Development of Gaussian and Mean Curvature Map Calculation for Surface Point Clouds

To calculate Gaussian and mean curvature maps for a given surface point cloud, a comprehensive MATLAB program can be developed. The implementation typically utilizes surface fitting algorithms such as moving least squares or local polynomial approximation to reconstruct the surface geometry from discrete points. Key mathematical formulations involve calculating first and second fundamental forms using partial derivatives of the fitted surface, where Gaussian curvature (K) is computed as the product of principal curvatures, and mean curvature (H) as their average. The program architecture may include functions for: - Point cloud preprocessing (noise filtering, normal estimation) - Local surface fitting using covariance analysis or k-nearest neighbors - Curvature computation through eigenvalue decomposition of the shape operator - Parameter customization for neighborhood size and fitting degree The implementation can incorporate interactive GUI components allowing users to adjust calculation parameters like search radius and smoothing factors. Visualization modules can generate color-mapped curvature plots, isoline displays, and 3D surface renderings with curvature overlays. For optimal performance, the code may leverage vectorized operations and spatial indexing techniques for efficient large-scale point cloud processing. This MATLAB solution provides researchers and engineers with a robust tool for quantitative surface analysis, supporting applications in computer vision, reverse engineering, and geometric modeling. The program's modular design facilitates integration with additional point cloud processing pipelines and export capabilities for further analysis in other software platforms.