# Bijective Rigid Motions of the 2D Cartesian Grid

Published in *DGCI*, 2016

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2016) Bijective Rigid Motions of the 2D Cartesian Grid. In: Normand N., Guédon J., Autrusseau F. (eds) Discrete Geometry for Computer Imagery. DGCI 2016. Lecture Notes in Computer Science, vol 9647. Springer, pp 359-371, doi:10.1007/978-3-319-32360-2_28

**Author(s):** K. Pluta, P. Romon, Y. Kenmochi, N. Passat

**Abstract:** Rigid motions are fundamental operations in image processing. While they are bijective and isometric in \(\mathbb{R}^2\), they lose these properties when digitized in \(\mathbb{Z}^2\). To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on \(\mathbb{Z}^2\), initially proposed by Nouvel and Rémila for rotations on \(\mathbb{Z}^2\). This allows us to study bijective rigid motions on \(\mathbb{Z}^2\), and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of \(\mathbb{Z}^2\) is bijective.

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**Errata is not provided but several typos and mistakes were corrected in the journal version of this paper** (see **Bijective Digitized Rigid Motions on Subsets of the Plane**)