Laguerre Voronoi Diagram as a Model for Generating the Tessellation Patterns on the Sphere


We propose a model for generating tessellation patterns on the sphere using the spherical Laguerre Voronoi diagram which satisfies the real-world assumptions. The generator pushing model is presented to generate the tessellation dynamically. The simulations were done for the different distribution of spherical circles on the sphere, and the results show the tendency of the distribution of resulting spherical circles.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, vol. 501. Wiley, Amsterdam (2009)

    Google Scholar 

  2. 2.

    Aurenhammer, F., Klein, R., Lee, D.T.: Voronoi Diagrams and Delaunay Triangulations. World Scientific Publishing Co Inc, Singapore (2013)

    Google Scholar 

  3. 3.

    Mach, P., Koehl, P.: Capturing protein sequencestructure specificity using computational sequence design. Protein. Struct. Funct. Bioinf. 81(9), 1556–1570 (2013)

    Article  Google Scholar 

  4. 4.

    Telley, H., Liebling, T.M., Mocellin, A.: The Laguerre model of grain growth in two dimensions I. Cellular structures viewed as dynamical Laguerre tessellations. Philos. Magn. B 73(3), 395–408 (1996)

    Article  Google Scholar 

  5. 5.

    Telley, H., Liebling, T.M., Mocellin, A.: The Laguerre model of grain growth in two dimensions II. Examples of coarsening simulations. Philos. Magn. B 73(3), 409–427 (1996)

    Article  Google Scholar 

  6. 6.

    Aurenhammer, F.: Power diagram: properties, algorithms, and applications. SIAM J. Comput. 16, 78–96 (1987)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Imai, H., Iri, M., Murota, K.: Voronoi diagram in the Laguerre geometry and its applications. SIAM J. Comput. 14, 93–105 (1985)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Sugihara, K.: Three-dimensional convex hull as a fruitful source of diagrams. Theor. Comput. Sci. 235(2), 325–337 (2000)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chaidee, S., Sugihara, K.: Recognition of the spherical Laguerre Voronoi diagram. arXiv:1705.03911 (2017)

  10. 10.

    Chaidee, S., Sugihara, K.: Fitting spherical laguerre voronoi diagrams to real world tessellations using planar photographic images. In: Akiyama, J. (ed.) Discrete and Computational Geometry and Graphs (LNCS 9943), pp. 73–84. Springer, Berlin (2016)

    Google Scholar 

  11. 11.

    Chaidee, S., Sugihara, K.: Spherical Laguerre Voronoi diagram approximation to tessellations without generators, graphical models. (2017)

  12. 12.

    Pushpakumara, D.K.N.G.: Foral and fruit morphology and phenology of Artocarpus heterophyllus Lam. (Moraceae). Sri. Lankan J. Agric. Sci. 43, 82–196 (2006)

    Google Scholar 

  13. 13.

    Underhill, S.J.R., Critchley, C.: The physiology and anatomy of lychee (Litchi chinensis Sonn.) pericarp during fruit development. J. Horticult. Sci. 67(4), 437–444 (1992)

    Article  Google Scholar 

  14. 14.

    Roth, I.: Fruits of angiosperms-Encyclopedia of Plant Anatomy. Gebr. Borntraeger (1977)

  15. 15.

    Reeve, R.M.: Fruit histogenesis in Rubus strigosus. I. Outer epidermis, parenchyma, and receptacle. Am. J. Bot. 41(2), 152–160 (1954)

    Article  Google Scholar 

  16. 16.

    Sugihara, K.: Laguerre Voronoi diagram on the sphere. J. Geom. Graph. 6(1), 69–81 (2002)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Chaidee, S., Sugihara, K.: Approximation of fruit skin patterns using spherical Voronoi diagrams. Pattern Anal. Appl. 20(3), 783–795 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Deserno, M.: How to generate equidistributed points on the surface of a sphere. P.-If Polymerforshung (Ed.) (2004)

Download references


This research is supported by Chiang Mai University, Thailand.

Author information



Corresponding author

Correspondence to Supanut Chaidee.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chaidee, S., Sugihara, K. Laguerre Voronoi Diagram as a Model for Generating the Tessellation Patterns on the Sphere. Graphs and Combinatorics 36, 371–385 (2020).

Download citation


  • Spherical Laguerre Voronoi diagram
  • Spherical tessellation
  • Pattern formations