```
/******************************************************************************
* @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 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)
*
*******************************************************************************/
#include <cassert> /// for std::assert
#include <iostream> /// for IO Operations
#include <vector> /// for std::vector
#include "./graham_scan_functions.hpp" /// for all the functions used
/*******************************************************************************
* @brief Self-test implementations
* @returns void
*******************************************************************************/
static void test() {
std::vector<geometry::grahamscan::Point> points = {
{0, 3}, {1, 1}, {2, 2}, {4, 4}, {0, 0}, {1, 2}, {3, 1}, {3, 3}};
std::vector<geometry::grahamscan::Point> expected_result = {
{0, 3}, {4, 4}, {3, 1}, {0, 0}};
std::vector<geometry::grahamscan::Point> derived_result;
std::vector<geometry::grahamscan::Point> res;
derived_result = geometry::grahamscan::convexHull(points, points.size());
std::cout << "1st test: ";
for (int i = 0; i < expected_result.size(); i++) {
assert(derived_result[i].x == expected_result[i].x);
assert(derived_result[i].y == expected_result[i].y);
}
std::cout << "passed!" << std::endl;
}
/*******************************************************************************
* @brief Main function
* @returns 0 on exit
*******************************************************************************/
int main() {
test(); // run self-test implementations
return 0;
}
```