Vietnam Journal of Mathematics

, Volume 46, Issue 4, pp 883–913 | Cite as

Minimal Surface Convex Hulls of Spheres

  • Josef Kallrath
  • Markus M. FreyEmail author


We present and solve a new computational geometry optimization problem. Spheres with given radii should be arranged such that (a) they do not overlap and (b) the surface area of the boundary of the convex hull enclosing the spheres is minimized. An additional constraint could be to fit the spheres into a specified geometry, e.g., a rectangular solid. To tackle the problem, we derive closed non-convex NLP models for this sphere arrangement or sphere packing problem. For two spheres, we prove that the minimal area of the boundary of the convex hull is identical to the sum of the surface areas of the two spheres. For special configurations of spheres we provide theoretical insights and we compute analytically minimal-area configurations. Numerically, we have solved problems containing up to 200 spheres.


Packing problem Convex hull minimization Isoperimetric inequality Computational geometry Non-convex nonlinear programming Global optimization 

Mathematics Subject Classification (2010)

51 90 



We thank Julius Näumann (Student, TU Darmstadt, Darmstadt, Germany) for producing the graphics software for this paper. Thanks are directed to Dr. Jens Schulz (Lufthansa Systems GmbH, Berlin, Germany), Prof. Dr. Julia Kallrath (Hochschule Darmstadt, Darmstadt, Germany), Julius Näumann, Dr. Fritz Näumann (Consultant, Weisenheim am Berg, Germany), and Dr. Wolfgang Heinecke (Edenkoben, Germany) for their careful reading of and feedback on the manuscript.


  1. 1.
    Bisschop, J., Meeraus, A.: On the development of a general algebraic modeling system in a strategic planning environment. In: Goffin, J.-L., Rousseau, J. (eds.) Applications. Mathematical Programming Studies, vol. 20, pp. 1–29. Springer, Berlin (1982)Google Scholar
  2. 2.
    Boissonnat, J.-D., Cérézo, A., Devillers, O., Duquesne, J., Yvinec, M.: An algorithm for constructing the convex hull of a set of spheres in dimension d. Comput. Geom. 6, 123–130 (1996)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bussieck, M.R., Meeraus, A.: General algebraic modeling system (GAMS). In: Kallrath, J (ed.) Modeling Languages in Mathematical Optimization. Applied Optimization, vol. 88, pp. 137–157. Springer, Boston (2004)Google Scholar
  4. 4.
    Bussieck, M.R., Meeraus, A.: Algebraic modeling for IP and MIP (GAMS). Ann. Oper. Res. 149, 49–56 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kim, B., Kim, K.-J.: Computing the convex hull for a set of spheres on a GPU. In: Proceedings of the 11th ACM SIGGRAPH International Conference on Virtual-Reality Continuum and its Applications in Industry, pp. 345–345. Singapore (2012)Google Scholar
  6. 6.
    Kim, B., Kim, K.-J., Kim, Y.J.: Approximating the convex hull for a set of spheres. KIPS Trans. Comput. Commun. Syst. 3, 1–6 (2014)CrossRefGoogle Scholar
  7. 7.
    Chen, D.Z.: Sphere packing problem. In: Kao, M. (ed.) Encyclopedia of Algorithms. Springer, Boston (2001)Google Scholar
  8. 8.
    Conway, J., Sloane, N.J.A.: Sphere Packings, Lattices, and Groups. Grundlehren der mathematischen Wissenschaften, vol. 290. Springer, New York (1999)CrossRefGoogle Scholar
  9. 9.
    Corporation, G.D.: General Algebraic Modeling System (GAMS) Release 24.9.2. Washington, DC (2017)Google Scholar
  10. 10.
    Costa, A., Hansen, P., Liberti, L.: On the impact of symmetry-breaking constraints on spatial branch-and-bound for circle packing in a square. Discret. Appl. Math. 161, 96–106 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Devillers, O., Golin, M.J.: Incremental algorithms for finding the convex hulls of circles and the lower envelopes of parabolas. Inf. Process. Lett. 56, 157–164 (1995)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hadwiger, H.: Die isoperimetrische Ungleichung im Raum. Elemente Math. 3, 25–38 (1948)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hifi, M., M’Hallah, R.: A literature review on circle and sphere packing problems: models and methodologies. Adv. Oper. Res. 2009, 150624 (2009)zbMATHGoogle Scholar
  14. 14.
    Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim. 43, 299–328 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett. 5, 453–466 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kallrath, J., Frey, M.T.: Packing circles into perimeter-minimizing convex hulls. J. Glob. Optim. (submitted revised version after 1st review) (2017)Google Scholar
  17. 17.
    Kallrath, J., Milone, E.F.: Eclipsing Binary Stars: Modeling and Analysis. Springer, New York (1999)CrossRefGoogle Scholar
  18. 18.
    Karavelas, M.I., Tzanaki, E.: Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes. In: Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry, pp. 397–406. ACM, New York (2011)Google Scholar
  19. 19.
    Li, S., Zhao, J., Lu, P., Xie, Y.: Maximum packing density of basic 3D objects. Chin. Sci. Bull. 55, 114–119 (2010)CrossRefGoogle Scholar
  20. 20.
    MacLean, K.J.M.: A Geometric Analysis of the Platonic Solids and other Semi-Regular Polyhedra. Kenneth James Michael MacLean, Ann Arbor (2006)Google Scholar
  21. 21.
    Markót, M.C.: Interval methods for verifying structural optimality of circle packing configurations in the unit square. J. Comput. Appl. Math. 199, 353–357 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Markót, M. C., Csendes, T.: A new verified optimization technique for the “packing circles in a unit square” problems. SIAM J. Optim. 16, 193–219 (2005)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Marshall, G.W., Hudson, T.S.: Dense binary sphere packings. Contrib. Algebra Geom. 51, 337–344 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Martello, S., Pisinger, D., Vigo, D.: The three-dimensional bin packing problem. Oper. Res. 48, 256–267 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rappaport, D.: Minimum polygon transversals of line segments. Int. J. Comput. Geom. Appl. 5, 243–256 (1995)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rogers, C.A.: The packing of equal spheres. Proc. Lond. Math. Soc. s3–8, 609–620 (1958)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Nonconvex Optimization And Its Applications Series. Kluwer Academic Publishers, Dordrecht (2002)CrossRefGoogle Scholar
  28. 28.
    Wilson, R.E., Devinney, E.J.: Realization of accurate close-binary light curves: Application to MR. Cygni. Astrophys. J. 166, 605–619 (1971)CrossRefGoogle Scholar
  29. 29.
    Zong, C.: Strange Phenomena in Convex and Discrete Geometry. Springer, New York (1996)CrossRefGoogle Scholar
  30. 30.
    Zong, C.: Sphere Packings, vol. 1. Springer, New York (1999)Google Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Weisenheim am BergGermany
  2. 2.Department of AstronomyUniversity of FloridaGainesvilleUSA
  3. 3.BASF SEAdvanced Business Analytics, G-FSS/OAOLudwigshafenGermany
  4. 4.TUM-School of ManagementTechnische Universität MünchenMunichGermany

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