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Find an Integer point on a line segment with given two ends

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  • Difficulty Level : Easy
  • Last Updated : 06 Jul, 2022

Given two points pointU and pointV in XY-space, we need to find a point which has integer coordinates and lies on a line going through points pointU and pointV. Examples:

If  pointU = (1, -1 and pointV = (-4, 1)
then equation of line which goes 
through these two points is,
2X + 5Y = -3
One point with integer co-ordinate which
satisfies above equation is (6, -3)

We can see that once we found the equation of line, this problem can be treated as Extended Euclid algorithm problem, where we know A, B, C in AX + BY = C and we want to find out the value of X and Y from the equation. In above Extended Euclid equation, C is gcd of A and B, so after finding out the line equation from given two points if C is not a multiple of gcd(A, B) then we can conclude that there is no possible integer coordinate on the specified line. If C is a multiple of g, then we can scale up the founded X and Y coefficients to satisfy the actual equation, which will be our final answer. 

CPP




// C++ program to get Integer point on a line
#include <bits/stdc++.h>
using namespace std;
 
// Utility method for extended Euclidean Algorithm
int gcdExtended(int a, int b, int *x, int *y)
{
    // Base Case
    if (a == 0)
    {
        *x = 0;
        *y = 1;
        return b;
    }
 
    int x1, y1; // To store results of recursive call
    int gcd = gcdExtended(b%a, a, &x1, &y1);
 
    // Update x and y using results of recursive
    // call
    *x = y1 - (b/a) * x1;
    *y = x1;
 
    return gcd;
}
 
// method prints integer point on a line with two
// points U and V.
void printIntegerPoint(int pointU[], int pointV[])
{
    // Getting coefficient of line
    int A = (pointU[1] - pointV[1]);
    int B = (pointV[0] - pointU[0]);
    int C = (pointU[0] * (pointU[1] - pointV[1]) +
            pointU[1] * (pointV[0] - pointU[0]));
 
    int x, y; // To be assigned a value by gcdExtended()
    int g = gcdExtended(A, B, &x, &y);
 
    // if C is not divisible by g, then no solution
    // is available
    if (C % g != 0)
        cout << "No possible integer point\n";
 
    else
 
        // scaling up x and y to satisfy actual answer
        cout << "Integer Point : " << (x * C/g) << " "
            << (y * C/g) << endl;
}
 
// Driver code to test above methods
int main()
{
    int pointU[] = {1, -1};
    int pointV[] = {-4, 1};
 
    printIntegerPoint(pointU, pointV);
    return 0;
}

Java




// Java program to get Integer point on a line
// Utility method for extended Euclidean Algorithm
class GFG {
 
  public static int x;
  public static int y;
 
  // Function for extended Euclidean Algorithm
  static int gcdExtended(int a, int b)
  {
 
    // Base Case
    if (a == 0) {
      x = 0;
      y = 1;
      return b;
    }
 
    // To store results of recursive call
    int gcd = gcdExtended(b % a, a);
    int x1 = x;
    int y1 = y;
 
    // Update x and y using results of recursive
    // call
    int tmp = b / a;
    x = y1 - tmp * x1;
    y = x1;
 
    return gcd;
  }
 
  // method prints integer point on a line with two
  // points U and V.
  public static void printIntegerPoint(int[] pointU,
                                       int[] pointV)
  {
    // Getting coefficient of line
    int A = (pointU[1] - pointV[1]);
    int B = (pointV[0] - pointU[0]);
    int C = (pointU[0] * (pointU[1] - pointV[1])
             + pointU[1] * (pointV[0] - pointU[0]));
 
    x = 0; // To be assigned a value by gcdExtended()
    y = 0;
    int g = gcdExtended(A, B);
 
    // if C is not divisible by g, then no solution
    // is available
    if (C % g != 0) {
      System.out.print("No possible integer point\n");
    }
 
    else {
 
      // scaling up x and y to satisfy actual answer
      System.out.print("Integer Point : ");
      System.out.print((x * C / g));
      System.out.print(" ");
      System.out.print((y * C / g));
      System.out.print("\n");
    }
  }
 
  // Driver code to test above methods
  public static void main(String[] args)
  {
    int[] pointU = { 1, -1 };
    int[] pointV = { -4, 1 };
 
    printIntegerPoint(pointU, pointV);
  }
}
 
// The code is contributed by phasing17

C#




using System;
 
public static class GFG {
    // C++ program to get Integer point on a line
    // Utility method for extended Euclidean Algorithm
    public static int gcdExtended(int a, int b, ref int x,
                                  ref int y)
    {
        // Base Case
        if (a == 0) {
            x = 0;
            y = 1;
            return b;
        }
 
        int x1 = 0; // To store results of recursive call
        int y1 = 0;
        int gcd = gcdExtended(b % a, a, ref x1, ref y1);
 
        // Update x and y using results of recursive
        // call
        x = y1 - (b / a) * x1;
        y = x1;
 
        return gcd;
    }
 
    // method prints integer point on a line with two
    // points U and V.
    public static void printIntegerPoint(int[] pointU,
                                         int[] pointV)
    {
        // Getting coefficient of line
        int A = (pointU[1] - pointV[1]);
        int B = (pointV[0] - pointU[0]);
        int C = (pointU[0] * (pointU[1] - pointV[1])
                 + pointU[1] * (pointV[0] - pointU[0]));
 
        int x
            = 0; // To be assigned a value by gcdExtended()
        int y = 0;
        int g = gcdExtended(A, B, ref x, ref y);
 
        // if C is not divisible by g, then no solution
        // is available
        if (C % g != 0) {
            Console.Write("No possible integer point\n");
        }
 
        else {
 
            // scaling up x and y to satisfy actual answer
            Console.Write("Integer Point : ");
            Console.Write((x * C / g));
            Console.Write(" ");
            Console.Write((y * C / g));
            Console.Write("\n");
        }
    }
 
    // Driver code to test above methods
    public static void Main()
    {
        int[] pointU = { 1, -1 };
        int[] pointV = { -4, 1 };
 
        printIntegerPoint(pointU, pointV);
    }
}
 
// The code is contributed by Aarti_Rathi

Output

Integer Point : 6 -3

Time complexity: O(logn)
Auxiliary Space: O(logn)

This article is contributed by Aarti_Rathi and Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.


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