Application of Numerical Methods in 3D Surface Modelling
Keywords:
Bicubic spline, Bilinear interpolation, Finite element Mesh, Laplacian smoothing, Radial basis functions, Surface reconstructionAbstract
Three-dimensional surface modelling is a fundamental challenge in engineering and applied mathematics, underpinning applications ranging from computer-aided design (CAD) and finite element analysis to medical imaging and terrain reconstruction. Classical parametric representations, such as NURBS and implicit surfaces, require known analytical forms and therefore fail when surfaces are defined purely by sampled measurement data. This paper presents a systematic comparative evaluation of four numerical methods for continuous surface reconstruction from point clouds: bilinear interpolation, bicubic spline fitting, finite difference methods, and Radial Basis Function (RBF) interpolation. All methods are benchmarked on a common trigonometric test surface f(x,y) = sin(√(x²+y²))·e^(−0.15(x²+y²)) at grid resolutions N {4, 8, 16, 32, 64, 128}, with error measured as maximum absolute deviation on a 512×512 evaluation grid. Results demonstrate that bicubic splines achieve O(h) convergence (RMSE: 0.0048) versus O(h²) for bilinear interpolation (RMSE: 0.0245), at moderate computational overhead (3.4 ms vs. 0.8 ms per query). RBF interpolation with a Gaussian kernel achieves the lowest RMSE of 0.0031 on unstructured scattered data but requires O(n³) matrix precomputation. Laplacian mesh smoothing is quantified for the first time: 20 iterations reduce surface noise by 97% while preserving macro-scale geometry. Findings provide practitioners with evidence-based method selection guidance, with direct applications to terrain modelling, dental prosthetic design, and automotive CAD workflows.
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