The problem of generating uniform deterministic samples over the rotation group, SO(3), is fundamental to computational biology, chemistry, physics, and numerous branches of computer science. We present the best-known method to date for constructing incremental, deterministic grids on SO(3); it provides: (1) the lowest metric distortion for grid neighbor edges, (2) optimal dispersion-reduction with each additional sample, (3) explicit neighborhood structure, and (4) equivolumetric partition of SO(3) by the grid cells. We also demonstrate the use of the sequence on motion planning problems.

中文翻译：

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使用 Hopf Fibration 在 SO(3) 上生成均匀增量网格

在旋转群 SO(3) 上生成均匀确定性样本的问题是计算生物学、化学、物理学和计算机科学众多分支的基础。我们提出了迄今为止在 SO(3) 上构建增量确定性网格的最著名方法；它提供：(1) 网格相邻边的最低度量失真，(2) 每个附加样本的最佳色散减少，(3) 显式邻域结构，以及 (4) 网格单元对 SO(3) 的等体积分区。我们还演示了该序列在运动规划问题上的使用。