Implement Gift Wrapping Algorithm in Two Dimensions using C++

Gift Wrapping Algorithm

The Gift Wrapping algorithm is also known as Jarvis's march. It is a method for calculating the convex hull of a set of points in a plane. It is essential to find the smallest convex polygon that encloses all the points.

Why We Use Gift Wrapping Algorithm?

Below are the following reasons to use this algorithm:

• Easy to Understand: It work like wrapping a string around points.
• Good for Small Data Sets: When fewer points make up the convex hull.
• Accurate: Never misses a point in the convex hull.
• Useful in Graphics & Robotic: Helps in pathfinding and object detection.
• Works in 3D too: It can be adapted for three-dimensional shapes.

Pseudocode of Gift Wrapping Algorithm

Following is the pseudocode of the gift-wrapping algorithm:

Algorithm Jarvis(S)
   Input: S = set of points
   Output: P = list of convex hull points in counter-clockwise order

   pointOnHull := leftmost point in S
   i := 0
   repeat
      P[i] := pointOnHull
      endpoint := S[0]

      for each point S[j] in S do
          if (endpoint == pointOnHull) or (S[j] is to the left of the line from P[i] to endpoint) then
              endpoint := S[j]
      
      i := i + 1
      pointOnHull := endpoint

   until endpoint == P[0] // completed the hull
   return P

Explanations

Leftmost point: The point that is guarantees to be a part of the convex hull and has smallest x-coordinate (and smallest y if there is a tie).

"is to the left": Makes sure we wrap around the outer edge of the points and responds to the orientation test (cross product > 0).

The algorithm iterates around the outermost points, choosing the "most counter-clockwise" point at each step until it reaches the starting point again.

C++ Implementation of Gift-wrapping Algorithm in Two Dimensions

Following is the C++ implementation of the gift-wrapping algorithm in two dimensions:

#include <iostream>
using namespace std;
#define INF 10000
struct P {
   int x;
   int y;
};
int orient(P a, P b, P c) {
   int v = (b.y - a.y) * (c.x - b.x) - (b.x - a.x) * (c.y - b.y);
   if (v == 0)
      return 0; // colinear
   return (v > 0) ? 1 : 2; // clock or counterclock wise
}
void convexHull(P points[], int m) {
   if (m < 3) //at least three points required
      return;
   int n[m];
   for (int i = 0; i < m; i++)
      n[i] = -1;
   int l = 0; //initialize result.
   for (int i = 1; i < m; i++)
      if (points[i].x < points[l].x)
         l = i; //find left most point
   int p = l, q;
   do {
      q = (p + 1) % m;
      for (int i = 0; i < m; i++)
         if (orient(points[p], points[i], points[q]) == 2)
            q = i;
      n[p] = q;
      p = q;
   } while (p != l);
   for (int i = 0; i < m; i++) {
      if (n[i] != -1)
         cout << "(" << points[i].x << ", " << points[i].y << ")\n";
   }
}
int main() {
   P points[] = {
      {0, 4}, {2, 1}, {2, 3}, {4, 1}, {3, 0}, {1, 1}, {7, 6}
   };
   cout << "The points in the convex hull are: ";
   int n = sizeof(points) / sizeof(points[0]);
   convexHull(points, n);
   return 0;
}

Following is the output of the above code:

The points in the convex hull are: (0, 4)
(4, 1)
(3, 0)
(1, 1)
(7, 6)