## 4 Methodology
### Extract Feature Paths
#### 4.1.2 Generate Feature Sections via Gaussian Mapping  
According to the geometric relationship between the feature section \(S\) and the feature path \(\Gamma\), solid features can be classified into two groups, as shown in Figure 4.  
Given the widespread application of solid features in CAD software, our study exclusively addresses the reconstruction of the first type of solid features. As is shown in Figure 4a, feature section \(S\) is consistently perpendicular to the tangent direction of the Frenet-Serret frame along the path \(\Gamma\). Moreover, at any intersection point \(v_{i}\) of the feature section \(S\) with the manifold mesh surface, it aligns orthogonally with the tangent direction of the Darboux frame at that specific location. Consequently, the parametric expression for the initial feature section \(S^{\prime}\) can be established using the principal direction vector \(\mathbf{n}_{1}\), correlated with the maximum curvature \(K_{1}\) of the primary curvature at \(v_{i}\), the normal vector \(\mathbf{n}\) at \(v_{i}\), and \(v_{i}\): \(S^{\prime}=v_{i}+\lambda\mathbf{n}_{1}+\mu\mathbf{n}\), where \(\lambda\) and \(\mu\) range from 0 to 1. According to [PERSON] et al. [36], cross-section Gaussian mapping is characterized by utilizing the cross-section normal vector at a specific point on the manifold grid surface as the normal vector of the Gaussian sphere's great circle (Gaussian sphere polar axis). Gaussian mapping has been introduced that effectively generates significant cross-sections at any point on the manifold mesh, approximating the local sweep. This novel concept offers an innovative idea for us to create the initial feature section \(S^{\prime}\). It is imperative to carefully select the initial sampling points on the manifold mesh surface. The curvature radius vector \(\mathbf{e}\) at vertex \(v_{i}\) is classified as concave (\(-\)) if it is oriented opposite to the normal vector, and as convex (\(+\)) if it aligns with the normal vector. Given that a vertex possesses two principal curvatures, the concave-convex characteristics of vertex \(v_{i}\) on the mesh are determined as follows: (\(-\),\(-\)) convex feature; (\(+\),\(-\)) or (\(-\),\(+\)) saddle feature; (\(+\),\(+\)) concave feature; (0,0) flat. Inaccurate selection of initial sampling points can result in a reduction in the precision of the Gaussian mapping. Hence, the following selection principles are defined:  
1. Sampling at points where the concave and convex features are (0,0) is prohibited, meaning sampling the cross-section of the segmented area to be reconstructed is not allowed;
2. Aim to sample at concave features (\(+\),\(+\)) or convex features (\(-\),\(-\)), ensuring that the discrepancy between the two principal curvature values meets the requirement: \(|K_{1}-K_{2}|\geq[K]\). Here, \([K]\) represents the main curvature difference threshold set at 0.005 [36]. Under the condition that the aforementioned principles are met, RANSAC [11] can be employed to determine the initial sampling point. Subsequently, the initial characteristic section \(S^{\prime}\) is defined based on the primary curvature and primary direction of the sampling point. The direction of \(S^{\prime}\) undergoes continuous iterative optimization through Gaussian  
Figure 4: Among the two types of solid features, the spatial positioning relationship between the feature section \(S\) and the feature path \(\Gamma\): (**a**) feature section \(S\) and the feature path \(\Gamma\) are perpendicular; (**b**) each feature section \(S\) is parallel to the initial feature sections. mapping. To execute Gaussian mapping effectively, it is crucial to identify the intersection of \(S^{\prime}\) with the triangles and locate all intersections between \(S^{\prime}\) and the mesh. When a plane intersects with a triangle, it can occur at only two instances: intersecting with a vertex or an edge of the triangle.