Computing the Maximum Overlap of Two Convex Polygons Under Translations.

Abstract : Let P be a convex polygon in the plane with n vertices and let Q be a convex polygon with m vertices. We prove that the maximum number of combinatorially distinct placements of Q with respect to P under translations is O(n^2+m^2+min(nm^2+n^2m)), and we give an example showing that this bound is tight in the worst case. Second, we present an O((n+m)log(n+m)) algorithm for determining a translation of Q that maximizes the area of overlap of P and Q. We also prove that the position which translates the centroid of Q on the centroid of P always realizes an overlap of 9/25 of the maximum overlap and that this overlap may be as small as 4/9 of the maximum.
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https://hal.inria.fr/inria-00413175
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Mark De Berg, Olivier Devillers, Marc Van Kreveld, Otfried Schwarzkopf, Monique Teillaud. Computing the Maximum Overlap of Two Convex Polygons Under Translations.. Theory of Computing Systems, Springer Verlag, 1998, 31, pp.613-628. ⟨10.1007/PL00005845⟩. ⟨inria-00413175⟩

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