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

Aligning Two Convex Figures to Minimize Area or Perimeter

  • Hee-Kap Ahn
  • Otfried Cheong


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 PQ. 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.


Computational geometry Optimization Convex hull Area Perimeter 


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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

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