arXiv:2611.12642 · v1

Non-Uniqueness in the Coordinate-Based Normalization of Tetrahedral Similarity Classes

首次提交 2026-11-10 00:18

版本更新 2026-11-10 00:18

雷达观测 2026-11-10 15:22

摘要

Recently, a new coordinate-based normalization has been introduced in [1] to represent tetrahedral similarity classes by normalized coordinates, enabling the study of LEB orbits as a discrete dynamical system. However, the uniqueness of this representation is not guaranteed when two or more edges adjacent to the selected longest edge have the same minimum length and remain admissible under the normalization criteria. In this paper, we demonstrate that different admissible choices may lead to distinct normalized coordinate representations of the same tetrahedral similarity class. We first present an explicit counterexample in which two admissible vertex labelings yield different normalized coordinate representations related by reflection, despite corresponding to the same similarity class. We then provide a systematic classification of the nine possible edge configurations arising in the selection of the canonical face when the longest edge of the tetrahedron is unique, and analyze the corresponding geometric and algebraic conditions. Our analysis identifies the configurations in which the normalization is non-unique and distinguishes them from those in which the ambiguity is resolved by the length criterion, angle criterion, or geometric symmetries. We further show that this non-uniqueness leads to an overcounting of distinct similarity classes.