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.
@InProceedings{zhu_et_al:LIPIcs.SoCG.2023.61, author = {Zhu, Honglin and Kweon, Hyuk Jun}, title = {{Maximum Overlap Area of a Convex Polyhedron and a Convex Polygon Under Translation}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {61:1--61:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.61}, URN = {urn:nbn:de:0030-drops-179116}, doi = {10.4230/LIPIcs.SoCG.2023.61}, annote = {Keywords: computational geometry, shape matching, arrangement} }
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