MATLAB Program for Triangular Meshing Using Linear Interpolation

Triangular meshing is a technique that converts discrete point sets or surfaces into grid structures composed of triangles, widely applied in computer graphics, finite element analysis, and 3D reconstruction. The linear interpolation-based triangular meshing method is particularly suitable for processing data points with smooth surfaces, efficiently generating high-quality triangular meshes. Implementing such an algorithm in MATLAB typically involves several key steps. First, the program needs to read or generate raw data points, which may originate from measurement data or mathematical functions. For implementation, this can be achieved using MATLAB's built-in functions like `load` for external data or `rand`/`meshgrid` for synthetic point generation. Next, the algorithm constructs an initial triangulation using methods like Delaunay triangulation through MATLAB's `delaunayTriangulation` function, which ensures generated triangles are as equilateral as possible and avoids overly acute triangles. Subsequently, linear interpolation is employed to estimate values between triangle vertices. By performing interpolation within each triangle, denser grid points can be generated to smoothly represent surfaces. MATLAB's `griddata` or `scatteredInterpolant` functions with 'linear' method specification can facilitate this process, particularly suitable for irregularly distributed data points. These functions automatically handle the interpolation computation across triangular elements. Finally, the program typically optimizes the generated mesh through operations like edge flipping or node adjustment to reduce sliver triangles and improve mesh quality. This can be implemented using MATLAB's mesh optimization routines or custom algorithms checking triangle aspect ratios. The output triangular mesh can be used for visualization (via `trisurf` or `trimesh` functions) or numerical computations. The advantage of this method lies in its computational efficiency and effective preservation of original data's geometric characteristics, making it applicable to various scenarios from medical imaging to terrain modeling.