#### Graham Scan Functions

```/******************************************************************************
* @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)
* @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
* either as the intersection of all convex sets containing a given subset of a
* 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.
*
* 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.
*
* ### Implementation
*
* Sort points
* We first find the bottom-most point. The idea is to pre-process
* points be sorting them with respect to the bottom-most point. Once the points
* 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.
*
* 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 orientation of these points
* (considering them in same order) is not counterclockwise, we discard c,
* otherwise we keep it.
*
* @author [Lajat Manekar](https://github.com/Lazeeez)
*
*******************************************************************************/
#include <algorithm>  /// for std::swap
#include <cstdlib>    /// for mathematics and datatype conversion
#include <iostream>   /// for IO operations
#include <stack>      /// for std::stack
#include <vector>     /// for std::vector

/******************************************************************************
* @namespace geometry
* @brief geometric algorithms
*******************************************************************************/
namespace geometry {

/******************************************************************************
* @namespace graham scan
* @brief convex hull algorithm
*******************************************************************************/
namespace grahamscan {

/******************************************************************************
* @struct Point
* @brief for X and Y co-ordinates of the co-ordinate.
*******************************************************************************/
struct Point {
int x, y;
};

// A global point needed for sorting points with reference
// to  the first point Used in compare function of qsort()

Point p0;

/******************************************************************************
* @brief A utility function to find next to top in a stack.
* @param S Stack to be used for the process.
* @returns @param Point Co-ordinates of the Point <int, int>
*******************************************************************************/
Point nextToTop(std::stack<Point> *S) {
Point p = S->top();
S->pop();
Point res = S->top();
S->push(p);
return res;
}

/******************************************************************************
* @brief A utility function to return square of distance between p1 and p2.
* @param p1 Co-ordinates of Point 1 <int, int>.
* @param p2 Co-ordinates of Point 2 <int, int>.
* @returns @param int distance between p1 and p2.
*******************************************************************************/
int distSq(Point p1, Point p2) {
return (p1.x - p2.x) * (p1.x - p2.x) + (p1.y - p2.y) * (p1.y - p2.y);
}

/******************************************************************************
* @brief To find orientation of ordered triplet (p, q, r).
* @param p Co-ordinates of Point p <int, int>.
* @param q Co-ordinates of Point q <int, int>.
* @param r Co-ordinates of Point r <int, int>.
* @returns @param int 0 --> p, q and r are collinear, 1 --> Clockwise,
* 2 --> Counterclockwise
*******************************************************************************/
int orientation(Point p, Point q, Point r) {
int val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y);

if (val == 0) {
return 0;  // collinear
}
return (val > 0) ? 1 : 2;  // clock or counter-clock wise
}

/******************************************************************************
* @brief A function used by library function qsort() to sort an array of
* points with respect to the first point
* @param vp1 Co-ordinates of Point 1 <int, int>.
* @param vp2 Co-ordinates of Point 2 <int, int>.
* @returns @param int distance between p1 and p2.
*******************************************************************************/
int compare(const void *vp1, const void *vp2) {
auto *p1 = static_cast<const Point *>(vp1);
auto *p2 = static_cast<const Point *>(vp2);

// Find orientation
int o = orientation(p0, *p1, *p2);
if (o == 0) {
return (distSq(p0, *p2) >= distSq(p0, *p1)) ? -1 : 1;
}

return (o == 2) ? -1 : 1;
}

/******************************************************************************
* @brief Prints convex hull of a set of n points.
* @param points vector of Point<int, int> with co-ordinates.
* @param size Size of the vector.
* @returns @param vector vector of Conver Hull.
*******************************************************************************/
std::vector<Point> convexHull(std::vector<Point> points, uint64_t size) {
// Find the bottom-most point
int ymin = points[0].y, min = 0;
for (int i = 1; i < size; i++) {
int y = points[i].y;

// Pick the bottom-most or chose the left-most point in case of tie
if ((y < ymin) || (ymin == y && points[i].x < points[min].x)) {
ymin = points[i].y, min = i;
}
}

// Place the bottom-most point at first position
std::swap(points[0], points[min]);

// Sort n-1 points with respect to the first point. A point p1 comes
// before p2 in sorted output if p2 has larger polar angle
// (in counterclockwise direction) than p1.
p0 = points[0];
qsort(&points[1], size - 1, sizeof(Point), compare);

// If two or more points make same angle with p0, Remove all but the one
// that is farthest from p0 Remember that, in above sorting, our criteria
// was to keep the farthest point at the end when more than one points have
// same angle.
int m = 1;  // Initialize size of modified array
for (int i = 1; i < size; i++) {
// Keep removing i while angle of i and i+1 is same with respect to p0
while (i < size - 1 && orientation(p0, points[i], points[i + 1]) == 0) {
i++;
}

points[m] = points[i];
m++;  // Update size of modified array
}

// If modified array of points has less than 3 points, convex hull is not
// possible
if (m < 3) {
return {};
};

// Create an empty stack and push first three points to it.
std::stack<Point> St;
St.push(points[0]);
St.push(points[1]);
St.push(points[2]);

// Process remaining n-3 points
for (int i = 3; i < m; i++) {
// Keep removing top while the angle formed by
// points next-to-top, top, and points[i] makes
// a non-left turn
while (St.size() > 1 &&
orientation(nextToTop(&St), St.top(), points[i]) != 2) {
St.pop();
}
St.push(points[i]);
}

std::vector<Point> result;
// Now stack has the output points, push them into the resultant vector
while (!St.empty()) {
Point p = St.top();
result.push_back(p);
St.pop();
}

return result;  // return resultant vector with Convex Hull co-ordinates.
}
}  // namespace grahamscan
}  // namespace geometry
```