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 + by + c = 0) ?

Note: I want to be able to do this computationally. So the equivalent for slope-intercept form would be something like:

double dx, dy; double m, b;  dx = x2 - x1; dy = y2 - y1; m = dy/dx; b = y1; 

Obviously, this is very simple, but I haven't been able to find the solution for the general equation form (which is more useful since it can do vertical lines). I already looked in my linear algebra book and two books on computational geometry (both too advanced to explain this).

Answer 1

If you start from the equation y-y1 = (y2-y1)/(x2-x1) * (x-x1) (which is the equation of the line defined by two points), through some manipulation you can get (y1-y2) * x + (x2-x1) * y + (x1-x2)*y1 + (y2-y1)*x1 = 0, and you can recognize that:

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

Answer 2

Get the tangent by subtracting the two points (x2-x1, y2-y1). Normalize it and rotate by 90 degrees to get the normal vector (a,b). Take the dot product with one of the points to get the constant, c.