Skip to main content
SpringerLink
Log in

On the Connectivity and Smoothness of Discrete Spherical Circles

  • Conference paper
  • First Online:
Combinatorial Image Analysis (IWCIA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 9448))

Included in the following conference series:

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.

V.E. Brimkov—On leave from the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The meaning of thickness of some discretization is discussed in Sect. 2.

  2. 2.

    For formal definitions and details on tunnels and gaps in discrete objects, we refer to [14]. In what follows, gap-free means 0-gap-free, since a 0-gap-free surface is also 1- and 2-gap-free.

  3. 3.

    The term offset is equivalent to thickness in the context of our work.

  4. 4.

    While the meaning of smoothness of discrete curve is perhaps intuitively clear, this will be formally defined and discussed in Sect. 3.2.

References

  1. Andres, E., Jacob, M.: The discrete analytical hyperspheres. IEEE Trans. Visual Comput. Graphics 3(1), 75–86 (1997)

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Anton, F.: Voronoi diagrams of semi-algebraic sets. Ph.D. thesis, University of British Columbia, Vancouver, British Columbia, Canada (2004)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  5. Arrondo, E., Sendra, J., Sendra, J.: Genus formula for generalized offset curves. J. Pure Appl. Algebr. 136(3), 199–209 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. Bresenham, J.E.: Algorithm for computer control of a digital plotter. IBM Syst. J. 4(1), 25–30 (1965)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  12. Brimkov, V.E., Barneva, R.P., Brimkov, B.: Connected distance-based rasterization of objects in arbitrary dimension. Graph. Models 73, 323–334 (2011)

    Article  Google Scholar 

  13. Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity–a review. Discrete Appl. Math. 155(4), 468–495 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Cohen-Or, D., Kaufman, A.: Fundamentals of surface voxelization. Graphics Model. Image Process. 57(6), 453–461 (1995)

    Article  Google Scholar 

  17. Cohen-Or, D., Kaufman, A.: 3D line voxelization and connectivity control. IEEE Comput. Graph. Appl. 17(6), 80–87 (1997)

    Article  Google Scholar 

  18. Cox, D., Little, J., OShea, D.: Using Algebraic Geometry. Springer, New York (2005)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Gouraud, H.: Continuous shading of curved surfaces. IEEE Trans. Comput. 20(6), 623–629 (1971)

    Article  MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  23. Kaufman, A.: Efficient algorithms for 3d scan-conversion of parametric curves, surfaces, and volumes. SIGGRAPH Comput. Graph. 21(4), 171–179 (1987)

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Montani, C., Scopigno, R.: Spheres-to-voxels conversion. In: Graphics Gems. Academic Press, pp. 327–334 (1990)

    Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Ranita Biswas & Partha Bhowmick

  2. Mathematics Department, SUNY Buffalo State, 1300 Elmwood Avenue, Buffalo, NY, 14222, USA

    Valentin E. Brimkov

Authors
  1. Ranita Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Partha Bhowmick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valentin E. Brimkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ranita Biswas .

Editor information

Editors and Affiliations

  1. SUNY Fredonia, Fredonia, New York, USA

    Reneta P. Barneva

  2. Advanced Computing and Microelectronics, Indian Statistical Institute, Kolkata, Germany

    Bhargab B. Bhattacharya

  3. SUNY Buffalo State, Buffalo, New York, USA

    Valentin E. Brimkov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Biswas, R., Bhowmick, P., Brimkov, V.E. (2015). On the Connectivity and Smoothness of Discrete Spherical Circles. In: Barneva, R., Bhattacharya, B., Brimkov, V. (eds) Combinatorial Image Analysis. IWCIA 2015. Lecture Notes in Computer Science(), vol 9448. Springer, Cham. https://doi.org/10.1007/978-3-319-26145-4_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26145-4_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26144-7

  • Online ISBN: 978-3-319-26145-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation