The generalized winding number (GWN) is a scalar field that supports robust containment queries on curved geometry, including non-watertight, overlapping, and nested boundary representations. While queries can be easily parallelized over samples, direct evaluation on parametric curves and surfaces remains costly for large and complex models. Fast, state-of-the-art GWN approaches leverage a spatial index to approximate the GWN, typically coupled with a Taylor expansion which approximates the GWN contribution for far clusters of geometric primitives. However, such methods operate only on discrete inputs such as triangle meshes and point clouds, and would introduce containment errors near boundaries if applied to curved input. We extend support for fast GWN evaluation over arbitrary collections of NURBS curves in 2D and trimmed NURBS patches in 3D via a Bounding Volume Hierarchy that stores efficiently precomputed moment data in the hierarchy nodes. When querying the hierarchy, approximations for far clusters are used alongside direct evaluation for nearby NURBS primitives, achieving sub-linear complexity while preserving the geometric features in the vicinity of the query point. Central to our performance improvements is an adaptive subdivision strategy for NURBS primitives during a preprocessing phase, creating better spatial partitions while retaining the same accuracy for containment decisions as a direct evaluation. We demonstrate the performance and accuracy of our approach across a large collection of 2D and 3D datasets.

翻译：广义环绕数（GWN）是一种标量场，能够对曲面几何（包括非水密、重叠和嵌套边界表示）进行鲁棒的包含性查询。虽然查询可以轻松地在样本上并行化，但对于大型复杂模型，对参数曲线和曲面进行直接计算仍然代价高昂。当前最先进的快速GWN方法利用空间索引来近似GWN，通常结合泰勒展开来近似远离几何基元簇的GWN贡献。然而，此类方法仅适用于离散输入（如三角网格和点云），若应用于曲线输入，则会在边界附近引入包含错误。我们将快速GWN评估的支持扩展到2D中任意NURBS曲线集合和3D中裁剪NURBS曲面片集合，通过一种层次包围盒结构在节点中高效存储预计算的矩数据。在查询层次结构时，远处簇的近似值与邻近NURBS基元的直接评估结合使用，实现了次线性复杂度，同时保留了查询点附近的几何特征。我们性能改进的核心在于预处理阶段对NURBS基元采用自适应细分策略，该方法在保持包含判定精度与直接评估相同的同时，创建了更优的空间划分。我们通过大量2D和3D数据集展示了所提方法的性能与精度。