MATLAB Code Implementation for 3D Curve Fitting

Implementing 3D curve fitting in MATLAB primarily involves two typical scenarios: explicit fitting based on parametric equations (such as polynomial models) and non-parametric fitting based on scattered points (such as interpolation or smoothing methods). Below is an analysis of common technical approaches: Polynomial Parametric Fitting Suitable for cases where the mathematical form of the curve is known (e.g., quadratic surfaces). Use the `polyfitn` toolkit or custom least squares method to establish polynomial relationships between x/y/z coordinates. Important consideration: overfitting - high-degree polynomials may perfectly match data but exhibit poor generalization. Non-Parametric Fitting For scattered point data without explicit mathematical models: Spline interpolation (`scatteredInterpolant`) ensures the curve passes through all sample points but is sensitive to noise; Smooth fitting (using `fit` function with `lowess` method) can suppress noise but may lose fine details; Radial Basis Function (RBF) is suitable for non-uniformly distributed data, allowing control over fitting smoothness through kernel function adjustments. Special Case Handling For closed curves, normalize parameter t as a periodic variable; For large-scale data, apply `pca` dimensionality reduction first to avoid the curse of dimensionality; For missing values, use `fillmissing` for preprocessing before fitting. Key Selection Criteria depend on data characteristics and application scenarios: choose interpolation for accuracy priority, smoothing for noise resistance requirements, and parametric methods when model interpretability is crucial. Code Implementation Notes: - For polynomial fitting: Use `polyfitn(xyz_data, degree)` to obtain coefficients, then `polyvaln` for evaluation - For scattered data: Create interpolant with `F = scatteredInterpolant(x,y,z)` and evaluate with `F(xq,yq)` - For smooth fits: Apply `fit(xyz_data, 'lowess', 'Span', 0.3)` where span controls smoothness - Always validate results with cross-validation or residual analysis to prevent overfitting