a=b\), \(\Delta\theta>0\), and \(0\leq\theta_{i}\leq 360\). Figure 9b shows the point-cloud of Sik in accordance with Equation (2), where \(a=2\), \(b=2\), \(\theta_{0}=0^{\circ}\), \(\Delta\theta=10^{\circ}\), \(n=37\), \(P_{0x}=3\), and \(P_{0y}=3\). The third main motif called Utasa which takes shape of two lines intersected at mid-point. Thus, the point-cloud of Utasa can be created by using line Equation (A1) which is described in Section 3.1.  
The fourth main motif called Ayusi cannot be created by using the parametric-equation-based approach. Thus, the Ayusi can be modeled using the algorithmic approach.  
Figure 10 shows the point-cloud of Ayusi in accordance with the algorithm described in Section 3.2, for \(\alpha=0^{\circ}\), \(d=10\), \(P_{cx}=0\), \(P_{cy}=2\), and where \(r_{i}\) and \(\rho_{i}\) are systematically adjusted for \(i=1,2,\ldots,85\). Some motifs can be created by combining both the parametric-equation-based approach and algorithmic approach, the example is shown in Figure 11. In Figure 11 the point-cloud of Series1 is created using the algorithmic approach and Series2 is created by using the parametric-equation-based approach.  
Figure 10: Point-clouds of Ayusi created using the algorithmic approach.  
Figure 9: Point-clouds created using the parametric-equation-based approach. (**a**) Moreu; (**b**) Sik.