Efficient Iterative Method for Intersecting Two NURBS Curves

This paper presents an efficient iterative algorithm using rational function interpolation for NURBS curve intersection. Traditional methods, like Newton’s and Secant methods, are sensitive to initial values and struggle with tangential cases, while implicitization and subdivision methods exhibit high computational complexity or instability. To address these shortcomings, we developed a novel iterative strategy that integrates rational function interpolation with a subdivision method, which significantly improves both the efficiency and stability of the intersection calculation. The algorithm first constructs a rational interpolation function, constrained by two-point derivatives, to approximate the curves’ parametric equations. It then derives a corresponding iterative formula from a quadratic equation to achieve rapid convergence. We performed extensive experiments, including scenarios with crossing and tangential intersections, alongside a benchmark test involving 4 000 curves of varying degrees. Results showed that for tangential intersections, our algorithm is approximately 200% more efficient than comparable iterative methods. Furthermore, numerical examples demonstrate that the overall intersection efficiency is 5 to 50 times that of the OCCT and 21% to 180% higher than the ACIS, confirming the algorithm’s effectiveness.