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

Hello There, You are doing right! First you find slope using your formula m= dy/ dx , than now you need to find y intercept using any one point (x1,y1) or (x2,y2) and your slope, plug those values in point slope equation! Y= my+b, or you can use equation y-y1=m(x-x1), one’s you have this than you can convert in to standard form ax+by+c=0, I hope this helps!

Answer 2

I would think the formula you need is (y - y₂)(x₂ - x₁) = (x - x₂)(y₂ - y₁)

Since there is no division in this form, vertical lines are not a problem.

You can massage this form around a bit to fit the needs of your computer program.

BTW, this is just the 2 point formula massaged around: (y₂ - y₁)/(x₂ - x₁) = (y - y₂)/(x - x₂)