On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing

  • Ranita BiswasEmail author
  • Partha Bhowmick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9647)


Although the concept of functional plane for naive plane is studied and reported in the literature in great detail, no similar study is yet found for naive sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is ‘mono-jumping’ but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane.


Naive sphere Quadraginta octants Symmetry groups Functional plane Projective geometry 


  1. 1.
    Aveneau, L., Andres, E., Mora, F.: Expressing discrete geometry using the conformal model. In: AGACSE 2012, La Rochelle, France, July 2012Google Scholar
  2. 2.
    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
  3. 3.
    Biswas, R., Bhowmick, P.: On finding spherical geodesic paths and circles in \(\mathbb{Z}^3\). In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) DGCI 2014. LNCS, vol. 8668, pp. 396–409. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Biswas, R., Bhowmick, P.: On different topological classes of spherical geodesic paths and circles in \({\mathbb{{Z}}}^3\). Theoret. Comput. Sci. 605, 146–163 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biswas, R., Bhowmick, P., Brimkov, V.E.: On the connectivity and smoothness of discrete spherical circles. In: Barneva, R.P., et al. (eds.) IWCIA 2015. LNCS, vol. 9448, pp. 86–100. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  6. 6.
    Biswas, R., Bhowmick, P.: From prima quadraginta octant to lattice sphere through primitive integer operations. Theoret. Comput. Sci. (2015, in press).
  7. 7.
    Bresenham, J.E.: Algorithm for for computer control of a digital plotter. IBM Syst. J. 4(1), 25–30 (1965)CrossRefGoogle Scholar
  8. 8.
    Brimkov, V.E., Barneva, R.P.: Graceful planes and thin tunnel-free meshes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 53–64. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Brimkov, V.E., Barneva, R.P.: Graceful planes and lines. Theoret. Comput. Sci. 283(1), 151–170 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brimkov, V.E., Barneva, R.P.: Connectivity of discrete planes. Theoret. Comput. Sci. 319(1–3), 203–227 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brimkov, V.E., Barneva, R.P.: Plane digitization and related combinatorial problems. Discrete Appl. Math. 147(2–3), 169–186 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity–a review. Discrete Appl. Math. 155(4), 468–495 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Graph. Models Image Process. 57(6), 453–461 (1995)CrossRefGoogle Scholar
  14. 14.
    Cohen-Or, D., Kaufman, A.: 3D line voxelization and connectivity control. IEEE Comput. Graph. Appl. 17(6), 80–87 (1997)CrossRefGoogle Scholar
  15. 15.
    Gouraud, H.: Continuous shading of curved surfaces. IEEE Trans. Comput. 20(6), 623–629 (1971)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kaufman, A.: Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes. In: SIGGRAPH 1987, pp. 171–179 (1987)Google Scholar
  17. 17.
    Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, San Francisco (2004)zbMATHGoogle Scholar
  18. 18.
    Mukhopadhyay, J., Das, P.P., Chattopadhyay, S., Bhowmick, P., Chatterji, B.N.: Digital Geometry in Image Processing. Chapman and Hall/CRC, Boca Ration (2013)zbMATHGoogle Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyKharagpurIndia

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