Summary: Topology preservation is a property of rigid motions in \({\mathbb R^2}\), but not in \({\mathbb{Z}}^2\). In this article, given a binary object \({\mathsf{X}} \subset{\mathbb{Z}}^2\) and a rational rigid motion \(\mathcal{R} \), we propose a method for building a binary object \(\mathsf{X}_{\mathcal{R}}\subset \mathbb{Z}^2\) resulting from the application of \({\mathcal{R}}\) on a binary object \(\mathsf{X}\). Our purpose is to preserve the homotopy type between \(\mathsf{X}\) and \(\mathsf X_{\mathcal R}\). To this end, we formulate the construction of \(\mathsf X_{\mathcal R}\) from \(\mathsf X\) as an optimization problem in the space of cellular complexes with the notion of collapse on complexes. More precisely, we define a cellular space \(\mathbb H\) by superimposition of two cubical spaces \(\mathbb F\) and \(\mathbb G\) corresponding to the canonical Cartesian grid of \(\mathbb Z^2\) where \(\mathsf X\) is defined, and the Cartesian grid induced by the rigid motion \({\mathcal R}\), respectively. The object \(\mathsf X_{\mathcal R}\) is then computed by building a homotopic transformation within the space \(\mathbb H\), starting from the cubical complex in \(\mathbb G\) resulting from the rigid motion of \(\mathsf X\) with respect to \({\mathcal R}\) and ending at a complex fitting \(\mathsf X_{\mathcal R}\) in \(\mathbb F\) that can be embedded back into \(\mathbb Z^2\).
For the entire collection see [Zbl 1476.68014].