We present an efficient algorithm for computing the precise Hausdorff Distance (HD) between two freeform surfaces. The algorithm is based on a hybrid Bounding Volume Hierarchy (BVH), where osculating toroidal patches (stored in the leaf nodes) provide geometric properties essential for the HD computation in high precision. Intrinsic features from the osculating geometry resolve computational issues in handling the cross-boundary problem for composite surfaces, which leads to the acceleration of HD algorithm with a solution (within machine precision) to the exact HD. The HD computation for general freeform surfaces is discussed, where we focus on the computational issues in handling the local geometry across surface boundaries or around surface corners that appear as the result of gluing multiple patches together in the modeling of generic composite surfaces. We also discuss how to switch from an approximation stage to the final step of computing the precise HD using numerical improvements and confirming the correctness of the HD computation result. The main advantage of our algorithm is in the high precision of HD computation result. As the best cases of the proposed torus-based approach, we also consider the acceleration of HD computation for freeform surfaces of revolution and linear extrusion, where we can support real-time computation even for deformable surfaces. The acceleration is mainly due to a fast biarc approximation to the planar profile curves of the simple surfaces, each generated by rotating or translating a planar curve. We demonstrate the effectiveness of the proposed approach using experimental results.

我们提出了一种有效的算法，用于计算两个自由曲面之间的精确Hausdorff距离（HD）。该算法基于混合边界体积层次结构（BVH），其中，环形环形面片（存储在叶节点中）的紧密接触为高精度计算提供了高清计算必不可少的几何特性。紧密几何关系的内在特征解决了处理复合曲面的跨边界问题时的计算问题，这导致了HD算法的加速（在机器精度范围内）达到了精确的HD。讨论了一般自由曲面的HD计算，我们将重点放在计算问题上，这些问题是在处理表面边界或表面拐角周围的局部几何图形时出现的，这些几何图形是由于在通用复合曲面的建模中将多个面块粘合在一起而出现的。我们还将讨论如何从近似阶段切换到使用数值改进来计算精确HD的最终步骤，并确认HD计算结果的正确性。我们算法的主要优点是高清计算结果的高精度。作为所提出的基于环面法的最佳案例，我们还考虑了自由旋转和线性挤压曲面的HD计算的加速，即使在可变形曲面的情况下，我们也可以支持实时计算。加速主要是由于对简单表面的平面轮廓曲线的快速偏置近似，每个曲线都是通过旋转或平移平面曲线而生成的。我们使用实验结果证明了该方法的有效性。