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 placement of Q that makes the centroids of Q and P coincide realizes an overlap of at least 9/25 of the maximum possible overlap. As an upper bound, we show an example where the overlap in this placement is 4/9 of the maximum possible overlap.
This work was supported by ESPRIT Basic Research Action No. 7141 (project ALCOM II: Algorithms and Complexity). M.d.B. and O.S. were supported by the Netherlands' Organisation for Scientific Research (NWO). O.S. also acknowledges partial support by Pohang University of Science and Technology Grant P96005, 1996.
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de Berg, M., Devillers, O., van Kreveld, M., Schwarzkopf, O., Teillaud, M. (1996). Computing the maximum overlap of two convex polygons under translations. In: Asano, T., Igarashi, Y., Nagamochi, H., Miyano, S., Suri, S. (eds) Algorithms and Computation. ISAAC 1996. Lecture Notes in Computer Science, vol 1178. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0009488
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DOI: https://doi.org/10.1007/BFb0009488
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