An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices
Abstract
We study an integer-coordinate encoding of boundary incidences for finite subsets of the regular hexagonal tiling. Each selected cell contributes six ordered records with endpoints in two role-distinguished classes, point and tail. Identical records from neighboring cells cancel, and the surviving records form a disjoint union of cycles, including the exterior boundaries of disconnected components and the boundaries of holes. We derive the actual even- and odd-row record formulas from the A--B--C transforms used in Hexagones 2.0, prove exact shared-edge identities and the global role-qualified incidence structure, and thereby obtain an unconditional correctness theorem for the abstract coordinate construction. We compare the encoding with an independent exact cube-coordinate half-edge hashing implementation, proving an incidence isomorphism and checking 504 finite regions. We give an expected-linear-time indexed algorithm and distinguish it from the historical fixed-capacity C traversal, whose sentinel and closure conventions do not inherit the theorem automatically. The claim is a compact coordinate realization of a known planar boundary principle, not historical priority or a new three-axis lattice.
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BibTeX
@misc{mullhaupt2026coordinate,
title = {{An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices}},
author = {Philippe Mullhaupt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25451},
eprint = {2610.25451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25451},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices
AU - Philippe Mullhaupt
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25451
UR - https://arxiv.org/abs/2610.25451
AB - We study an integer-coordinate encoding of boundary incidences for finite subsets of the regular hexagonal tiling. Each selected cell contributes six ordered records with endpoints in two role-distinguished classes, point and tail. Identical records from neighboring cells cancel, and the surviving records form a disjoint union of cycles, including the exterior boundaries of disconnected components and the boundaries of holes. We derive the actual even- and odd-row record formulas from the A--B--C transforms used in Hexagones 2.0, prove exact shared-edge identities and the global role-qualified incidence structure, and thereby obtain an unconditional correctness theorem for the abstract coordinate construction. We compare the encoding with an independent exact cube-coordinate half-edge hashing implementation, proving an incidence isomorphism and checking 504 finite regions. We give an expected-linear-time indexed algorithm and distinguish it from the historical fixed-capacity C traversal, whose sentinel and closure conventions do not inherit the theorem automatically. The claim is a compact coordinate realization of a known planar boundary principle, not historical priority or a new three-axis lattice.
ER -