Algorithmica

, Volume 62, Issue 1–2, pp 464–479 | Cite as

Aligning Two Convex Figures to Minimize Area or Perimeter

Article

Abstract

Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕP of P that minimizes the convex hull of ϕPQ. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕP and Q to intersect or we restrict their interiors to remain disjoint; and we either allow reorienting P or require its orientation to be fixed. In the case without reorientations, we achieve exact near-linear time algorithms for all versions of the problem. In the case with reorientations, we compute a (1+ε)-approximation in time O(ε−1/2log n+ε−3/2log a(1/ε)) if the two sets are convex polygons with n vertices in total, where a∈{0,1,2} depending on the version of the problem.

Keywords

Computational geometry Optimization Convex hull Area Perimeter 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51, 606–635 (2004) CrossRefMATHMathSciNetGoogle 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. Comput. Geom., Theory Appl. 33, 152–164 (2006) MATHMathSciNetGoogle 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. Comput. Geom., Theory Appl. 37, 3–15 (2007) MATHMathSciNetGoogle Scholar
  4. 4.
    Ahn, H.-K., Brass, P., Shin, C.-S.: Maximum overlap and minimum convex hull of two convex polyhedra under translations. Comput. Geom., Theory Appl. 40, 171–177 (2008) MATHMathSciNetGoogle Scholar
  5. 5.
    Alt, H., Hurtado, F.: Packing convex polygons into rectangular boxes. In: Proc. Japanese Conference on Discrete and Computational Geometry 2000. LNCS, vol. 2098, pp. 67–80. Springer, Berlin (2001) CrossRefGoogle Scholar
  6. 6.
    Alt, H., Blömer, J., Godau, M., Wagener, H.: Approximation of convex polygons. In: Proc. 17th International Colloquium on Automata, Languages and Programming. LNCS, vol. 443, pp. 703–716. Springer, Berlin (1990) CrossRefGoogle Scholar
  7. 7.
    Alt, H., Fuchs, U., Rote, G., Weber, G.: Matching convex shapes with respect to the symmetric difference. Algorithmica 21, 89–103 (1998) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    de Berg, M., Cheong, O., Devillers, O., van Kreveld, M., Teillaud, M.: Computing the maximum overlap of two convex polygons under translations. Theory Comput. Syst. 31, 613–628 (1998) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    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. Int. J. Comput. Geom. Appl. 19, 533–556 (2009) CrossRefMATHGoogle Scholar
  10. 10.
    do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976) MATHGoogle Scholar
  11. 11.
    Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10, 227–236 (1974); Erratum in J. Approx. Theory 26, 192–193 (1979) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Egeblad, J., Nielsen, B.K., Brazil, M.: Translational packing of arbitrary polytopes. Comput. Geom., Theory Appl. 42, 269–288 (2009) MATHMathSciNetGoogle Scholar
  13. 13.
    Lee, H.C., Woo, T.C.: Determining in linear time the minimum area convex hull of two polygons. IIE Trans. 20, 338–345 (1988) CrossRefGoogle Scholar
  14. 14.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, Berlin (2002) MATHGoogle Scholar
  15. 15.
    Milenkovic, V.J.: Translational polygon containment and minimum enclosure using linear programming based restriction. In: Proc. 28th Annual ACM Symposium on Theory of Computation, pp. 109–118 (1996) Google Scholar
  16. 16.
    Milenkovic, V.J.: Rotational polygon containment and minimum enclosure. In: Proc. 14th Annual ACM Symposium on Computational Geometry, pp. 1–8 (1998) Google Scholar
  17. 17.
    Milenkovic, V.J.: Rotational polygon containment and minimum enclosure using only robust 2D constructions. Comput. Geom., Theory Appl. 13, 3–19 (1999) MATHMathSciNetGoogle Scholar
  18. 18.
    Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Comput. Vis. Image Underst. 64, 53–61 (1996) CrossRefGoogle Scholar
  19. 19.
    Sugihara, K., Sawai, M., Sano, H., Kim, D.-S., Kim, D.: Disk packing for the estimation of the size of a wire bundle. Jpn. J. Ind. Appl. Math. 21, 259–278 (2004) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Tang, K., Wang, C.C.L., Chen, D.Z.: Minimum area convex packing of two convex polygons. Int. J. Comput. Geom. Appl. 16, 41–74 (2006) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Yaglom, I.M., Boltyanskii, V.G.: Convex Figures. Holt, Rinehart and Winston, New York (1961) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Dept. of Computer Science and EngineeringPOSTECHPohangKorea
  2. 2.Dept. of Computer ScienceKAISTDaejeonKorea

Personalised recommendations