A reliable method for finding intersections of 2D NURBS curves

Abstract

A reliable method is presented to compute the intersection of two-dimensional NURBS curves. Two innovative verification criteria are introduced: an intersection existence criterion based on geometric method, and an intersection uniqueness verification criterion founded on Hodograph theory. The combination of these criteria enables precise identification of parametric subintervals containing a unique intersection point. Experimental results demonstrate that compared to traditional Bézier clipping algorithms and piecewise conversion methods, our algorithm significantly improves the computational efficiency and numerical stability of NURBS curves intersection calculations. The method can find intersections in non-singular cases and shows superior robustness, particularly in scenarios with high-degree curves and dense intersections. We also discuss how to deal with singular cases stably. This approach provides both theoretical foundation and practical tools for high-precision intersection computation in complex geometric modeling, path planning, and engineering simulation applications.