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geometry/graham_scan_algorithm.cpp
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/******************************************************************************
* @file
* @brief Implementation of the [Convex
*Hull](https://en.wikipedia.org/wiki/Convex_hull) implementation using [Graham
*Scan](https://en.wikipedia.org/wiki/Graham_scan)
* Hull](https://en.wikipedia.org/wiki/Convex_hull) implementation using [Graham
* Scan](https://en.wikipedia.org/wiki/Graham_scan)
* @details
* In geometry, the convex hull or convex envelope or convex closure of a shape
* is the smallest convex set that contains it. The convex hull may be defined
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* Euclidean space, or equivalently as the set of all convex combinations of
* points in the subset. For a bounded subset of the plane, the convex hull may
* be visualized as the shape enclosed by a rubber band stretched around the
*subset.
* subset.
*
* The worst case time complexity of Jarvis’s Algorithm is O(n^2). Using
* Graham’s scan algorithm, we can find Convex Hull in O(nLogn) time.
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* are sorted, they form a simple closed path.
* The sorting criteria is to use the orientation to compare angles without
* actually computing them (See the compare() function below) because
*computation of actual angles would be inefficient since trigonometric
*functions are not simple to evaluate.
* computation of actual angles would be inefficient since trigonometric
* functions are not simple to evaluate.
*
* Accept or Reject Points
* Once we have the closed path, the next step is to traverse the path and
* remove concave points on this path using orientation. The first two points in
* sorted array are always part of Convex Hull. For remaining points, we keep
* track of recent three points, and find the angle formed by them. Let the
*three points be prev(p), curr(c) and next(n). If the orientation of these
*points (considering them in the same order) is not counterclockwise, we
*discard c, otherwise we keep it.
* three points be prev(p), curr(c) and next(n). If the orientation of these
* points (considering them in the same order) is not counterclockwise, we
* discard c, otherwise we keep it.
*
* @author [Lajat Manekar](https://github.com/Lazeeez)
*