## 4 Methodology
### Extract Feature Paths
#### 4.1.2 Generate Feature Sections via Gaussian Mapping
This process initiates from the initial sampling point vi on the initial characteristic section \(S^{\prime}\) and traverses the neighboring points in the primary direction until returning to point \(v_{i}\). The discussion will explore two scenarios. If a point \(v_{j}\) is present on \(S^{\prime}\) among the first-order neighborhood points of \(v_{i}\), then the new point \(v_{j}\) should be adopted as the starting point for further exploration of its neighbors. It is crucial to avoid returning to \(v_{i}\) to prevent the algorithm from getting stuck in an infinite loop. On the other hand, if there are no points on \(S^{\prime}\) within the first-order neighborhood of \(v_{i}\), each of \(v_{i}\)'s first-order neighbors should be successively substituted into the parametric equation of \(S^{\prime}\). When two adjacent first-order neighbors \(v_{m}\) and \(v_{n}\) result in a negative product upon substitution into the equation, it indicates that the edge \(\overline{v_{m}v_{n}}\) intersects with \(S^{\prime}\). The point of intersection \(v_{i}\) is determined, and its normal vector \(\mathbf{n}_{l}\) is calculated using linear interpolation between the normal vectors of \(v_{m}\) and \(v_{n}\) for Gaussian mapping:  
\[\mathbf{n}_{l}=\frac{a\mathbf{n}_{m}+b\mathbf{n}_{n}}{\|a\mathbf{n}_{m}+b \mathbf{n}_{n}\|} \tag{2}\]  
where \(\mathbf{n}_{m}\) and \(\mathbf{n}_{n}\) represent the normal vectors of points \(v_{m}\) and \(v_{n}\), while \(a\) and \(b\) denote the distances from \(v_{l}\) to \(v_{m}\) and \(v_{n}\), respectively. To create cross-sectional Gaussian mapping, the initial points of the normal vectors of intersections between \(S^{\prime}\) and the manifold mesh are translated to the center of the Gaussian sphere. Given the discrete distribution characteristics of the endpoints of all normal vectors, it becomes imperative to conduct plane fitting based on the least squares method. Subsequently, the angle between the normal vector \(\mathbf{n}_{f}\) of the fitting plane \(S_{f}\) and the normal vector \(\mathbf{n}_{s}\) of the original section \(S^{\prime}\) is examined for iterative corrections. Compute the cosine of the angle between \(\mathbf{n}_{f}\) and \(\mathbf{n}_{s}\) represented as \(\cos\left\langle\mathbf{n}_{f},\mathbf{n}_{s}\right\rangle\), and proceed with the following discussion:  
(1) If \(\left(1-\cos\left\langle\mathbf{n}_{f},\mathbf{n}_{s}\right\rangle\right)\geq 0.0001\), adjustments to the orientation of the fitting plane are necessary. This involves translating the initial section normal vector \(\mathbf{n}_{s}\) to the center of the Gaussian sphere, utilizing it as the polar axis of the sphere to compute the normal vector and angle at each intersection point. The included angle \(\theta_{i}\) of the axis is then determined, followed by calculating the mean \(E\) and standard deviation \(\sigma\) of all included angles. Subsequently, a Gaussian probability distribution function is established, and normal vectors beyond \(\pm 3\sigma\) are eliminated. The process continues by refitting the remaining normal vectors until the specified criteria are satisfied. (2) If \(\left(1-\cos\left\langle\mathbf{n}_{f},\mathbf{n}_{s}\right\rangle\right)<0.0001\), the current fitting plane is accepted as the definitive feature section.