Abstract
Given two compact convex sets P and Q in the plane, we consider the problem of finding a placement ϕ P of P that minimizes the convex hull of ϕ P∪Q. We study eight versions of the problem: we consider minimizing either the area or the perimeter of the convex hull; we either allow ϕ P 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51, 606–635 (2004)
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)
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)
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)
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)
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)
Alt, H., Fuchs, U., Rote, G., Weber, G.: Matching convex shapes with respect to the symmetric difference. Algorithmica 21, 89–103 (1998)
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)
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)
do Carmo, M.: Differential Geometry of Curves and Surfaces. Prentice-Hall, New York (1976)
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)
Egeblad, J., Nielsen, B.K., Brazil, M.: Translational packing of arbitrary polytopes. Comput. Geom., Theory Appl. 42, 269–288 (2009)
Lee, H.C., Woo, T.C.: Determining in linear time the minimum area convex hull of two polygons. IIE Trans. 20, 338–345 (1988)
Matoušek, J.: Lectures on Discrete Geometry. Springer, Berlin (2002)
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)
Milenkovic, V.J.: Rotational polygon containment and minimum enclosure. In: Proc. 14th Annual ACM Symposium on Computational Geometry, pp. 1–8 (1998)
Milenkovic, V.J.: Rotational polygon containment and minimum enclosure using only robust 2D constructions. Comput. Geom., Theory Appl. 13, 3–19 (1999)
Mount, D.M., Silverman, R., Wu, A.Y.: On the area of overlap of translated polygons. Comput. Vis. Image Underst. 64, 53–61 (1996)
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)
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)
Yaglom, I.M., Boltyanskii, V.G.: Convex Figures. Holt, Rinehart and Winston, New York (1961)
Author information
Authors and Affiliations
Corresponding author
Additional information
H.-K. Ahn was supported by the National Research Foundation of Korea Grant funded by the Korean Government (MEST) (NRF-2009-0067195). O. Cheong was supported by Mid-career Researcher Program through NRF grant funded by the MEST (No. R01-2008-000-11607-0).
Rights and permissions
About this article
Cite this article
Ahn, HK., Cheong, O. Aligning Two Convex Figures to Minimize Area or Perimeter. Algorithmica 62, 464–479 (2012). https://doi.org/10.1007/s00453-010-9466-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-010-9466-1