Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation

Authors Honglin Zhu , Hyuk Jun Kweon

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Honglin Zhu
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Hyuk Jun Kweon
  • Massachusetts Institute of Technology, Cambridge, MA, USA


This paper is the result of the MIT SPUR 2022, a summer undergraduate research program organized by the MIT math department. The authors would like to thank the faculty advisors David Jerison and Ankur Moitra for their support and the math department for providing this research opportunity. We thank the anonymous referees for providing helpful comments that increased the quality of this paper.

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Honglin Zhu and Hyuk Jun Kweon. Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 61:1-61:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Let P be a convex polyhedron and Q be a convex polygon with n vertices in total in three-dimensional space. We present a deterministic algorithm that finds a translation vector v ∈ ℝ³ maximizing the overlap area |P ∩ (Q + v)| in O(n log² n) time. We then apply our algorithm to solve two related problems. We give an O(n log³ n) time algorithm that finds the maximum overlap area of three convex polygons with n vertices in total. We also give an O(n log² n) time algorithm that minimizes the symmetric difference of two convex polygons under scaling and translation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • computational geometry
  • shape matching
  • arrangement


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