# Camera Path Generation for Triangular Mesh Using Toroidal Patches
## 3 Camera Path Generation
### Preprocessing
Figure 4 shows camera paths generated using our method for the Rocker model. The red curve represents the original third-order periodic B-spline curve derived from the control points; the blue curve shows the camera gaze path, and the green curve shows the camera position path. Algorithm 2 provides the procedure for generating these camera position and gaze paths. Figure 3: Example of control point selection process: (**a**) grids with their weights, (**b**) selected grids with maximum weight, and (**c**) selected control points for generating periodic B-spline curve. Figure 5 presents the results of applying the complete process to the triangular mesh model. Figure 5a shows the smoothing result [23], and Figure 5b shows faces with different colors according to their toroidal patches' relationship. Figure 5c displays the mesh face weights, while Figure 5d illustrates mesh portions within each cylinder grid, with colors derived from the weights in Figure 5c. In both figures, faces closer to red indicate higher weights, while those closer to green indicate lower weights. Figure 5e displays the camera gaze path in blue, which is generated using our method. ## 4 Experimental Result and Discussion

The triangular meshes were stored and represented using the PMP library [31], which employs a half-edge-based data structure. To ensure consistent experimental results, all input meshes were enclosed within an axis-aligned bounding box (AABB) and resized

Figure 4: Camera path visualization for the Rocker model. The camera gaze path (shown in blue) and the camera position path (shown in green) are generated based on the initial curve (shown in red). Figure 5: Camera path generation workflow: (**a**) smoothing result of the input mesh, (**b**) colored faces according to included toroidal patch, (**c**) colored faces according to their weight, (**d**) colored faces according to corresponding cylinder grid’s weight, and (**e**) camera gaze path generation result. so that the diagonal length of its AABB equals 1. For the toroidal patch generation, the tolerance between the toroidal patch and the corresponding face(s) was set to \(10^{-3}\). The bounding cylinder \(C_{h,r}(u,v)\) was uniformly divided into \(m\) and \(n\) segments, with \(m=11\) and \(n=20\). Table 1 displays the count of toroidal patches corresponding to each of the ten selected models. The number of toroidal patches does not exhibit a direct proportionality to the face count of the models, suggesting that toroidal patch distribution is influenced more by geometric features, such as local curvature variations, than by the sheer number of mesh faces. For example, despite having a higher triangle count than the Haris model, the Venus model generates fewer toroidal patches, implying a lower curvature variation in the model. This discrepancy arises from variations in the models' geometric characteristics, particularly their local curvature variation. For instance, while the Venus model has more triangles than the Haris model, it yields fewer toroidal patches, indicating fewer variations in its curvature. The Rocker model exhibits the highest ratio of the toroidal patches to faces at 13%, this is because many toroidal patches are formed by a single triangle of the model. This phenomenon is due to the model's large flat areas compared to other models, preventing the merging of adjacent faces into toroidal patches. Additionally, the model's relatively low face count amplifies the effect of single-face toroidal patches on this ratio, resulting in the highest percentage observed among the models. Table 2 provides comprehensive data for the camera path, which includes the radius and height of the bounding cylinder \(C_{h,r}(u,v)\). The amplitude ratio \(A\) indicates the proportion of the cylinder height occupied by the curve, with a higher value suggesting an expanded vertical field of view. An analysis of the number of extreme points of the camera path and the \(A\) offers a perspective on the distribution of the model's geometric characteristics.