The geometry processing of a point cloud 2-manifold (or point cloud surface) heavily depends on the discretization of differential geometry properties such as Gaussian curvature, mean curvature, principal curvature, and principal directions. Most of the existing algorithms indirectly compute these differential geometry properties by seeking a local approximation surface or fitting point clouds with certain polynomial functions and then applying the curvature formulas in classical differential geometry. This paper initially proposed a new discretized Laplace-Beltrami operator by applying an inherent distance parameter, which acts as the foundation for precisely estimating the mean curvature. Subsequently, the estimated mean curvature was taken as a strong constraint condition for estimating the Gaussian curvatures, principal curvatures, and principal directions by determining an optimal ellipse. The proposed methods are mainly based on the heat kernel function and do not require local surface reconstruction, thus belonging to truly mesh-free methods. We demonstrated the correctness of the estimated curvatures in both analytic and non-analytic models. Various experiments indicated that the proposed methods have high accuracy. As an exemplary application, we utilized the mean curvature for detecting features of point clouds.

Automated Fiber Placement (AFP) components are affected by environmental and technological factors in the actual production process, leading to the generation of defects and posing serious harm to the fiber placement components and the overall assembly. Due to the confidentiality of AFP and forming technology, as well as the expensive and time-consuming manufacturing process, it is difficult to obtain sample datasets related to defect images, resulting in the problem of small sample learning. This article proposes a method of expanding the dataset through virtual sample generation and using it as input training samples for the target detection network. In response to the characteristics of fiber placement defects, a coordinate attention mechanism was added to YOLOv5s to optimize the design. The final target detection network model was named Place YOLO, with an accuracy value of 0.941 and a mean Average Precision (mAP) of 0.833, which is 1.8% higher than that of the original YOLOv5s.