Determine which side of a line a point lies?

Question

I've got a line through points (x1,y1) and (x2, y2). I'd like to see if point (x3, y3) lies to the "left" or "right" of said line. How would I do so?

Answer 1

Here's a suggestion:

Determine the equation of the straight line that goes through points 1 & 2. Standard form: y = mx + b.

Next, substitute x₃ into this equation to get the y coordinate of the point on this line for x = x_3.

Let's call this point 4, with coordinates (x₃,y₄).

Calculate the vertical distance between point 3 and point 4:

If y₃ - y₄ > 0, then point 3 lies above the line (to the "left").

If y₃ - y₄ < 0, then point 3 lies below the line (to the "right").

Hope that helps.

Answer 2

To determine which side of the line from A = (x1,y1) to B = (x2,y2) a point P = (x,y) falls on you need to compute the value:

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

If d < 0, then the point lies on one side of the line, and if d > 0, then it lies on the other side. If d=0, then the point lies exactly on the line

To see whether points on the left side of the line are those with positive or negative values compute the value for d for a point you know is to the left of the line, such as (x1-1,y1) and then compare the sign with the point you are interested in.