Constructing Non-Interpolating Quadratic and Cubic B-Spline Curves Using de Boor Evaluation Algorithm

In B-spline curve modeling, interpolation serves as a fundamental task, yet non-interpolating curves prove equally valuable in certain applications. The de Boor evaluation algorithm represents a widely adopted computational method for generating non-interpolating quadratic and cubic B-spline curves. This approach enables the creation of smooth curves under specified control vertex conditions while bypassing interpolation calculations. The algorithm operates through recursive knot insertion and basis function evaluation, where the curve point at parameter value t is computed by iteratively combining control points weighted by B-spline basis functions. For implementation, key functions typically include: - knot vector validation and normalization - basis function calculation using Cox-de Boor recursion - coordinate computation through weighted averaging of control points While this method may not suit all scenarios, it provides an effective alternative for generating satisfactory curves in cases where exact interpolation isn't required. The algorithm's efficiency stems from its localized control property, where each curve segment depends only on a limited set of neighboring control points, making it particularly suitable for real-time graphical applications and CAD systems requiring smooth curve approximations.