# Honeycomb geometry: Rigid Motions on the Hexagonal Grid

Published in DGCI, 2017

Recommended citation: Pluta K., Romon P., Kenmochi Y., Passat N. (2017) Honeycomb geometry: Rigid Motions on the Hexagonal Grid. In: Discrete Geometry for Computer Imagery. DGCI 2017. Lecture Notes in Computer Science, vol 10502. Springer, pp 33-45, doi:10.1007/978-3-319-66272-5_4

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

Abstract: Euclidean rotations in $\mathbb{R}^2$ are bijective and isometric maps, but they lose generally these properties when digitized in discrete spaces. In particular, the topological and geometrical defects of digitized rigid motions on the square grid have been studied. In this context, the main problem is related to the incompatibility between the square grid and rotations; in general, one has to accept either relatively high loss of information or non-exactness of the applied digitized rigid motion. Motivated by these considerations, we study digitized rigid motions on the hexagonal grid. We establish a framework for studying digitized rigid motions in the hexagonal grid—previously proposed for the square grid and known as neighborhood motion maps. This allows us to study non-injective digitized rigid motions on the hexagonal grid and to compare the loss of information between digitized rigid motions defined on the two grids.

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