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

This MATLAB program computes the Gaussian curvature and mean curvature of three-dimensional surfaces represented as triangular meshes. The implementation begins by importing surface data into MATLAB and converting it into a triangulated mesh structure using functions like triangulation or delaunayTriangulation. For each vertex in the mesh, the program calculates curvature properties through discrete differential geometry algorithms. Key computational steps include estimating vertex normals using adjacent face normals, computing curvature tensors through covariance matrix analysis of neighboring vertices, and deriving principal curvatures (κ1 and κ2) via eigenvalue decomposition. The Gaussian curvature (K) is calculated as the product of principal curvatures (K = κ1 × κ2), while mean curvature (H) is obtained as their average (H = (κ1 + κ2)/2). The program incorporates numerical optimization techniques such as the finite element method for smooth curvature estimation and uses discrete differential geometry approaches for accurate curvature computation on irregular meshes. Results can be visualized through color mapping on the surface using patch or trisurf functions, with curvature values represented as vertex colors. Alternatively, surface plots with curvature-based height modulation provide intuitive geometric insights. For enhanced performance, the implementation includes mesh preprocessing steps like normal vector smoothing and curvature regularization to handle noisy input data. The code structure allows for customizable parameters including neighborhood size selection for local curvature estimation and interpolation methods for smooth curvature field generation.