Stacking and Bundling Two Convex Polygons

  • Hee-Kap Ahn
  • Otfried Cheong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3827)


Given two compact convex sets C1 and C2 in the plane, we consider the problem of finding a placement ϕC1 of C1 that minimizes the area of the convex hull of ϕC1 ∪ C2. We first consider the case where ϕC1 and C2 are allowed to intersect (as in “stacking” two flat objects in a convex box), and then add the restriction that their interior has to remain disjoint (as when “bundling” two convex objects together into a tight bundle). In both cases, we consider both the case where we are allowed to reorient C1, and where the orientation is fixed. In the case without reorientations, we achieve exact near-linear time algorithms, in the case with reorientations we compute a (1 + ε)-approximation in time  O(ε− 1/2 log n + ε− 3/2 log ε− 1/2), if two sets are convex polygons with n vertices in total.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating Extent Measures of Points. Journal of the ACM 51, 606–635 (2004)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Ahn, H.-K., Brass, P., Cheong, O., Na, H.-S., Shin, C.-S., Vigneron, A.: Inscribing an axially symmetric polygon and other approximation algorithms for planar convex sets. To appear in Comput. Geom. Theory Appl.Google Scholar
  3. 3.
    Ahn, H.-K., Cheong, O., Park, C.-D., Shin, C.-S., Vigneron, A.: Maximizing the overlap of two planar convex sets under rigid motions. In: Proc. 21st Annu. Symp. Comput. geometry, pp. 356–363 (2005)Google Scholar
  4. 4.
    Alt, H., Blömer, J., Godau, M., Wagener, H.: Approximation of convex polygons. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 703–716. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  5. 5.
    Alt, H., Fuchs, U., Rote, G., Weber, G.: Matching convex shapes with respect to the symmetric difference. Algorithmica 21, 89–103 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    de Berg, M., Cabello, S., Giannopoulos, P., Knauer, C., van Oostrum, R., Veltkamp, R.C.: Maximizing the area of overlap of two unions of disks under rigid motion. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 138–149. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    de Berg, M., Cheong, O., Devillers, O., van Kreveld, M., Teillaud, M.: Computing the maximum overlap of two convex polygons under translations. Theo. Comp. Sci. 31, 613–628 (1998)MATHGoogle Scholar
  8. 8.
    Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory 10, 227–236 (1974); Erratum in J. Approx. Theory 26, 192–193 (1979) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)MATHGoogle Scholar
  10. 10.
    Milenkovic, V.J.: Rotational polygon containment and minimum enclosure. In: Proc. 14th Annu. Symp. Comput. geometry, pp. 1–8 (1998)Google Scholar
  11. 11.
    Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Computer Vision and Image Understanding: CVIU 64(1), 53–61 (1996)CrossRefGoogle Scholar
  12. 12.
    Sugihara, K., Sawai, M., Sano, H., Kim, D.-S., Kim, D.: Disk Packing for the Estimation of the Size of a Wire Bundle. Japan Journal of Industrial and Applied Mathematics 21(3), 259–278 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yaglom, I.M., Boltyanskii, V.G.: Convex figures. Holt, Rinehart and Winston, New York (1961)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Otfried Cheong
    • 1
  1. 1.Division of Computer ScienceKorea Advanced Institute of Science and TechnologyDaejeonKorea

Personalised recommendations