Largest Similar Copies of Convex Polygons in Polygonal Domains

Authors Taekang Eom, Seungjun Lee, Hee-Kap Ahn

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Taekang Eom
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Seungjun Lee
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Hee-Kap Ahn
  • Department of Computer Science and Engineering, Graduate School of Artificial Intelligence, Pohang University of Science and Technology, South Korea

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Taekang Eom, Seungjun Lee, and Hee-Kap Ahn. Largest Similar Copies of Convex Polygons in Polygonal Domains. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 19:1-19:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Given a convex polygon with k vertices and a polygonal domain consisting of polygonal obstacles with n vertices in total in the plane, we study the optimization problem of finding a largest similar copy of the polygon that can be placed in the polygonal domain without intersecting the obstacles. We present an upper bound O(k²n²λ₄(k)) on the number of combinatorial changes occurred to the underlying structure during the rotation of the polygon, together with an O(k²n²λ₄(k)log n)-time deterministic algorithm for the problem. This improves upon the previously best known results by Chew and Kedem [SoCG89, CGTA93] and Sharir and Toledo [SoCG91, CGTA94] on the problem in more than 27 years. Our result also improves the time complexity of the high-clearance motion planning algorithm by Chew and Kedem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Polygon placement
  • Largest similar copy
  • Polygonal domain


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