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The Number of Cylindrical Shells

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Given a set P of n points in three dimensions, a cylindrical shell (or zone cylinder) is formed by two circular cylinders with the same axis such that all points of P are between the two cylinders. We prove that the number of cylindrical shells enclosing P passing through combinatorially different subsets of P has size Ω(n 3) and O(n 4) (the previously known bound was O(n 5)). As a consequence, the minimum enclosing shell can be found in O(n 4) time.


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Correspondence to Olivier Devillers.

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Devillers, O. The Number of Cylindrical Shells. Discrete Comput Geom 30, 453–458 (2003). https://doi.org/10.1007/s00454-003-2818-8

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  • Cylindrical Shell
  • Circular Cylinder