B-Spline Curve Fitting for Arbitrary Point Coordinates in MATLAB

Implementing B-spline curve fitting in MATLAB is a fundamental numerical computing task, particularly valuable in engineering modeling and graphics processing applications. For any given set of coordinate points, MATLAB's computational capabilities combined with the mathematical definition of B-spline curves enable the generation of smooth and precise fitted curves. The core concept of B-spline curves involves describing curves through linear combinations of control points and basis functions. The fitting process typically includes these key steps: First, determine the curve order (degree), which governs the smoothness of the resulting curve. Second, calculate the knot vector distribution based on the input coordinate points, as the knot vector directly influences the curve's shape. Finally, adjust control points using least-squares optimization or other numerical methods to ensure the curve closely approximates the given coordinates. In MATLAB implementation, developers can utilize built-in functions like `spapi` for spline interpolation or `csape` for complete spline approximation, which provide quick solutions for standard applications. However, for scenarios requiring customized control over fitting precision or specific curve characteristics, programmers can manually construct the basis function matrix and solve the corresponding linear system. This approach involves creating a coefficient matrix using B-spline basis functions evaluated at parameter values, then employing the backslash operator (\) or `lsqnonneg` for constrained least-squares solutions to compute optimal control points. Key programming considerations include: - Using `aptknt` or `optknt` functions for optimal knot placement - Implementing de Boor's algorithm for basis function evaluation - Applying `spcol` for generating collocation matrices in custom implementations Through appropriate selection of control point quantity and knot vector distribution, B-spline curves can flexibly adapt to various complex point set patterns, making them essential tools for data interpolation, path planning, and geometric design applications. The implementation allows balancing computational efficiency with accuracy requirements by adjusting the number of control points relative to input data points.