How to tell whether a point is to the right or left side of a line?

Question

I have a set of points. I want to separate them into 2 distinct sets. To do this, I choose two points (*a* and *b*) and draw an imaginary line between them. Now I want to have all points that are left from this line in one set and those that are right from this line in the other set. How can I tell for any given point *z* whether it is in the left or in the right set? I tried to calculate the angle between *a-z-b* – angles smaller than 180 are on the right hand side, greater than 180 on the left hand side – but because of the definition of ArcCos, the calculated angles are always smaller than 180°. Is there a formula to calculate angles greater than 180° (or any other formula to chose right or left side)?

Answer 1

By "left" and "right", I assume you mean relative to the x - coordinate, on an x-y plane. I also assume that the straight line you have drawn goes to infinity in both directions.

If the equation of your line is y = mx + b, then x = (y - b)/m, where of course m = slope, except in the special case of a horizontal line (m=0), where of course y = b, or in the special case of a vertical line, where x is a constant, and y can be anything (undefined slope).

Case 1: If your line is neither vertical nor horizontal:

For any point p1 (x1, y1) in your set of points to be classified, there is a corresponding point p2 (x2, y2) on your line, such that y1 = y2. Then the corresponding x coordinate on your line for the point to be classified is then x2 = (y1 - b)/m.

Points on the left satisfy: x1 < x2.

Points on the right satisfy: x1 > x2.

Points on the line satisfy x1 = x2.

Case 2: If your line is vertical, located at x = V, y can be anything:

Points on the left satisfy: x1 < V.

Points on the right satisfy: x1 > V.

Points on the line satisfy x1 = V.

Case 3: If your line is horizontal, located at y = H, x can be anything:

Hence, the concept of "left" and "right" in this case is indeterminate.

UNLESS, you now decide to change your perspective and define "left" = "above" and "right" = "below".

Points above satisfy: y1 > H.

Points below satisfy: y1 < H.

Points on the line satisfy y1 = H.