Computation of Gaussian Curvature and Mean Curvature for Surface Point Clouds

Computing and visualizing Gaussian curvature and mean curvature for surface point clouds in MATLAB represents a crucial task in geometric processing. Gaussian curvature characterizes the local bending properties of a surface, while mean curvature describes the average bending degree at a given point. The following introduces a common implementation approach.

First, construct a triangular mesh or establish local neighborhoods based on the point cloud data. Point clouds typically consist of 3D coordinates, requiring local geometric feature fitting for each point to compute curvatures. Common methods include least-squares fitting of local surfaces or constructing covariance matrices using neighboring points. In MATLAB code implementation, the `pcread` function can load point cloud data, while `pcsegdist` or custom k-nearest neighbor searches help establish local neighborhoods using `knnsearch`.

For Gaussian curvature computation, normal vector variations between points and their neighborhoods typically estimate principal curvatures, with Gaussian curvature being the product of the two principal curvatures. Mean curvature is calculated as the arithmetic mean of the principal curvatures. MATLAB's built-in matrix operations (`eig` for eigenvalue decomposition) and optimization tools (`lsqnonlin` for nonlinear least squares) enable efficient implementation. A practical code approach involves computing the covariance matrix for each point's neighborhood and performing eigenvalue analysis to derive curvature properties.

Finally, for intuitive result visualization, curvature values can be mapped to colors using color mapping techniques. For instance, the `scatter3` function plots point clouds with curvature-mapped coloring via the `CData` property, while `patch` functions visualize mesh-based representations with `facevertexcdata`. This facilitates rapid identification of convex/concave regions or abnormal points on surfaces.

This methodology finds broad applications in 3D modeling and medical image processing, enabling effective analysis of surface geometric characteristics through programmable curvature computation pipelines.