# Honeycomb Geometry: Rigid Motions on the Hexagonal Grid

## Abstract

Euclidean rotations in \(\mathbb {R}^2\) are bijective and isometric maps, but they generally lose these properties when digitized in discrete spaces. In particular, the topological and geometric defects of digitized rigid motions on the square grid have been studied. This 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 facts, 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.

## References

- 1.Nouvel, B., Rémila, E.: Configurations induced by discrete rotations: periodicity and quasi-periodicity properties. Discrete Appl. Math.
**147**, 325–343 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Kong, T., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process.
**48**, 357–393 (1989)CrossRefGoogle Scholar - 3.Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
- 4.Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process.
**46**, 141–161 (1989)CrossRefGoogle Scholar - 5.Middleton, L., Sivaswamy, J.: Hexagonal Image Processing: A Practical Approach. Advances in Pattern Recognition. Springer, Berlin (2005)zbMATHGoogle Scholar
- 6.Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar
- 7.Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. Image Process.
**4**, 1213–1222 (1995)CrossRefGoogle Scholar - 8.Pluta, K., Romon, P., Kenmochi, Y., Passat, N.: Bijective digitized rigid motions on subsets of the plane. J. Math. Imaging Vis.
**59**, 84–105 (2017)MathSciNetCrossRefGoogle Scholar - 9.Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Topology-preserving conditions for 2D digital images under rigid transformations. J. Math. Imaging Vis.
**49**, 418–433 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Nouvel, B., Rémila, É.: On colorations induced by discrete rotations. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 174–183. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-39966-7_16 CrossRefGoogle Scholar
- 11.Gilder, J.: Integer-sided triangles with an angle of 60\(^\circ \). Math. Gaz.
**66**, 261–266 (1982)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Gordon, R.A.: Properties of Eisenstein triples. Math. Mag.
**85**, 12–25 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 13.Berthé, V., Nouvel, B.: Discrete rotations and symbolic dynamics. Theoret. Comput. Sci.
**380**, 276–285 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Thibault, Y.: Rotations in 2D and 3D discrete spaces. Ph.D. thesis, Université Paris-Est (2010)Google Scholar