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Update graham_scan_algorithm.cpp

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geometry/graham_scan_algorithm.cpp
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/****************************************************************************** /******************************************************************************
* @file * @file
* @brief Implementation of the [Convex * @brief Implementation of the [Convex
*Hull](https://en.wikipedia.org/wiki/Convex_hull) implementation using [Graham * Hull](https://en.wikipedia.org/wiki/Convex_hull) implementation using [Graham
*Scan](https://en.wikipedia.org/wiki/Graham_scan) * Scan](https://en.wikipedia.org/wiki/Graham_scan)
* @details * @details
* In geometry, the convex hull or convex envelope or convex closure of a shape * 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 * 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 * 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 * 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 * 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 * 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. * 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. * are sorted, they form a simple closed path.
* The sorting criteria is to use the orientation to compare angles without * The sorting criteria is to use the orientation to compare angles without
* actually computing them (See the compare() function below) because * actually computing them (See the compare() function below) because
*computation of actual angles would be inefficient since trigonometric * computation of actual angles would be inefficient since trigonometric
*functions are not simple to evaluate. * functions are not simple to evaluate.
* *
* Accept or Reject Points * Accept or Reject Points
* Once we have the closed path, the next step is to traverse the path and * 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 * 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 * 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 * 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 * 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 * points (considering them in the same order) is not counterclockwise, we
*discard c, otherwise we keep it. * discard c, otherwise we keep it.
* *
* @author [Lajat Manekar](https://github.com/Lazeeez) * @author [Lajat Manekar](https://github.com/Lazeeez)
* *