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Number of triangles that can be formed with given N points

  • Difficulty Level : Medium
  • Last Updated : 28 Mar, 2019

Given X and Y coordinates of N points on a Cartesian plane. The task is to find the number of possible triangles with the non-zero area that can be formed by joining each point to every other point.

Examples:

Input: P[] = {{0, 0}, {2, 0}, {1, 1}, {2, 2}}
Output: 3
Possible triangles can be [(0, 0}, (2, 0), (1, 1)], 
[(0, 0), (2, 0), (2, 2)] and [(1, 1), (2, 2), (2, 0)]

Input : P[] = {{0, 0}, {2, 0}, {1, 1}}
Output : 1

A Naive approach has been already discussed in Number of possible Triangles in a Cartesian coordinate system

Efficient Approach: Consider a point Z and find its slope with every other point. Now, if two points are having the same slope with point Z that means the 3 points are collinear and they cannot form a triangle. Hence, the number of triangles having Z as one of its points is the number of ways of choosing 2 points from the remaining points and then subtracting the number of ways of choosing 2 points from points having the same slope with Z. Since Z can be any point among N points, we have to iterate one more loop.

Below is the implementation of above approach:

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
  
// This function returns the required number
// of triangles
int countTriangles(pair<int, int> P[], int N)
{
    // Hash Map to store the frequency of
    // slope corresponding to a point (X, Y)
    map<pair<int, int>, int> mp;
    int ans = 0;
  
    // Iterate over all possible points
    for (int i = 0; i < N; i++) {
        mp.clear();
  
        // Calculate slope of all elements
        // with current element
        for (int j = i + 1; j < N; j++) {
            int X = P[i].first - P[j].first;
            int Y = P[i].second - P[j].second;
  
            // find the slope with reduced
            // fraction
            int g = __gcd(X, Y);
            X /= g;
            Y /= g;
            mp[{ X, Y }]++;
        }
        int num = N - (i + 1);
  
        // Total number of ways to form a triangle
        // having one point as current element
        ans += (num * (num - 1)) / 2;
  
        // Subtracting the total number of ways to
        // form a triangle having the same slope or are
        // collinear
        for (auto j : mp)
            ans -= (j.second * (j.second - 1)) / 2;
    }
    return ans;
}
  
// Driver Code to test above function
int main()
{
    pair<int, int> P[] = { { 0, 0 }, { 2, 0 }, { 1, 1 }, { 2, 2 } };
    int N = sizeof(P) / sizeof(P[0]);
    cout << countTriangles(P, N) << endl;
    return 0;
}

Python3




# Python3 implementation of the above approach 
from collections import defaultdict
from math import gcd
  
# This function returns the 
# required number of triangles 
def countTriangles(P, N): 
  
    # Hash Map to store the frequency of 
    # slope corresponding to a point (X, Y) 
    mp = defaultdict(lambda:0
    ans = 0
  
    # Iterate over all possible points 
    for i in range(0, N): 
        mp.clear() 
  
        # Calculate slope of all elements 
        # with current element 
        for j in range(i + 1, N): 
            X = P[i][0] - P[j][0
            Y = P[i][1] - P[j][1
  
            # find the slope with reduced 
            # fraction 
            g = gcd(X, Y) 
            X //=
            Y //=
            mp[(X, Y)] += 1
          
        num = N - (i + 1
  
        # Total number of ways to form a triangle 
        # having one point as current element 
        ans += (num * (num - 1)) // 2
  
        # Subtracting the total number of 
        # ways to form a triangle having 
        # the same slope or are collinear 
        for j in mp: 
            ans -= (mp[j] * (mp[j] - 1)) // 2
      
    return ans 
  
# Driver Code
if __name__ == "__main__"
  
    P = [[0, 0], [2, 0], [1, 1], [2, 2]] 
    N = len(P) 
    print(countTriangles(P, N))
      
# This code is contributed by Rituraj Jain
Output:
3



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