# Camera Path Generation for Triangular Mesh Using Toroidal Patches
## 2 Related Work  
[PERSON] et al. [5] proposed a method that uses entropy to automatically determine viewpoints for generating camera paths for 3D models. They proposed general guidelines for automated camera path generation using their method. [PERSON] et al. [6] proposed a technique for generating camera paths that can move smoothly around 3D point cloud models. Their approach involves estimating motion direction by maximizing photometric entropy, avoiding obstacles, and controlling optical flow to prevent sudden camera jumps. This method was developed to tackle the difficulties associated with observing large-scale models. Figure 1: Camera path generation examples in 3d modeling software: (**a**) process for camera path in Blender 3.5, (**b**) process for camera path in 3 ds Max 2024. Curvature is used to analyze the regional shape of a triangular mesh and select viewpoints for generating camera paths [7; 8; 9; 10]. [PERSON] et al. [7] introduced a method for automatic camera path generation that relied on view saliency. They adjusted the camera's velocity, magnification, and minimization along this path. However, the cost of computing the camera path depends on the resolution of the view sphere and the method used to determine the stable view area. [PERSON] et al. [8] developed a method for generating camera paths that are tailored to preview videos featuring mesh sequences. While [PERSON] et al. [7] focused on examining static models, [PERSON] et al. [8] expanded the scope to include the dynamic movements of models in terms of animation. [PERSON] et al. [9] improved upon [PERSON] et al. [8]'s strategy by incorporating motion information through frame transitions. However, this approach cannot handle mesh sequences that undergo topological changes over time. [PERSON] et al. [10] suggested an algorithm that incorporates user feedback during the path generation process, building on the insights of [PERSON] et al. [8] and [PERSON] et al. [9]. When calculating curvature for triangular meshes, outliers with significantly larger or smaller values compared to nearby points may occur because of noise from 3D scanning. In contrast to previous studies, our method can reduce the potential of outlier occurrence by applying smoothing and further mitigate the impact of outliers by incorporating toroidal patches. To enable the smooth observation of 3D models from online databases while streaming, some researchers proposed a path-planning method that considers network bandwidth [11; 12]. [PERSON] et al. [11; 12] developed a streaming framework incorporating techniques like halting the camera, delaying its movement, or reducing the mesh quality when planning the camera path. These guarantee a seamless preview experience by accounting for model transmission speed. While this framework considers network bandwidth for camera path generation, its use is limited to scenarios with predefined viewpoints. [PERSON] et al. [13] developed an automated path generation methodology to ensure collision-free camera movement while maintaining the visibility of a specific point in 3D space. In their study, the accuracy and magnitude of sampling the 3D space correlate with the scene's size. When planning routes for large areas, it is important to consider the performance limitations arising from this correlation. [PERSON] et al. [14] presented a path-planning technique that prioritizes minimizing path curvature. They demonstrated the efficacy of the Pythagorean-Hodograph in mitigating viewer motion sickness by seamlessly linking key-frames from a first-person view. [PERSON] et al. [17] utilized the concept of curvature to define and evaluate the sensation of smoothness on surfaces of revolution, where third-order B-spline curves represent the profile curve of these surfaces. Several noteworthy contributions have emerged for approximating differential geometry on triangular meshes. [PERSON] [18] proposed a method that utilizes polyhedral approximation to estimate the principal curvature and direction of vertices in a triangular mesh. [PERSON] et al.