AI 中文总结
Hex9 是参考椭球上的拟连续六边形DGGS，通过明确几何一致性要求确定网格结构，实现无需预先坐标参考系统的单元寻址，层级5时99%单元面积偏差极小。
AI 中文摘要
离散全球网格系统（DGGS）通常基于预先定义的坐标参考系统设计，继承了该系统的固有妥协。Hex9 颠覆了设计思路：我们提出，分层网格要满足几何一致性需具备内在可定向性、平面模式传输、顶点闭合、细化交换性等要求，且这些要求从本质上决定了网格的结构。可允许的单元基元为三角形；可允许的种子是二维球面（S²）的八面三角剖分；可允许的细化方式唯一为孔径9，且仅在手性上存在差异，每一种手性对应唯一的六边形对偶晶格取向，该取向由机器验证的穷举枚举确定。最终形成的结构是移位孔径9的六边形层次结构，其中每个单元都带有仅由构造产生的唯一地址：截断至层级L时，该地址属于OGC Topic 21意义下的DGGS区域标识符；取极限时，可通过函数从地址恢复参考椭球上的任意精确点。该寻址方式为拟连续——除测度为零的接缝集外，所有位置均可恢复——而非严格ISO 19111意义上的连续；同一对象兼具区域标识符和位置恢复坐标的功能，无需预先输入坐标参考系统。可分离的几何实现方式为：分析性八面体基投影结合最优传输推导的面积校正变换，将网格置于WGS84参考椭球上，单元面积呈拟均匀分布：在层级5时，708588个单元中的99%的面积与理想值偏差在0.005%以内，残余偏差仅存在于拓扑所需的6个八面体顶点处。该组合网格与投影及参考椭球无关；仅面积校正变换与参考体相关，且可针对任意椭球（陆地或行星）重新计算。
英文摘要
Discrete global grid systems are conventionally designed over a prior coordinate reference system and inherit its compromises. Hex9 inverts the direction of design: we ask what requirements a hierarchical grid must satisfy to be geometrically coherent -- intrinsic orientability, flat mode transport, vertex closure, refinement commutativity -- and show that these requirements essentially determine the grid. The admissible cell primitive is the triangle; the admissible seed is the octahedral triangulation of S^2; admissible refinement is uniquely aperture 9, and uniquely up to chirality, with exactly one orientation of the hexagonal dual lattice per chirality -- fixed by a machine-verified exhaustive enumeration. The surviving structure is a shifted-aperture-9 hexagonal hierarchy in which every cell carries a unique address derived from the construction alone: truncated at level L, the address is a DGGS zonal identifier in the sense of OGC Topic 21; carried to the limit, a function recovers from it a point on the reference ellipsoid to arbitrary precision. The addressing is quasi-continuous -- position is recoverable everywhere except on a measure-zero set of seams -- rather than continuous in the strict ISO 19111 sense; the same object serves as both zonal identifier and position-recovery coordinate, with no prior coordinate reference system as input. A separable geometric realisation -- an analytical octahedral base projection composed with an optimal-transport-derived area-correcting warp -- places the grid on WGS84 with quasi-uniform cell areas: at level 5, 99% of the 708,588 cells lie within 0.005% of ideal area, with residual deviation confined to the six octahedral vertices required by the topology. The combinatorial grid is projection- and ellipsoid-independent; only the warp is specific to the reference body, and is recomputable for any ellipsoid, terrestrial or planetary.
Comments41 pages, 18 figures. Reference implementation and machine-verified enumeration/measurement scripts: https://github.com/MrBenGriffin/hex9 (Python, tag: paper_v2) and https://github.com/MrBenGriffin/libhex9 (C/C++)
