Abstract
There are various tessellations of the plane. There are three regular ones, each of them using a sole regular tile. The square grid is self-dual, and the two others, the hexagonal and triangular grids are duals of each other. There are eight semi-regular tessellations, they are based on more than one type of tiles. In this paper, we are interested to their dual tessellations. We show a general method to obtain coordinate system to address the tiles of these tessellations. The properties of the coordinate systems used to address the tiles are playing crucial roles. For some of those grids, including the tetrille tiling D(6, 4, 3, 4) (also called deltoidal trihexagonal tiling and it is the dual of the rhombihexadeltille, T(6, 4, 3, 4) tiling), the rhombille tiling, D(6, 3, 6, 3) (that is the dual of the hexadeltille T(6, 3, 6, 3), also known as trihexagonal tiling) and the kisquadrille tiling D(8, 8, 4) (it is also called tetrakis square tiling and it is the dual of the truncated quadrille tiling T(8, 8, 4) which is also known as Khalimsky grid) we give detailed descriptions. Moreover, we are also presenting formulae to compute the digital, i.e., path-based distance based on the length of a/the shortest path(s) through neighbor tiles for these specific grids.
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References
Borgefors, G.: A semiregular image grid. J. Vis. Commun. Image Repres. 1(2), 127–136 (1990)
Conway, J.H., Burgiel, H., Goodman-Strass, C.: The symmetries of things. AK Peters, (2008)
Grünbaum, B., Shephard, G.C.: Tilings by regular polygons. Math. Mag. 50(5), 227–247 (1977)
Her, I.: A symmetrical coordinate frame on the hexagonal grid for computer graphics and vision. ASME J. Mech. Des. 115(3), 447–449 (1993)
Her, I.: Geometric transformations on the hexagonal grid. IEEE Trans. Image Process. 4(9), 1213–1222 (1995)
Kirby, M., Umble, R.: Edge tessellations and stamp folding puzzles. Math. Mag. 84(4), 283–289 (2011)
Kovács, G., Nagy, B., Turgay, N.D.: Distance on the Cairo pattern. Pattern Recogn. Lett. 145, 141–146 (2021)
Kovács, G., Nagy, B., Vizvári, B.: On weighted distances on the Khalimsky grid. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) DGCI 2016. LNCS, vol. 9647, pp. 372–384. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32360-2_29
Kovács, G., Nagy, B., Vizvári, B.: Weighted distances on the trihexagonal grid. In: Kropatsch, W.G., Artner, N.M., Janusch, I. (eds.) DGCI 2017. LNCS, vol. 10502, pp. 82–93. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66272-5_8
Kovalevsky, V.A.: Geometry of locally finite spaces: Computer agreeable topology and algorithms for computer imagery. Bärbel Kovalevski, Berlin (2008)
Luczak, E., Rosenfeld, A.: Distance on a hexagonal grid. IEEE Trans. Comput. 5, 532–533 (1976)
Nagy, B., Abuhmaidan, K.: A continuous coordinate system for the plane by triangular symmetry. Symmetry 11(2), paper 191 (2019)
Nagy, B.: Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids. Annals Math. Artif. Intell. 75(1–2), 117–134 (2014). https://doi.org/10.1007/s10472-014-9404-z
Nagy, B.: Finding shortest path with neighbourhood sequences in triangular grids. In: International Symposium on Image and Signal Processing and Analysis conjunction with 23rd International Conference on Information Technology Interfaces 2001, pp. 55–60. IEEE, Pula (2001)
Nagy, B.: Generalized triangular grids in digital geometry. Acta Mathematica Academiae Paedagogicae NyÃregyháziensis 20, 63–78 (2004)
Nagy, B.: Weighted distances on a triangular grid. In: Barneva, R., Brimkov, V.E., Šlapal, J. (eds.) IWCIA 2014. LNCS, vol. 8466, pp. 37–50. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-07148-0_5
Räbinä, J., Kettunen, L., Mönkölä, S., Rossi, T.: Generalized wave propagation problems and discrete exterior calculus. ESAIM Math. Modell. Numer. Anal. 52(3), 1195–1218 (2018)
Radványi, A.G.: On the rectangular grid representation of general CNN networks. Int. J. Circuit Theory Appl. 30(2–3), 181–193 (2002)
Stephenson, J.: Ising model with antiferromagnetic next-nearest-neighbor coupling: spin correlations and disorder points. Phys. Rev. B 1(11), 4405–4409 (1970)
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Saadat, M., Nagy, B. (2021). Digital Geometry on the Dual of Some Semi-regular Tessellations. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2021. Lecture Notes in Computer Science(), vol 12708. Springer, Cham. https://doi.org/10.1007/978-3-030-76657-3_20
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