Overlap Region Extraction of Two NURBS Surfaces

The detection and computation of the overlap region between two NURBS surfaces, as a special case of the intersection problem, are essential components of CAD systems, directly influencing the robustness of the entire system. Despite their importance, efficient, topologically correct, and numerically robust algorithms for detecting overlap regions remain lacking. To address this issue, we propose an optimization approach for computing the overlap region between two NURBS surfaces within a given error threshold. Based on a bilevel optimization framework, our algorithm first employs cubic Bézier simplices to approximate the boundary of the overlap region. The boundary points of the overlap region are computed iteratively, followed by a Delaunay triangulation to establish the boundary topology. Additional refinement of the boundary edge is applied to ensure the topological correctness and maintain the precision of the overlap region within the specified error threshold. Our main contribution lies in the development of a novel and robust algorithm to calculate the boundary of the overlap region. This approach differs from previous overlap computation methods, which seldom account for error thresholds and are difficult to implement in floating-point arithmetic in CAD systems. We demonstrate the robustness and topological accuracy of our method through extensive experiments on a diverse set of complex examples with varying error thresholds.