d维立方体和单形的十六条空间填充曲线和遍历
Sixteen space-filling curves and traversals for d-dimensional cubes and simplices
摘要：
本文描述了16种不同的方法来递归地遍历d维空间，这种方法对于任何维数都是定义良好的。这些遍历中的每一个都具有可能对某些应用有益的不同属性。有些穿越是新的，有些在原则上是已知的，但没有被充分描述为任何数量的维度，有些穿越是已知的。这篇文章是第一次在一个一致的符号系统中展示它们。此外，本文还提供了一些工具，用于按照每次遍历访问的顺序枚举规则网格中的点。特别是，我们涵盖：基于将立方体细分为2^d子立方体的五个不连续遍历：Z-遍历（Morton索引）、U-遍历、格雷码遍历、双格雷码遍历和由内向外遍历;基于将单形细分为2^d子单形的两个不连续遍历：Hill-Z遍历和Maehara反射遍历;基于将立方体细分为2^d子立方体的五个连续遍历：Base-camp Hilbert曲线、Harmonious Hilbert曲线、Alfa Hilbert曲线、Beta Hilbert曲线和Butz-Hilbert曲线;基于将立方体细分为3^d子立方体的四种连续遍历：Peano曲线、Coil曲线、Half-coil曲线和Meurthe曲线。所有这些遍历都是自相似的，因为在任何递归细分级别上，立方体或单纯形的每个子立方体或子单纯形中的遍历可以通过缩放、平移、旋转、反射和/或反转完整单位立方体或单纯形的遍历来获得。
英文摘要：
This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain applications. Some of the traversals are novel, some have been known in principle but had not been described adequately for any number of dimensions, some of the traversals have been known. This article is the first to present them all in a consistent notation system. Furthermore, with this article, tools are provided to enumerate points in a regular grid in the order in which they are visited by each traversal. In particular, we cover: five discontinuous traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and Inside-out traversal; two discontinuous traversals based on subdividing simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected traversal; five continuous traversals based on subdividing cubes into 2^d subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four continuous traversals based on subdividing cubes into 3^d subcubes: the Peano curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these traversals are self-similar in the sense that the traversal in each of the subcubes or subsimplices of a cube or simplex, on any level of recursive subdivision, can be obtained by scaling, translating, rotating, reflecting and/or reversing the traversal of the complete unit cube or simplex.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
