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The Visual Computer

, Volume 35, Issue 6–8, pp 899–907 | Cite as

Acquiring periodic tilings of regular polygons from images

  • José Ezequiel Soto Sánchez
  • Asla Medeiros e Sá
  • Luiz Henrique de FigueiredoEmail author
Original Article
  • 121 Downloads

Abstract

We describe how we have acquired geometrical models of many periodic tilings of regular polygons from two large collections of images. These models are based on a simplification of the representation recently proposed by us that uses complex numbers. We also describe an algorithm for deciding when two representations give the same tiling, which was used to identify coincidences in these collections.

Keywords

Tilings Tessellations Geometrical models 

Notes

Acknowledgements

We thank Sá and Sá and Galebach for making their collections of tilings freely available at their websites. The first author is partially supported by a CNPq doctoral scholarship. The third author is partially supported by a CNPq research grant. This research was done in the Visgraf Computer Graphics laboratory at IMPA. Visgraf is supported by the funding agencies FINEP, CNPq, and FAPERJ, and also by gifts from IBM Brasil, Microsoft, NVIDIA, and other companies.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Bradley, G.H.: Algorithms for Hermite and Smith normal matrices and linear Diophantine equations. Math. Comput. 25, 897–907 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chavey, D.: Tilings by regular polygons. II. A catalog of tilings. Comput. Math. Appl. 17, 147–165 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Conway, J.H., Burgiel, H., Goodman-Strauss, C.: The Symmetries of Things. CRC Press (2008)Google Scholar
  4. 4.
    Galebach, B.: \(n\)-uniform tilings. http://probabilitysports.com/tilings.html
  5. 5.
    Grünbaum, B., Shephard, G.C.: Tilings by regular polygons. Math. Mag. 50, 227–247 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman, New York (1989)zbMATHGoogle Scholar
  7. 7.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Hilbert, D., Cohn-Vossen, S.: Geometry and the Imagination. Chelsea, New York (1952)zbMATHGoogle Scholar
  9. 9.
    Kaplan, C.S.: Islamic star patterns from polygons in contact. In: Proceedings of the Graphics Interface 2005, pp. 177–185 (2005)Google Scholar
  10. 10.
    Kaplan, C.S.: Introductory tiling theory for computer graphics. Synth. Lect. Comput. Gr. Anim. 4(1), 1–113 (2009)zbMATHGoogle Scholar
  11. 11.
    Kawarabayashi, K.I., Mohar, B.: Graph and map isomorphism and all polyhedral embeddings in linear time. In: STOC’08, pp. 471–480. ACM (2008)Google Scholar
  12. 12.
    Lenngren, N.: \(k\)-uniform tilings by regular polygons. Tech. Rep. U.U.D.M. project report 2009:23, Uppsala University (2009)Google Scholar
  13. 13.
    Liu, S., Ng, T., Sunkavalli, K., Do, M.N., Shechtman, E., Carr, N.: PatchMatch-based automatic lattice detection for near-regular textures. In: Proceedings of ICCV 2015, pp. 181–189 (2015)Google Scholar
  14. 14.
    Liu, Y., Hel-Or, H., Kaplan, C.S., Gool, L.J.V.: Computational symmetry in computer vision and computer graphics. Found. Trends Comput. Gr. Vis. 5(1–2), 1–195 (2010)zbMATHGoogle Scholar
  15. 15.
    McKay, B.D., Piperno, A.: Practical graph isomorphism, II. J. Symb. Comput. 60, 94–112 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nasri, A., Benslimane, R.: Parametric shape grammar formalism for Moorish geometric design analysis and generation. J. Comput. Cultural Herit. 10, 1–20 (2017)CrossRefGoogle Scholar
  17. 17.
    Soto Sánchez, J.E., Sá, A.M., de Figueiredo, L.H.: Periodic tilings of regular polygons. http://www.impa.br/~cheque/tiling/
  18. 18.
    Sá, A.M., Echavarria, K.R., Arnold, D.: Dual joints for 3d-structures. Vis. Comput. 30(12), 1321–1331 (2014)CrossRefGoogle Scholar
  19. 19.
    Sá, A.M., de Figueiredo, L.H., Soto Sánchez, J.E.: Synthesizing periodic tilings of regular polygons. In: Proceedings of SIBGRAPI 2018, pp. 17–24. IEEE Computer Press (2018)Google Scholar
  20. 20.
    Sá, R., Sá, A.M.: Sobre malhas arquimedianas. Editora Olhares, São Paulo (2017)Google Scholar
  21. 21.
    The On-Line Encyclopedia of Integer Sequences: A299780. https://oeis.org//A299780
  22. 22.
    Wikipedia: Euclidean tilings by convex regular polygons. https://en.wikipedia.org/wiki/Euclidean_tilings_by_convex_regular_polygons

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.FGV EMApRio de JaneiroBrazil

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