feat: Added Graham Scan Algorithm. (#1836)

* Implementated Grahamscan Algorithm for Convex Hull

* Update graham_scan_algorithm.cpp

* Update graham_scan_functions.h

* Update graham_scan_algorithm.cpp

* Update graham_scan_functions.h

* Update graham_scan_algorithm.cpp

* Update and rename graham_scan_functions.h to graham_scan_functions.hpp

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_functions.hpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_functions.hpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* updating DIRECTORY.md

* clang-format and clang-tidy fixes for e89e4c8c

* clang-format and clang-tidy fixes for 7df4778f

* Fix #1

* Update graham_scan_functions.hpp

* Delete composite_simpson_rule.cpp

* Delete inverse_fast_fourier_transform.cpp

* Fix #2

* updating DIRECTORY.md

* clang-format and clang-tidy fixes for 69b6832b

* Fix #3

* clang-format and clang-tidy fixes for 1c05ca7c

* Update graham_scan_functions.hpp

* Fix #4

* clang-format and clang-tidy fixes for 2957fd21

* Create composite_simpson_rule.cpp

* updating DIRECTORY.md

* Create inverse_fast_fourier_transform.cpp

* updating DIRECTORY.md

* clang-format and clang-tidy fixes for 405d21a5

* clang-format and clang-tidy fixes for 333ef5ca

* Update geometry/graham_scan_functions.hpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_functions.hpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* Update geometry/graham_scan_algorithm.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* clang-format and clang-tidy fixes for ee4cb635

* Update graham_scan_algorithm.cpp

* Update graham_scan_functions.hpp

* clang-format and clang-tidy fixes for f2f69234

* Update graham_scan_functions.hpp

* Create partition_problem.cpp

* Update partition_problem.cpp

* Delete partition_problem.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

geometry/graham_scan_algorithm.cpp

@@ -0,0 +1,76 @@
+/******************************************************************************
+ * @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;
+}