Orbit Computation for Atomically Generated Subgroups of Isometries of $\mathbb{Z}^n$

Isometries and their induced symmetries are ubiquitous in the world. Taking a computational perspective, this paper considers isometries of $\mathbb{Z}^n$ (since values are discrete in digital computers), and tackles the problem of orbit computation under various isometry subgroup actions on $\mathbb{Z}^n$. Rather than just conceptually, we aim for a practical algorithm that can partition any finite subset of $\mathbb{Z}^n$ based on the orbit relation. In this paper, instead of all subgroups of isometries, we focus on a special class of subgroups, namely atomically generated subgroups. This newly introduced notion is key to inheriting the semidirect-product structure from the whole group of isometries, and in turn, the semidirect-product structure is key to our proposed algorithm for efficient orbit computation.