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Antipodal pair is pair of vertices x,y such that we can draw parallel tangents to convex-hull H through vertex x and y without intersecting H.

I found many algorithms to find such pairs, but i am not able to derive the upper bound on number of possible pairs.

Can somebody give the upper bound for convex-hull of n numbers and prove it?

CodeLover
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  • See Computational Geometry by Preparata & Shamos, theorem 4.18. As you rotate a tangent line around the polygon, it touches every vertex in turn (N moves); at the same time, the antipodal vertex also advances on the outline (N moves, there is no backtracking). There are exactly N pairs, when no edges are parallel. With parallel edges (at most N/2 pairs), the number of antipodal pairs does not exceed 3N/2. –  Sep 25 '14 at 21:19

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See Computational Geometry by Preparata & Shamos, theorem 4.18.

As you rotate a tangent line around the polygon, it touches every vertex in turn (N moves); at the same time, the antipodal vertex also advances on the outline (N moves, there is no backtracking).

When the are no parallel edges, there are exactly N pairs (no move on one side coincides with a move on the other). When there are parallel edges, an extra pair is possible and the total number is N + P, where P is the number of parallel edge pairs, at most N/2.

  • I am not able to find the book. can you please mention the proof of this that will be so helpfull. thanks – CodeLover Sep 26 '14 at 04:11
  • No, the text is too long. But the principle is obvious. –  Sep 26 '14 at 07:31
  • please upload pic or that text, you don't need to type. – CodeLover Sep 26 '14 at 08:05
  • I claim that it suffices to enumerate the pairs by means of cases 2 and 3 only (cases 1 bring no additional pair). Cases 2 gives two pairs and cases 4 gives four of them. But by removing duplicates you get 1 and 2 pairs respectively. Total `(N-P)+2P = N+P`. –  Sep 26 '14 at 08:30