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Finding a "complete" convex hull in less than O(n^2)

Here's an O(log h)-time algorithm that, given a convex hull with h vertices in sorted order and a query point, tests whether the query point lies on the hull. From the hull, compute a point in the interior by averaging three of its vertices. Call this point the origin. Partition the plane into wedges bounded by rays from the origin through hull vertices. Use binary search with an orientation test to determine which wedge the query point belongs to. Test whether it lies on that wedge's hull segment.

If you've already implemented the Graham scan, then it is possible to retain collinear points just by changing the test, but you need to determine angles relative to an origin point that is not collinear with two other hull points. Such a point can be obtained by taking the mean of three input points that are not collinear.

Categories : Algorithm

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