# Convex Hull Example in Data Structures

Data Structure AlgorithmsAnalysis of AlgorithmsAlgorithms

Here we will see one example on convex hull. Suppose we have a set of points. We have to make a polygon by taking less amount of points, that will cover all given points. In this section we will see the Jarvis March algorithm to get the convex hull.

Jarvis March algorithm is used to detect the corner points of a convex hull from a given set of data points.

Starting from left most point of the data set, we keep the points in the convex hull by anti-clockwise rotation. From a current point, we can choose the next point by checking the orientations of those points from current point. When the angle is largest, the point is chosen. After completing all points, when the next point is the start point, stop the algorithm.

Input − Set of points: {(-7,8), (-4,6), (2,6), (6,4), (8,6), (7,-2), (4,-6), (8,-7),(0,0), (3,-2),(6,-10),(0,-6),(-9,-5),(-8,-2),(-8,0),(-10,3),(-2,2),(-10,4)}

Output − Boundary points of convex hull are −

(-9, -5) (6, -10) (8, -7) (8, 6) (-7, 8) (-10, 4) (-10, 3)

## Algorithm

findConvexHull(points, n)
Input: The points, number of points.
Output: Corner points of convex hull.
Begin
start := points[0]
for each point i, do
if points[i].x < start.x, then // get the left most point
start := points[i]
done
current := start
add start point to the result set.
define colPts set to store collinear points
while true, do //start an infinite loop
next := points[i]
for all points i except 0th point, do
if points[i] = current, then
skip the next part, go for next iteration
val := cross product of current, next, points[i]
if val > 0, then
next := points[i]
clear the colPts array
else if cal = 0, then
if next is closer to current than points[i], then
next := points[i]
else
done
add all items in the colPts into the result
if next = start, then
break the loop
insert next into the result
current := next
done
return result
End

## Example

Live Demo

#include<iostream>
#include<set>
#include<vector>
using namespace std;
struct point{ //define points for 2d plane
int x, y;
bool operator==(point p2){
if(x == p2.x && y == p2.y)
return 1;
return 0;
}
bool operator<(const point &p2)const{ //dummy compare function used to sort in set
return true;
}
};
int crossProduct(point a, point b, point c){ //finds the place of c from ab vector
int y1 = a.y - b.y;
int y2 = a.y - c.y;
int x1 = a.x - b.x;
int x2 = a.x - c.x;
return y2*x1 - y1*x2; //if result < 0, c in the left, > 0, c in the right, = 0, a,b,c are collinear
}
int distance(point a, point b, point c){
int y1 = a.y - b.y;
int y2 = a.y - c.y;
int x1 = a.x - b.x;
int x2 = a.x - c.x;
int item1 = (y1*y1 + x1*x1);
int item2 = (y2*y2 + x2*x2);
if(item1 == item2)
return 0; //when b and c are in same distance from a
else if(item1 < item2)
return -1; //when b is closer to a
return 1; //when c is closer to a
}
set<point> findConvexHull(point points[], int n){
point start = points[0];
for(int i = 1; i<n; i++){ //find the left most point for starting
if(points[i].x < start.x)
start = points[i];
}
point current = start;
set<point> result; //set is used to avoid entry of duplicate points
result.insert(start);
vector<point> *collinearPoints = new vector<point>;
while(true){
point nextTarget = points[0];
for(int i = 1; i<n; i++){
if(points[i] == current) //when selected point is current point, ignore rest part
continue;
int val = crossProduct(current, nextTarget, points[i]);
if(val > 0){ //when ith point is on the left side
nextTarget = points[i];
collinearPoints = new vector<point>; //reset collinear points
}else if(val == 0){ //if three points are collinear
if(distance(current, nextTarget, points[i]) < 0){ //add closer one to collinear list
collinearPoints->push_back(nextTarget);
nextTarget = points[i];
}else{
collinearPoints->push_back(points[i]); //when ith point is closer or same as nextTarget
}
}
}
vector<point>::iterator it;
for(it = collinearPoints->begin(); it != collinearPoints->end(); it++){
result.insert(*it); //add allpoints in collinear points to result set
}
if(nextTarget == start) //when next point is start it means, the area covered
break;
result.insert(nextTarget);
current = nextTarget;
}
return result;
}
int main(){
point points[] = {
{-7,8},{-4,6},{2,6},{6,4},{8,6},{7,-2},{4,-6},{8,-7},{0,0},
{3,-2},{6,-10},{0,-6},{-9,-5},{-8,-2},{-8,0},{-10,3},{-2,2},{-10,4}};
int n = 18;
set<point> result;
result = findConvexHull(points, n);
cout << "Boundary points of convex hull are: "<<endl;
set<point>::iterator it;
for(it = result.begin(); it!=result.end(); it++)
cout << "(" << it->x << ", " <<it->y <<") ";
}

## Output

Boundary points of convex hull are:
(-9, -5) (6, -10) (8, -7) (8, 6) (-7, 8) (-10, 4) (-10, 3)
Published on 27-Aug-2019 13:34:33