Advertisement

On the Connectivity and Smoothness of Discrete Spherical Circles

  • Ranita Biswas
  • Partha Bhowmick
  • Valentin E. Brimkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9448)

Abstract

A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness.

Keywords

3D discrete circle Discrete sphere Spherical circle  Digital geometry 

References

  1. 1.
    Andres, E., Jacob, M.: The discrete analytical hyperspheres. IEEE Trans. Visual Comput. Graphics 3(1), 75–86 (1997)CrossRefGoogle Scholar
  2. 2.
    Andres, E.: Discrete circles, rings and spheres. Comput. Graphics 18(5), 695–706 (1994)CrossRefGoogle Scholar
  3. 3.
    Anton, F.: Voronoi diagrams of semi-algebraic sets. Ph.D. thesis, University of British Columbia, Vancouver, British Columbia, Canada (2004)Google Scholar
  4. 4.
    Anton, F., Emiris, I.Z., Mourrain, B., Teillaud, M.: The offset to an algebraic curve and an application to conics. In: Gervasi, O., Gavrilova, M.L., Kumar, V., Laganá, A., Lee, H.P., Mun, Y., Taniar, D., Tan, C.J.K. (eds.) ICCSA 2005. LNCS, vol. 3480, pp. 683–696. Springer, Heidelberg (2005) CrossRefGoogle Scholar
  5. 5.
    Arrondo, E., Sendra, J., Sendra, J.: Genus formula for generalized offset curves. J. Pure Appl. Algebr. 136(3), 199–209 (1999)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Aveneau, L., Andres, E., Mora, F.: Expressing discrete geometry using the conformal model. In: AGACSE (2012). http://hal.archives-ouvertes.fr/hal-00865103
  7. 7.
    Aveneau, L., Fuchs, L., Andres, E.: Digital geometry from a geometric algebra perspective. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) DGCI 2014. LNCS, vol. 8668, pp. 358–369. Springer, Heidelberg (2014) Google Scholar
  8. 8.
    Biswas, R., Bhowmick, P.: On finding spherical geodesic paths and circles in \(\mathbb{Z}\) \(^\text{3 }\). In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) DGCI 2014. LNCS, vol. 8668, pp. 396–409. Springer, Heidelberg (2014) Google Scholar
  9. 9.
    Bresenham, J.E.: Algorithm for computer control of a digital plotter. IBM Syst. J. 4(1), 25–30 (1965)CrossRefGoogle Scholar
  10. 10.
    Brimkov, V.E., Barneva, R.P.: On the polyhedral complexity of the integer points in a hyperball. Theor. Comput. Sci. 406(1–2), 24–30 (2008)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Brimkov, V.E., Barneva, R.P., Brimkov, B.: Minimal offsets that guarantee maximal or minimal connectivity of digital curves in nD. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 337–349. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  12. 12.
    Brimkov, V.E., Barneva, R.P., Brimkov, B.: Connected distance-based rasterization of objects in arbitrary dimension. Graph. Models 73, 323–334 (2011)CrossRefGoogle Scholar
  13. 13.
    Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity–a review. Discrete Appl. Math. 155(4), 468–495 (2007)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Brimkov, V.: Formulas for the number of \((n-2)\)-gaps of binary objects in arbitrary dimension. Discrete Appl. Math. 157(3), 452–463 (2009)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Chamizo, F., Cristobal, E.: The sphere problem and the \(L\)-functions. Acta Math. Hungar. 135(1–2), 97–115 (2012)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Graphics Model. Image Process. 57(6), 453–461 (1995)CrossRefGoogle Scholar
  17. 17.
    Cohen-Or, D., Kaufman, A.: 3D line voxelization and connectivity control. IEEE Comput. Graph. Appl. 17(6), 80–87 (1997)CrossRefGoogle Scholar
  18. 18.
    Cox, D., Little, J., OShea, D.: Using Algebraic Geometry. Springer, New York (2005) MATHGoogle Scholar
  19. 19.
    Debled-Rennesson, I., Domenjoud, E., Jamet, D.: Arithmetic discrete parabolas. In: Bebis, G., Boyle, R., Parvin, B., Koracin, D., Remagnino, P., Nefian, A., Meenakshisundaram, G., Pascucci, V., et al. (eds.) ISVC 2006. LNCS, vol. 4292, pp. 480–489. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  20. 20.
    Fiorio, C., Jamet, D., Toutant, J.L.: Discrete circles: an arithmetical approach with non-constant thickness. In: Vision Geometry XIV, Electronic Imaging, SPIE, vol. 6066, pp. 60660C.1–60660C.12 (2006)Google Scholar
  21. 21.
    Gouraud, H.: Continuous shading of curved surfaces. IEEE Trans. Comput. 20(6), 623–629 (1971)MATHCrossRefGoogle Scholar
  22. 22.
    Hoffmann, C., Vermeer, P.: Eliminating extraneous solutions for the sparse resultant and the mixed volume. J. Symbolic Geom. Appl. 1(1), 47–66 (1991)MATHMathSciNetGoogle Scholar
  23. 23.
    Kaufman, A.: Efficient algorithms for 3d scan-conversion of parametric curves, surfaces, and volumes. SIGGRAPH Comput. Graph. 21(4), 171–179 (1987)CrossRefGoogle Scholar
  24. 24.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)Google Scholar
  25. 25.
    Maehara, H.: On a sphere that passes through \(n\) lattice points. European J. Combin. 31(2), 617–621 (2010)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Montani, C., Scopigno, R.: Spheres-to-voxels conversion. In: Graphics Gems. Academic Press, pp. 327–334 (1990)Google Scholar
  27. 27.
    Mukhopadhyay, J., Das, P.P., Chattopadhyay, S., Bhowmick, P., Chatterji, B.N.: Digital Geometry in Image Processing. Chapman & Hall/CRC, Boca Ration, UK (2013)MATHGoogle Scholar
  28. 28.
    Toutant, J.L., Andres, E., Roussillon, T.: Digital circles, spheres and hyperspheres: From morphological models to analytical characterizations and topological properties. Discrete Appl. Math. 161(16–17), 2662–2677 (2013)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ranita Biswas
    • 1
  • Partha Bhowmick
    • 1
  • Valentin E. Brimkov
    • 2
  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia
  2. 2.Mathematics DepartmentSUNY Buffalo StateBuffaloUSA

Personalised recommendations