MATLAB Implementation of NURBS Surface Fitting with Code Descriptions

NURBS (Non-Uniform Rational B-Spline) surfaces are mathematical tools widely used in computer-aided design and geometric modeling, capable of flexibly representing complex surface shapes. Implementing NURBS surface fitting in MATLAB typically requires curve fitting techniques and parameterization methods to reconstruct surfaces from randomly generated point cloud data. ### Fundamental Approach to NURBS Surface Fitting Point Cloud Data Preparation: The first step involves generating or acquiring random point cloud data for fitting. This data typically consists of scattered points in 3D space with x, y, and z coordinates. In MATLAB implementation, this can be achieved using functions like rand() or randn() to generate sample data, or by importing external point cloud files. Parameterization Processing: Since NURBS surfaces are based on parametric equations, input points require parameterization. Common parameterization methods include uniform parameterization, chord-length parameterization, and centripetal parameterization to rationally distribute knot values in parameter space. MATLAB code would typically implement this through custom functions that calculate parameter values based on point distributions. Knot Vector Calculation: Determining the knot vector for the NURBS surface defines the support intervals for B-spline basis functions. Non-uniform knot distributions are commonly employed to adapt to data variation characteristics. In MATLAB, this can be implemented using algorithms like the de Boor algorithm for knot vector generation based on data points and desired surface continuity. Least Squares Fitting: Control points are solved through least squares methods to minimize the error between the fitted NURBS surface and original data points. MATLAB's robust matrix operations (using backslash operator or lsqnonlin function) efficiently handle this optimization problem by solving linear systems derived from the B-spline basis functions. Surface Evaluation: After fitting, surface shape can be further optimized by adjusting control points, weights, or knot vectors. Final visualization is achieved by sampling the parameter space using MATLAB's surf() or mesh() functions, often combined with nrbmak() and nrbplot() functions from specialized toolboxes. ### Extended Applications NURBS surface fitting finds applications not only in simulation data reconstruction but also in reverse engineering, medical image modeling, and industrial design surface optimization. By adjusting the degree of the surface or knot density, users can balance between fitting accuracy and computational complexity. MATLAB implementations may involve iterative refinement algorithms where surface properties are dynamically adjusted based on error metrics calculated using point-to-surface distance measurements. Key MATLAB functions that support this implementation include: bspline() for basis function calculation, spap2() for least-squares approximation, and custom routines for parameterization and knot vector generation. The implementation often requires careful handling of weight factors and rational B-spline computations to maintain surface quality and numerical stability.