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Layer the sphere

For accurate and additive voxelation by integer operation

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Abstract

Voxelation today is not only limited to discretization and representation of 3D objects, but has also been gaining tremendous importance in rapid prototyping through 3D printing. In this paper, we introduce a novel technique for discretization of a sphere in the integer space, which gives rise to a set of mathematically accurate, 3D-printable physical voxels up to the desired level of precision. The proposed technique is based on an interesting correspondence between the voxel set forming a discrete sphere and certain easy-to-compute integer intervals defined by voxel position and sphere dimension. It gives us several algorithmic leverages, such as computational sufficiency with simple integer operations and amenability to layer-by-layer additive fabrication. Theoretical analysis, prototype characteristics, and experimental results demonstrate its efficiency, versatility, and further prospects.

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References

  1. Andres, E.: Discrete circles, rings and spheres. Comput. Gr. 18(5), 695–706 (1994)

    Article  Google Scholar 

  2. Andres, E., Jacob, M.: The discrete analytical hyperspheres. IEEE TVCG 3(1), 75–86 (1997)

    Google Scholar 

  3. Augustin, C., Hungerbach, W.: Production of hollow spheres (HS) and hollow sphere structures (HSS). Mater. Lett. 63(1314), 1109–1112 (2009)

    Article  Google Scholar 

  4. Bera, S., Bhowmick, P., Bhattacharya, B.B.: A digital-geometric algorithm for generating a complete spherical surface in \({\mathbb{{Z}}}^3\). In: Proceedings of ICAA’14, LNCS, vol. 8321, pp. 49–61 (2014)

  5. Bresenham, J.E.: A linear algorithm for incremental digital display of circular arcs. CACM 20(2), 100–106 (1977)

    Article  MATH  Google Scholar 

  6. Brimkov, V.E., Barneva, R.P.: On the polyhedral complexity of the integer points in a hyperball. TCS 406(1–2), 24–30 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chamizo, F., Cristobal, E.: The sphere problem and the \(L\)-functions. Acta Mathematica Hungarica 135(1–2), 97–115 (2012)

  8. Chandru, V., Manohar, S., Prakash, C.E.: Voxel-based modeling for layered manufacturing. IEEE Comput. Gr. Appl. 15(6), 42–47 (1995)

    Article  Google Scholar 

  9. Cochran, J.K.: Ceramic hollow spheres and their applications. Curr. Opin. Solid State Mater. Sci. 3(5), 474–479 (1998)

    Article  Google Scholar 

  10. Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Gr. Models Image Process. 57(6), 453–461 (1995)

    Article  Google Scholar 

  11. Coxeter, H.S.M.: Regular Polytopes. Dover Publications, New York (1973)

  12. Desimone, J.M., Ermoshkin, A., Samulski, E.T.: Method and apparatus for three-dimensional fabrication. US Patent 20140361463 (2014)

  13. Fiorio, C., Jamet, D., Toutant, J.L.: Discrete circles: An arithmetical approach with non-constant thickness. In: Proceedings of Vision Geometry XIV, Electronic Imaging, SPIE, vol. 6066, p. 60660C (2006)

  14. Fiorio, C., Toutant, J.L.: Arithmetic discrete hyperspheres and separatingness. In: Proceedings of DGCI’06, pp. 425–436 (2006)

  15. Foley, J.D., Dam, A.V., Feiner, S.K., Hughes, J.F.: Computer Graphics: Principles and Practice. Addison-Wesley, New York (1993)

    Google Scholar 

  16. Ghahramani, M., Garibov, A., Agayev, T.: Production and quality control of radioactive yttrium microspheres for medical applications. Appl. Radiat. Isot. 85, 87–91 (2014)

    Article  Google Scholar 

  17. Guo, L., Dong, X., Cui, X., Cui, F., Shi, J.: Morphology and dispersivity modulation of hollow microporous spheres synthesized by a hard template route. Mater. Lett. 63(1314), 1141–1143 (2009)

    Article  Google Scholar 

  18. Hearn, E.: Chapter 9: Thin cylinders and shells. In: Mechanics of Materials 1, 3 edn. Butterworth-Heinemann, Oxford (1997)

  19. Hiller, J., Lipson, H.: Design and analysis of digital materials for physical 3D voxel printing. Rapid Prototyp. J. 15(2), 137–149 (2009)

    Article  Google Scholar 

  20. Hiller, J., Lipson, H.: Tunable digital material properties for 3D voxel printers. Rapid Prototyp. J. 16(4), 241–247 (2010)

    Article  Google Scholar 

  21. Hong, J.Y., Way, D.L., Shih, Z.C., Tai, W.K., Chang, C.C.: Inner engraving for the creation of a balanced lego sculpture. Vis. Comput. 1–10 (2015). doi:10.1007/s0037101510724

  22. Hoque, M.E. (ed.): Advanced Applications of Rapid Prototyping Technology in Modern Engg. InTech (2011)

  23. Jee, H.J., Sachs, E.: A visual simulation technique for 3D printing. Adv. Eng. Softw. 31(2), 97–106 (2000)

    Article  Google Scholar 

  24. Kamrani, A.K., Nasr, E.A.: Engineering Design and Rapid Prototyping. Springer, Boston (2009)

    Google Scholar 

  25. Kawashita, M., et al.: Preparation of ceramic microspheres for in situ radiotherapy of deep-seated cancer. Biomaterials 24(17), 2955–2963 (2003)

    Article  Google Scholar 

  26. Kim, O.: Rapid prototyping of electrically small spherical wire antennas. IEEE Trans. Antennas Propag. 62(7), 3839–3842 (2014)

    Article  Google Scholar 

  27. Klette, R.: Digital geometry: the birth of a new discipline. In: Davis, L.S. (ed.) Foundations of Image Understanding, pp. 33–71 (2001)

  28. Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)

    Google Scholar 

  29. Lipson, H., Pollack, J.B.: Automatic design and manufacture of robotic lifeforms. Nature 406, 974–978 (2000)

    Article  Google Scholar 

  30. Maehara, H.: On a sphere that passes through \(n\) lattice points. Eur. J. Comb. 31(2), 617–621 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Melchels, F.P., Domingos, M.A., Klein, T.J., Malda, J., Bartolo, P.J., Hutmacher, D.W.: Additive manufacturing of tissues and organs. Progr. Polym. Sci. 37(8), 1079–1104 (2012)

    Article  Google Scholar 

  32. Montani, C., Scopigno, R.: Graphics Gems (Chapter: Spheres-to-voxels conversion). pp. 327–334. Academic Press Professional Inc, San Diego (1990)

  33. Nanya, T., Yoshihara, H., Maekawa, T.: Reconstruction of complete 3D models by voxel integration. J. Adv. Mech. Desig. Sys. Manuf. 7, 362–376 (2013)

    Google Scholar 

  34. Pintus, R., Gobbetti, E., Cignoni, P., Scopigno, R.: Shape enhancement for rapid prototyping. Vis. Comput. 26, 831–840 (2010)

    Article  Google Scholar 

  35. Roget, B., Sitaraman, J.: Wall distance search algorithm using voxelized marching spheres. J. Comput. Phys. 241, 76–94 (2013)

    Article  Google Scholar 

  36. Medeiros e Sá, A., Rodriguez Echavarria, K., Arnold, D.: Dual joints for 3d-structures. Vis. Comput. 30, 1321–1331 (2014)

    Article  Google Scholar 

  37. Sene, F.F., Martinelli, J.R., Okuno, E.: Synthesis and characterization of phosphate glass microspheres for radiotherapy applications. J. Non-Cryst. Solids 354, 4887–4893 (2008)

    Article  Google Scholar 

  38. Steingart, R.C., Tzu-Wei, D.: Fabrication of non-homogeneous articles via additive manufacturing using 3D voxel-based models. US Patent 8509933 (2013)

  39. Toutant, J.L., Andres, E., Roussillon, T.: Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties. Discret. Appl. Math. 161, 2662–2677 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. Waag, U., Schneider, L., Uthman, P., Stephani, G.: Metallic hollow spheres: materials for the future. Metal Powder Rep. 55, 29–33 (2000)

    Article  Google Scholar 

  41. Zheng, M., Cao, J., Chang, X., Wang, J., Liu, J., Ma, X.: Preparation of oxide hollow spheres by colloidal carbon spheres. Mater. Lett. 60, 2991–2993 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Computer Science and Engineering Department, Indian Institute of Technology, Kharagpur, India

    Ranita Biswas & Partha Bhowmick

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  1. Ranita Biswas
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  2. Partha Bhowmick
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Correspondence to Partha Bhowmick.

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Biswas, R., Bhowmick, P. Layer the sphere. Vis Comput 31, 787–797 (2015). https://doi.org/10.1007/s00371-015-1101-3

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  • DOI: https://doi.org/10.1007/s00371-015-1101-3

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