Dynamic Convex hull | Adding Points to an Existing Convex Hull

Last Updated : 14 Sep, 2023

Given a convex hull, we need to add a given number of points to the convex hull and print the convex hull after every point addition. The points should be in anti-clockwise order after addition of every point.
Examples:

Input :   Convex Hull : (0, 0), (3, -1), (4, 5), (-1, 4)  Point to add : (100, 100)    Output :  New convex hull : (-1, 4) (0, 0) (3, -1) (100, 100)

We first check whether the point is inside the given convex hull or not. If it is, then nothing has to be done we directly return the given convex hull. If the point is outside the convex hull, we find the lower and upper tangents, and then merge the point with the given convex hull to find the new convex hull, as shown in the figure.

point_to_hull2

The red outline shows the new convex hull after merging the point and the given convex hull.
To find the upper tangent, we first choose a point on the hull that is nearest to the given point. Then while the line joining the point on the convex hull and the given point crosses the convex hull, we move anti-clockwise till we get the tangent line.

point_to_hull

The figure shows the moving of the point on the convex hull for finding the upper tangent.
Note: It is assumed here that the input of the initial convex hull is in the anti-clockwise order, otherwise we have to first sort them in anti-clockwise order then apply the following code.

Code:

CPP

// C++ program to add given a point p to a given // convex hull. The program assumes that the // point of given convex hull are in anti-clockwise // order. #include<bits/stdc++.h> using namespace std;  // checks whether the point crosses the convex hull // or not int orientation(pair<int, int> a, pair<int, int> b,                 pair<int, int> c) {     int res = (b.second-a.second)*(c.first-b.first) -               (c.second-b.second)*(b.first-a.first);      if (res == 0)         return 0;     if (res > 0)         return 1;     return -1; }  // Returns the square of distance between two input points int sqDist(pair<int, int> p1, pair<int, int> p2) {     return (p1.first-p2.first)*(p1.first-p2.first) +            (p1.second-p2.second)*(p1.second-p2.second); }  // Checks whether the point is inside the convex hull or not bool inside(vector<pair<int, int>> a, pair<int, int> p) {     // Initialize the centroid of the convex hull     pair<int, int> mid = {0, 0};      int n = a.size();      // Multiplying with n to avoid floating point     // arithmetic.     p.first *= n;     p.second *= n;     for (int i=0; i<n; i++)     {         mid.first += a[i].first;         mid.second += a[i].second;         a[i].first *= n;         a[i].second *= n;     }      // if the mid and the given point lies always     // on the same side w.r.t every edge of the     // convex hull, then the point lies inside     // the convex hull     for (int i=0, j; i<n; i++)     {         j = (i+1)%n;         int x1 = a[i].first, x2 = a[j].first;         int y1 = a[i].second, y2 = a[j].second;         int a1 = y1-y2;         int b1 = x2-x1;         int c1 = x1*y2-y1*x2;         int for_mid = a1*mid.first+b1*mid.second+c1;         int for_p = a1*p.first+b1*p.second+c1;         if (for_mid*for_p < 0)             return false;     }      return true; }  // Adds a point p to given convex hull a[] void addPoint(vector<pair<int, int>> &a, pair<int, int> p) {     // If point is inside p     if (inside(a, p))         return;        // point having minimum distance from the point p     int ind = 0;     int n = a.size();     for (int i=1; i<n; i++)         if (sqDist(p, a[i]) < sqDist(p, a[ind]))             ind = i;      // Find the upper tangent     int up = ind;     while (orientation(p, a[up], a[(up+1)%n])>=0)         up = (up + 1) % n;      // Find the lower tangent     int low = ind;     while (orientation(p, a[low], a[(n+low-1)%n])<=0)         low = (n+low - 1) % n;      // Initialize result     vector<pair<int, int>>ret;      // making the final hull by traversing points     // from up to low of given convex hull.     int curr = up;     ret.push_back(a[curr]);     while (curr != low)     {         curr = (curr+1)%n;         ret.push_back(a[curr]);     }      // Modify the original vector     ret.push_back(p);     a.clear();     for (int i=0; i<ret.size(); i++)         a.push_back(ret[i]); }  // Driver code int main() {     // the set of points in the convex hull     vector<pair<int, int> > a;     a.push_back({0, 0});     a.push_back({3, -1});     a.push_back({4, 5});     a.push_back({-1, 4});     int n = a.size();      pair<int, int> p = {100, 100};     addPoint(a, p);      // Print the modified Convex Hull     for (auto e : a)         cout << "(" << e.first << ", "               << e.second << ") ";      return 0; }

Java

// Java program to add given a point p to a given // convex hull. The program assumes that the // point of given convex hull are in anti-clockwise // order.  import java.io.*; import java.util.*; class GFG  {    // checks whether the point crosses the convex hull   // or not   static int orientation(ArrayList<Integer> a,                          ArrayList<Integer> b,                          ArrayList<Integer> c)   {     int res = (b.get(1) - a.get(1)) * (c.get(0) - b.get(0)) -       (c.get(1) - b.get(1)) * (b.get(0)-a.get(0));     if (res == 0)       return 0;     if (res > 0)       return 1;     return -1;   }    // Returns the square of distance between two input points   static int sqDist(ArrayList<Integer>p1, ArrayList<Integer>p2)   {     return (p1.get(0) - p2.get(0)) * (p1.get(0) - p2.get(0)) +        (p1.get(1) - p2.get(1)) * (p1.get(1) - p2.get(1));   }    // Checks whether the point is inside the convex hull or not   static boolean inside(ArrayList<ArrayList<Integer>> A,ArrayList<Integer>p)   {      // Initialize the centroid of the convex hull     ArrayList<Integer> mid = new ArrayList<Integer>(Arrays.asList(0,0));      int n = A.size();       for (int i = 0; i < n; i++)     {       mid.set(0,mid.get(0) + A.get(i).get(0));       mid.set(1,mid.get(1) + A.get(i).get(1));      }      // if the mid and the given point lies always     // on the same side w.r.t every edge of the     // convex hull, then the point lies inside     // the convex hull     for (int i = 0, j; i < n; i++)     {       j = (i + 1) % n;       int x1 = A.get(i).get(0)*n, x2 = A.get(j).get(0)*n;       int y1 = A.get(i).get(1)*n, y2 = A.get(j).get(1)*n;       int a1 = y1 - y2;       int b1 = x2 - x1;       int c1 = x1 * y2 - y1 * x2;       int for_mid = a1 * mid.get(0) + b1 * mid.get(1) + c1;       int for_p = a1 * p.get(0) * n + b1 * p.get(1) * n + c1;       if (for_mid*for_p < 0)         return false;     }     return true;   }    // Adds a point p to given convex hull a[]   static void addPoint(ArrayList<ArrayList<Integer>> a,ArrayList<Integer> p)   {      // If point is inside p     if (inside(a, p))       return;      // point having minimum distance from the point p     int ind = 0;     int n = a.size();     for (int i = 1; i < n; i++)     {       if (sqDist(p, a.get(i)) < sqDist(p, a.get(ind)))       {         ind = i;       }     }      // Find the upper tangent     int up = ind;     while (orientation(p, a.get(up), a.get((up+1)%n))>=0)       up = (up + 1) % n;      // Find the lower tangent     int low = ind;     while (orientation(p, a.get(low), a.get((n+low-1)%n))<=0)       low = (n+low - 1) % n;      // Initialize result     ArrayList<ArrayList<Integer>> ret = new ArrayList<ArrayList<Integer>>();      // making the final hull by traversing points     // from up to low of given convex hull.     int curr = up;     ret.add(a.get(curr));      while (curr != low)     {       curr = (curr + 1) % n;       ret.add(a.get(curr));     }      // Modify the original vector      ret.add(p);     a.clear();     for (int i = 0; i < ret.size(); i++)     {       a.add(ret.get(i));     }   }    // Driver code   public static void main (String[] args)    {      // the set of points in the convex hull     ArrayList<ArrayList<Integer>> a = new ArrayList<ArrayList<Integer>>();      a.add(new ArrayList<Integer>(Arrays.asList(0, 0)));     a.add(new ArrayList<Integer>(Arrays.asList(3, -1)));     a.add(new ArrayList<Integer>(Arrays.asList(4, 5)));     a.add(new ArrayList<Integer>(Arrays.asList(-1, 4)));      int n = a.size();      ArrayList<Integer> p = new ArrayList<Integer>(Arrays.asList(100,100));      addPoint(a, p);     // Print the modified Convex Hull     for(ArrayList<Integer> e:a )     {       System.out.print("(" + e.get(0) + ", " + e.get(1) + ") ");     }   } }  // This code is contributed by rag2127

Python3

# Python 3 program to add given a point p to a given # convex hull. The program assumes that the # point of given convex hull are in anti-clockwise # order. import copy  # checks whether the point crosses the convex hull # or not def orientation(a, b, c):      res = ((b[1] - a[1]) * (c[0] - b[0]) -               (c[1] - b[1]) * (b[0] - a[0]))      if (res == 0):         return 0;     if (res > 0):         return 1;     return -1;  # Returns the square of distance between two input points def sqDist(p1, p2):      return ((p1[0] - p2[0]) * (p1[0] - p2[0]) +            (p1[1] - p2[1]) * (p1[1] - p2[1]));  # Checks whether the point is inside the convex hull or not def inside( a, p ):      # Initialize the centroid of the convex hull     mid = [0, 0]      n = len(a)      # Multiplying with n to avoid floating point     # arithmetic.     p[0] *= n;     p[1] *= n;     for i in range(n):                mid[0] += a[i][0];         mid[1] += a[i][1];         a[i][0] *= n;         a[i][1] *= n;          # if the mid and the given point lies always     # on the same side w.r.t every edge of the     # convex hull, then the point lies inside     # the convex hull     for i in range( n ):              j = (i + 1) % n;         x1 = a[i][0]         x2 = a[j][0]         y1 = a[i][1]         y2 = a[j][1]         a1 = y1 - y2;         b1 = x2 - x1;         c1 = x1 * y2 - y1 * x2;         for_mid = a1 * mid[0] + b1 * mid[1] + c1;         for_p = a1 * p[0] + b1*p[1]+c1;         if (for_mid*for_p < 0):             return False;         return True;  # Adds a point p to given convex hull a[] def addPoint( a, p):      # If point is inside p     arr= copy.deepcopy(a)     prr =p.copy()          if (inside(arr, prr)):         return;        # point having minimum distance from the point p     ind = 0;     n = len(a)     for i in range(1, n):         if (sqDist(p, a[i]) < sqDist(p, a[ind])):             ind = i      # Find the upper tangent     up = ind;     while (orientation(p, a[up], a[(up + 1) % n]) >= 0):         up = (up + 1) % n;      # Find the lower tangent     low = ind;     while (orientation(p, a[low], a[(n + low - 1) % n]) <= 0):         low = (n + low - 1) % n      # Initialize result     ret = []      # making the final hull by traversing points     # from up to low of given convex hull.     curr = up;     ret.append(a[curr]);     while (curr != low):         curr = (curr + 1) % n;         ret.append(a[curr]);      # Modify the original vector     ret.append(p);     a.clear();     for i in range(len(ret)):         a.append(ret[i]);  # Driver code if __name__ == "__main__":        # the set of points in the convex hull     a = []     a.append([0, 0]);     a.append([3, -1]);     a.append([4, 5]);     a.append([-1, 4]);     n = len(a)      p = [100, 100]     addPoint(a, p);      # Print the modified Convex Hull     for e in a :         print("(" , e[0], ", ",               e[1] , ") ",end=" ")  # This code is contributed by chitranayal

// C# program to add given a point p to a given // convex hull. The program assumes that the // point of given convex hull are in anti-clockwise // order.  using System; using System.Collections.Generic;  public class GFG{    // checks whether the point crosses the convex hull   // or not   static int orientation(List<int> a,List<int> b,List<int> c)   {     int res=(b[1]-a[1]) * (c[0]-b[0]) - (c[1]-b[1]) * (b[0]-a[0]);     if (res == 0)       return 0;     if (res > 0)       return 1;     return -1;   }   // Returns the square of distance between two input points   static int sqDist(List<int>p1, List<int>p2)   {     return (p1[0] - p2[0]) * (p1[0] - p2[0]) +        (p1[1] - p2[1]) * (p1[1] - p2[1]);   }    // Checks whether the point is inside the convex hull or not   static bool inside(List<List<int>> A,List<int>p)   {      // Initialize the centroid of the convex hull     List<int> mid = new List<int>(){0,0};      int n = A.Count;       for (int i = 0; i < n; i++)     {       mid[0]+=A[i][0];       mid[1]+=A[i][1];      }         // if the mid and the given point lies always     // on the same side w.r.t every edge of the     // convex hull, then the point lies inside     // the convex hull     for (int i = 0, j; i < n; i++)     {       j = (i + 1) % n;       int x1 = A[i][0]*n, x2 = A[j][0]*n;       int y1 = A[i][1]*n, y2 = A[j][1]*n;       int a1 = y1 - y2;       int b1 = x2 - x1;       int c1 = x1 * y2 - y1 * x2;       int for_mid = a1 * mid[0] + b1 * mid[1] + c1;       int for_p = a1 * p[0] * n + b1 * p[1] * n + c1;       if (for_mid*for_p < 0)         return false;     }     return true;   }    // Adds a point p to given convex hull a[]   static void addPoint(List<List<int>> a,List<int> p)   {      // If point is inside p     if (inside(a, p))       return;      // point having minimum distance from the point p     int ind = 0;     int n = a.Count;     for (int i = 1; i < n; i++)     {       if (sqDist(p, a[i]) < sqDist(p, a[ind]))       {         ind = i;       }     }      // Find the upper tangent     int up = ind;     while (orientation(p, a[up], a[(up+1)%n])>=0)       up = (up + 1) % n;      // Find the lower tangent     int low = ind;     while (orientation(p, a[low], a[(n+low-1)%n])<=0)       low = (n+low - 1) % n;      // Initialize result     List<List<int>> ret = new List<List<int>>();      // making the final hull by traversing points     // from up to low of given convex hull.     int curr = up;     ret.Add(a[curr]);      while (curr != low)     {       curr = (curr + 1) % n;       ret.Add(a[curr]);     }      // Modify the original vector      ret.Add(p);     a.Clear();     for (int i = 0; i < ret.Count; i++)     {       a.Add(ret[i]);     }   }    // Driver code     static public void Main (){     // the set of points in the convex hull     List<List<int>> a = new List<List<int>>();      a.Add(new List<int>(){0,0});     a.Add(new List<int>(){3,-1});     a.Add(new List<int>(){4,5});     a.Add(new List<int>(){-1,4});      int n=a.Count;     List<int> p = new List<int>(){100,100};     addPoint(a, p);     // Print the modified Convex Hull     foreach(List<int> e in a)     {       Console.Write("(" + e[0] + ", " + e[1] + ") ");     }    } }  // This code is contributed by avanitrachhadiya2155

JavaScript

<script> // Javascript program to add given a point p to a given // convex hull. The program assumes that the // point of given convex hull are in anti-clockwise // order.  // checks whether the point crosses the convex hull   // or not function orientation(a,b,c) {     let res = (b[1] - a[1]) * (c[0] - b[0]) -       (c[1] - b[1]) * (b[0]-a[0]);     if (res == 0)       return 0;     if (res > 0)       return 1;     return -1; }  // Returns the square of distance between two input points function sqDist(p1,p2) {     return (p1[0] - p2[0]) * (p1[0] - p2[0]) +       (p1[1] - p2[1]) * (p1[1] - p2[1]); }  // Checks whether the point is inside the convex hull or not function inside(A,p) {     // Initialize the centroid of the convex hull     let mid = [0,0];       let n = A.length;         for (let i = 0; i < n; i++)     {      mid[0]+=A[i][0];       mid[1]+=A[i][1];       }       // if the mid and the given point lies always     // on the same side w.r.t every edge of the     // convex hull, then the point lies inside     // the convex hull     for (let i = 0, j; i < n; i++)     {       j = (i + 1) % n;       let x1 = A[i][0]*n, x2 = A[j][0]*n;       let y1 = A[i][1]*n, y2 = A[j][1]*n;       let a1 = y1 - y2;       let b1 = x2 - x1;       let c1 = x1 * y2 - y1 * x2;       let for_mid = a1 * mid[0] + b1 * mid[1] + c1;       let for_p = a1 * p[0] * n + b1 * p[1] * n + c1;       if (for_mid*for_p < 0)         return false;     }     return true; }  // Adds a point p to given convex hull a[] function addPoint(a,p) {     // If point is inside p     if (inside(a, p))       return;       // point having minimum distance from the point p     let ind = 0;     let n = a.length;     for (let i = 1; i < n; i++)     {       if (sqDist(p, a[i]) < sqDist(p, a[ind]))       {         ind = i;       }     }       // Find the upper tangent     let up = ind;     while (orientation(p, a[up], a[(up+1)%n])>=0)       up = (up + 1) % n;       // Find the lower tangent     let low = ind;     while (orientation(p, a[low], a[(n+low-1)%n])<=0)       low = (n+low - 1) % n;       // Initialize result     let ret = [];       // making the final hull by traversing points     // from up to low of given convex hull.     let curr = up;     ret.push(a[curr]);       while (curr != low)     {       curr = (curr + 1) % n;       ret.push(a[curr]);     }       // Modify the original vector       ret.push(p);     a=[];     for (let i = 0; i < ret.length; i++)     {       a.push(ret[i]);     }     return a; }  // Driver code  // the set of points in the convex hull let a = [] a.push([0, 0]); a.push([3, -1]); a.push([4, 5]); a.push([-1, 4]);  let n=a.length; let p=[100,100]; a=addPoint(a, p); // Print the modified Convex Hull for(let e=0;e<a.length;e++ ) {     document.write("(" + a[e][0] + ", " + a[e][1] + ") "); }  // This code is contributed by ab2127 </script>

Output:

(-1, 4) (0, 0) (3, -1) (100, 100)

Time Complexity:
The time complexity of the above algorithm is O(n*q), where q is the number of points to be added.

Auxiliary Space: O(n), since n extra space has been taken.

This article is contributed by Aarti_Rathi and Amritya Vagmi and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to [email protected].

Deleting points from Convex Hull

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Dynamic Convex hull | Adding Points to an Existing Convex Hull

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