# Camera Path Generation for Triangular Mesh Using Toroidal Patches
## 3 Camera Path Generation
### Preprocessing
The maximum (\(W_{\max}\)) and average (\(W_{\text{avg}}\)) values associated with the number of toroidal patches allocated to the grids (with a 2-ring neighborhood relationship) also provided. Generally, as the ratio of toroidal patches to faces in Table 1 increases, there is a corresponding increase in \(W_{max}\), \(W_{avg}\) values in Table 2. However, this relationship may not always hold true, especially when considering N-Ring scalability. For example, while the ratio of toroidal patches to faces of the Rocker model exceeds that of the Haris model, \(W_{max}\) of the Rocker is lower. Figure 6 illustrates the camera's starting position on the camera path for each input model. This starting point is determined based on the \(C_{h,r}(u,v)\) grid with the maximum weight, where the model's curvature undergoes the most significant change. Figure 7 presents the generated camera gaze paths for the input models in 3D, depicted in blue. For clarity comparison, Figure 8 shows the generated camera gaze path (blue), the position path (green), and the initial third-order periodic B-spline curve (red) in the parameter domain of \(C_{h,r}(u,v)\). These paths are closely aligned with areas exhibiting significant curvature changes. In instances like Figure 7a,c,d,g, points with significant curvature \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **\# Face** & **\# Vertex** & **\# \(\mathbf{T_{p}}\)** & \(\mathbf{T_{p}}\)/Face** \\ \hline Bear & 20,188 & 10,096 & 1622 & 8\% \\ Rocker & 24,364 & 12,182 & 3064 & 13\% \\ Horse & 39,696 & 19,850 & 4393 & 11\% \\ Kitten & 55,304 & 27,652 & 2487 & 4\% \\ Bunny & 69,630 & 34,817 & 7200 & 10\% \\ Haris & 132,632 & 66,316 & 14,840 & 11\% \\ Venus & 201,514 & 100,759 & 7423 & 4\% \\ Popiersie & 300,000 & 150,002 & 24,594 & 8\% \\ Armadillo & 345,944 & 172,974 & 19,022 & 5\% \\ Pieta & 509,650 & 255,129 & 37,194 & 7\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical information of input models. changes are concentrated in specific areas compared to other models, leading to a more detailed observation of these areas. As shown in Figure 7e,f,h, the model has a flat area where the evaluated curvature is nearly zero. When this flat area is notably extensive, the camera path is designed to focus on this flat area. The substantial curvature variation between the flat and non-flat area boundary resulted in more toroidal patches allocated to the boundary faces than other areas. However, although Figure 7j exhibits similar characteristics, its path behaves differently. This difference is due to the numerous points with significant changes in curvature, present not only at the bottom but also throughout other areas of the model. In Figure 7b,j, points with significant changes in curvature are distributed throughout the model, guiding the camera path to capture the entire model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{\(C_{\mathbf{h,t}}(\mathbf{u,v})\)} & \multirow{2}{*}{\(A\)} & \multirow{2}{*}{\# **Extreme pts**} & \multirow{2}{*}{\(\mathbf{W_{\text{max}}}\)} & \multirow{2}{*}{\(\mathbf{W_{\text{avg}}}\)} \\  & \(\mathbf{r}\) & & & & & \\ \hline Bear & 0.27 & 0.80 & 30\% & 8 & 24.71 & 8.08 \\ Rocker & 0.45 & 0.44 & 29\% & 10 & 24.92 & 11.12 \\ Horse & 0.42 & 0.60 & 32\% & 8 & 18.15 & 5.76 \\ Kitten & 0.27 & 0.75 & 34\% & 8 & 12.48 & 5.53 \\ Bunny & 0.35 & 0.61 & 44\% & 10 & 19.20 & 8.25 \\ Haris & 0.36 & 0.73 & 39\% & 4 & 30.06 & 9.62 \\ Venus & 0.35 & 0.63 & 24\% & 6 & 9.56 & 5.26 \\ Popiersie & 0.26 & 0.80 & 30\% & 6 & 17.41 & 10.22 \\ Armadillo & 0.35 & 0.66 & 26\% & 10 & 11.13 & 5.52 \\ Pieta & 0.29 & 0.78 & 33\% & 10 & 23.00 & 7.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Generated cylinder and curve information. Figure 6: Snapshot of each model that the camera is at the starting point of the generated camera path.