Algorithm - GCD

http://www.geeksforgeeks.org/find-integer-point-line-segment-given-two-ends/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.

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.

//  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 c[], 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 divisble 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;

}