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.

Boolean operations for Boundary Representation (B-Rep) models are among the most commonly used functions in Computer Aided Design (CAD) systems. They are also one of the most delicate soft modules, with challenges arising from complex algorithmic flows and efficiency and accuracy issues, especially in extreme cases. Common issues encountered in processing complex models include low efficiency, missing results, and non-watertightness. In this paper, we propose a novel algorithm for efficient and accurate Boolean operations on B-Rep models. This is achieved by establishing a bijective mapping between B-Rep models and the corresponding triangle meshes with controllable approximation error, thus mapping B-Rep Boolean operations to mesh Boolean operations. By using conservative intersection detection on the mesh to locate all surface intersection curves and carefully handling degeneration and topology errors, we ensure that the results are consistently watertight and correct. We demonstrate the superior efficiency of the proposed method using the open-source geometry engine OCCT, the commercial engine ACIS, and the commercial software Rhino as benchmarks.

The surface/surface intersection technique serves as one of the most fundamental functions in modern Computer Aided Design (CAD) systems. Despite the long research history and successful applications of surface intersection algorithms in various CAD industrial software, challenges still exist in balancing computational efficiency, accuracy, as well as topology correctness. Specifically, most practical intersection algorithms fail to guarantee the correct topology of the intersection curve(s) when two surfaces are in near-critical positions, which brings instability to CAD systems. Even in one of the most successfully used commercial geometry engines ACIS, such complicated intersection topology can still be a tough nut to crack. In this paper, we present a practical topology guaranteed algorithm for computing the intersection loci of two B-spline surfaces. Our algorithm well treats all types of common and complicated intersection topology with practical efficiency, including those intersections with multiple branches or cross singularities, contacts in several isolated singular points or highorder contacts along a curve, as well as intersections along boundary curves. We present representative examples of these hard topology situations that challenge not only the open-source geometry engine OCCT but also the commercial engine ACIS. We compare our algorithm in both efficiency and topology correctness on plenty of common and complicated models with the open-source intersection package in SISL, OCCT, and the commercial engine ACIS.