Designing aperture [formula omitted] hexagonal grids and their indexing as factor rings of Eisenstein integers

Abstract

This paper proposes apertures of size 22k for hexagonal grids together with their indexing scheme, where k may be any positive integer. Hexagonal grid can be represented by an ideal I(2+ω) of a unit-ring Z(ω) of Eisenstein integers, where, for the set Z of integers and the cube root ω of 1, Z(ω) denotes {a+bω|a,b∈Z}, and, for any λ∈Z(ω), I(λ) denotes an ideal {κλ|κ∈Z(ω)} of Z(ω). We show that a factor ring I(2+ω)/I(2k(2+ω)) represents an aperture 22k hexagonal grid, and prove that it is isomorphic to a factor ring Z(ω)/I(2k). The latter represents a trigonal grid in a rhombus-shaped aperture with its edge length 2k, where we can address each of its 22k grid points as i+jω using two k-bit positive binary integers. We show that each grid point α=a+bω in I(2+ω)/I(2k(2+ω)) can be indexed by the address of its isomorphic image in Z(ω)/I(2k), and that this index is identical to the pair of the k-bit 2′s complements of a and b. Based on these findings, we clarify how indices may change for different resolutions, how to define the hierarchical indexing, how to geometrically interpret algebraic operations on indices, and, as one of the applications, how the quantization of trajectories of 106∼107 mobile objects on hexagonal grids may reduce the communication capacity required for their quasi-real-time simultaneous monitoring and sharing through the Internet. We also show how our indexing scheme can be applied to the design of DGGSs (Discrete Global Grid Systems) that cover the entire Earth's surface and 3D regular grids.