An explicit and rapid intersection algorithm of plane and NURBS surface in CNC surface machining

To address the challenge of plane-surface intersection in toolpath planning for CNC surface machining, a rapid intersection algorithm between a plane and a NURBS surface is proposed, which can provide an explicit Non-Uniform Rational B-Spline (NURBS) description of the intersection curve. By leveraging the dual-parameter property of NURBS surfaces, a series of fixed parameter values (usually values in the knot vector of this parameter) in one parametric direction are selected to generate NURBS curves in the other parametric direction. The bisection method is employed to compute the intersection points of these curves and the plane. Subsequently, the reverse engineering algorithm is applied to construct a NURBS curve that passes through these points, representing the desired intersection curve. The proposed algorithm significantly improves computational efficiency by reducing the problem to one of calculating the intersections of multiple NURBS curves and the plane. Furthermore, the explicit NURBS representation of the intersection curve facilitates subsequent NURBS interpolation. Simulations and experimental results validate the algorithm's effectiveness and accuracy, demonstrating its capability of deriving the intersection expression.