MATLAB Program for Computing Gaussian Curvature and Mean Curvature of 3D Surfaces

Computing Gaussian curvature and mean curvature for 3D surfaces in MATLAB is a common task involving differential geometry, applicable to fields such as computer graphics and medical image processing. Curvature calculation typically requires the following key steps: First, geometric information of the surface must be obtained. For mesh surfaces, vertex coordinates and connectivity relationships need to be known. For parametric surfaces, the parametric equations are required. Regardless of the representation, computation of first and second derivative information forms the foundation for curvature calculations. In MATLAB implementation, this can be achieved using structured grid data or parametric function handles. The calculation process utilizes two fundamental matrices: the first fundamental form and the second fundamental form. The first fundamental form describes the intrinsic geometry of the surface and contains three coefficients E, F, and G. The second fundamental form characterizes the surface's bending in space and contains coefficients L, M, and N. These coefficients are computed using first and second partial derivatives of the surface. In code implementation, these derivatives can be approximated using central difference methods or gradient functions for discrete data. Gaussian curvature represents the product of principal curvatures, reflecting the intrinsic bending properties of the surface. Mean curvature is the arithmetic average of principal curvatures, describing the extrinsic bending degree. In MATLAB implementation, finite difference methods or local fitting approaches (such as quadratic surface fitting) can be used to approximate derivatives, particularly suitable for discrete mesh data. The principal curvatures can be derived from the fundamental forms using eigenvalue decomposition. For practical applications, numerical stability must be considered. When surfaces are relatively flat or mesh quality is poor, calculation results may show abnormalities. Regularization techniques or improved numerical methods can enhance robustness. Common approaches include smoothing filters or weighted least squares fitting. Calculation results can be visualized using color mapping to represent curvature magnitudes across different regions, facilitating intuitive analysis through MATLAB's surface plotting and colormap functions.