# Packing circles into perimeter-minimizing convex hulls

- 24 Downloads

## Abstract

We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than \(10^{-4}\) using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most \(14\%\).

## Keywords

Global optimization Non-convex nonlinear programming Circular packing problem Convex hull Perimeter minimization Non-overlap constraints Computational geometry Isoperimetric inequality## Notes

### Acknowledgements

We thank Julius Näumann (Student, TU Darmstadt, Darmstadt, Germany), Prof. Dr. Julia Kallrath and Jan-Erik Justkowiak (Student, Hochschule Darmstadt, Darmstadt, Germany) and Dr. Fritz Näumann (Consultant, Weisenheim am Berg, Germany) for their careful reading of and feedback on the manuscript.

## Supplementary material

## References

- 1.Ahn, H.K., Cheong, O.: Aligning two convex figures to minimize area or perimeter. Algorithmica
**62**(1), 464–479 (2012)MathSciNetCrossRefGoogle Scholar - 2.Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T., Pankratov, A.: Optimal clustering of a pair of irregular objects. J. Global Optim.
**61**(3), 497–524 (2015)MathSciNetCrossRefGoogle Scholar - 3.Bezdek, K.: Lectures on Sphere Arrangements—the Discrete Geometric Side, vol. 32. Springer, New York (2013)CrossRefGoogle Scholar
- 4.Böröczky, J.R.: Finite Packing and Covering. Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
- 5.Castillo, I., Kampas, F., Pintér, J.: Solving circle packing problems by global optimization: numerical results and industrial application. Eur. J. Oper. Res.
**191**(3), 786–802 (2008)MathSciNetCrossRefGoogle Scholar - 6.Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups. A Series of Comprehensive Studies in Mathematics. Springer, Berlin (1988)CrossRefGoogle Scholar
- 7.Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res.
**44**(4), 145–159 (1990)MathSciNetCrossRefGoogle Scholar - 8.Hifi, M., M’Hallah, R.: Adaptive and restarting techniques-based algorithms for circular packing problems. Comput. Optim. Appl.
**39**, 17–35 (2008)MathSciNetCrossRefGoogle Scholar - 9.Hifi, M., Paschos, V., Zissimopoulos, V.: A simulated annealing approach for the circular cutting problem. Eur. J. Oper. Res.
**159**, 430–448 (2004)CrossRefGoogle Scholar - 10.Kallrath, J.: Combined strategic design and operative planning in the process industry. Comput. Chem. Eng.
**33**, 1983–1993 (2009)Google Scholar - 11.Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Global Optim.
**43**, 299–328 (2009)MathSciNetCrossRefGoogle Scholar - 12.Kallrath, J.: Polylithic modeling and solution approaches using algebraic modeling systems. Optim. Lett.
**5**, 453–466 (2011). https://doi.org/10.1007/s11590-011-0320-4 MathSciNetCrossRefzbMATHGoogle Scholar - 13.Kallrath, J.: Packing ellipsoids into volume-minimizing rectangular boxes. J. Global Optim.
**67**(1), 151–185 (2017). https://doi.org/10.1007/s10898-015-0348-6 MathSciNetCrossRefzbMATHGoogle Scholar - 14.Libeskind, S.: Euclidean and transformational geometry: a deductive inquiry. Jones & Bartlett Learning, LLC (2008). https://books.google.de/books?id=JiTse0Nm_-IC
- 15.Nash, E., Pir, A., Sottile, F., Ying, L.: Convex hull of two circles in \(r^3\). Tech. rep. Algebraic Geometry (2017)Google Scholar
- 16.Rappaport, D.: A convex hull algorithm for discs, an application. Comput. Geom. Theory Appl.
**1**(3), 171–187 (1992)MathSciNetCrossRefGoogle Scholar - 17.Stetsyuk, P., Romanova, T., Scheithauer, G.: On the global minimum in a balanced circular packing problem. Optim. Lett.
**10**, 1347–1360 (2016)MathSciNetCrossRefGoogle Scholar - 18.Stoyan, Y., Yaskov, G.: Packing congruent hyperspheres into a hypersphere. J. Global Optim.
**52**(4), 855–868 (2012)MathSciNetCrossRefGoogle Scholar - 19.Stoyan, Y., Yaskov, G.: Packing unequal circles into a strip of minimal length with a jump algorithm. Optim. Lett.
**8**(3), 949–970 (2014). https://doi.org/10.1007/s11590-013-0646-1 MathSciNetCrossRefzbMATHGoogle Scholar - 20.Thue, A.: Über die dichteste Zusammenstellung von kongruenten Kreisen in einer Ebene. J. Dybwad (1910). https://books.google.de/books?id=IUJyQwAACAAJ