By Barutello V., Terracini S.
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Extra info for Action minimizing orbits in the n-body problem with simple choreography constraint
The star-shaped polygon formed by the vertices and points on the stack at any stage along with q is referred to as the current visibility region Vc (q). Let bd(vj , vk ) denote the counterclockwise boundary of P from vj to vk . We also assume that vertices (and the endpoints of constructed edges) on bd(v0 , vi−1 ), which are found to be visible from q by the procedure, are pushed on a stack in the order they are encountered, where v0 and vi−1 are at the bottom and top of the stack, respectively.
Let us discuss the correctness of the algorithm. , vertices of V (q) are in sorted angular order with respect to q. The algorithm maintains an invariant that the vertices and points on the stack at any stage are in sorted angular order with respect to q. When the algorithm terminates, the current visibility region Vc (q) is V (q). The algorithm scans the vertices of P starting from v0 in counterclockwise order and checks in Step 2 whether the current vertex vi is in the sorted angular order with the vertices and points on the stack.
Since the cost for computing Ki from Ki−1 for all i is proportional to the number of corner points removed from K i−1 , the time taken for computing K1 , K2 , . . , Kn−1 is O(n). 5 Notes and Comments 43 right tangents for all vertices is O(n), as the tangents move around the common intersection region of interior half-planes once in counterclockwise order. Hence, the overall time complexity of the algorithm is O(n). We summarize the result in the following theorem. 4 The kernel of an n-sided simple polygon P can be computed in O(n) time.
Action minimizing orbits in the n-body problem with simple choreography constraint by Barutello V., Terracini S.