How can I find the general form equation of a line from two points?

Question

Given the input:

double x1,y1,x2,y2;

How can I find the general form equation (double a,b,c where ax + b

Answer 1

If you start from the equation of defining line from 2 points

(x - x1)/(x2 - x1) = (y - y1)/(y2 - y1)

you can end up with the next equation

x(y2 - y1) - y(x2 - x1) - x1*y2 + y1*x2 = 0

so the coefficients will be:

• a = y2 - y1
• b = -(x2 - x1) = x1 - x2
• c = y1*x2 - x1*y2

My implementation of the algorithm in C

inline v3 LineEquationFrom2Points(v2 P1, v2 P2) {
    v3 Result;

    Result.A = P2.y - P1.y;
    Result.B = -(P2.x - P1.x);
    Result.C = P1.y * P2.x - P1.x * P2.y;

    return(Result);
}