We present a highly efficient algorithm for computing the minimum distance between two solids of revolution, each of which is defined by a planar cross‐section region and a rotation axis. The boundary profile curve for the cross‐section is first approximated by a bounding volume hierarchy (BVH) of fat arcs. By rotating the fat arcs around the axis, we generate the BVH of fat tori that bounds the surface of revolution. The minimum distance between two solids of revolution is then computed very efficiently using the distance between fat tori, which can be boiled down to the minimum distance computation for circles in the three‐dimensional space. Our circle‐based approach to the solids of revolution has distinctive features of geometric simplification. The main advantage is in the effectiveness of our approach in handling the complex cases where the minimum distance is obtained in non‐convex regions of the solids under consideration. Though we are dealing with a geometric problem for solids, the algorithm actually works in a computational style similar to that of handling planar curves. Compared with conventional BVH‐based methods, our algorithm demonstrates outperformance in computing speed, often 10–100 times faster. Moreover, the minimum distance can be computed very efficiently for the solids of revolution under deformation, where the dynamic reconstruction of fat arcs dominates the overall computation time and takes a few milliseconds.

中文翻译：

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旋转固体的有效最小距离计算

我们提出了一种高效的算法来计算两个旋转体之间的最小距离，每个旋转体都由平面横截面区域和旋转轴定义。横截面的边界轮廓曲线首先由脂肪弧的边界体积层次（BVH）近似。通过围绕轴旋转脂肪弧，我们生成了包围旋转表面的脂肪环的 BVH。然后使用胖圆环之间的距离非常有效地计算两个旋转实体之间的最小距离，这可以归结为三维空间中圆的最小距离计算。我们基于圆的旋转实体方法具有几何简化的独特特征。主要优势在于我们的方法在处理复杂情况时的有效性，这些情况在所考虑的固体的非凸区域中获得最小距离。尽管我们正在处理实体的几何问题，但该算法实际上以类似于处理平面曲线的计算方式工作。与传统的基于 BVH 的方法相比，我们的算法在计算速度方面表现出色，通常快 10-100 倍。此外，对于变形下的旋转实体，可以非常有效地计算最小距离，其中脂肪弧的动态重建在整个计算时间中占主导地位，需要几毫秒。该算法实际上以类似于处理平面曲线的计算方式工作。与传统的基于 BVH 的方法相比，我们的算法在计算速度方面表现出色，通常快 10-100 倍。此外，对于变形下的旋转实体，可以非常有效地计算最小距离，其中脂肪弧的动态重建在整个计算时间中占主导地位，需要几毫秒。该算法实际上以类似于处理平面曲线的计算方式工作。与传统的基于 BVH 的方法相比，我们的算法在计算速度方面表现出优异的性能，通常快 10-100 倍。此外，对于变形下的旋转实体，可以非常有效地计算最小距离，其中脂肪弧的动态重建在整个计算时间中占主导地位，需要几毫秒。