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

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)?

Try this code which makes use of a cross product:

public bool isLeft(Point a, Point b, Point c){
     return ((b.X - a.X)*(c.Y - a.Y) - (b.Y - a.Y)*(c.X - a.X)) > 0;
}

Where a = line point 1; b = line point 2; c = point to check against.

If the formula is equal to 0, the points are colinear.

If the line is horizontal, then this returns true if the point is above the line.

Use the sign of the determinant of vectors (AB,AM), where M(X,Y) is the query point:

position = sign((Bx - Ax) * (Y - Ay) - (By - Ay) * (X - Ax))

It is 0 on the line, and +1 on one side, -1 on the other side.

You look at the sign of the determinant of

| x2-x1  x3-x1 |
| y2-y1  y3-y1 |

It will be positive for points on one side, and negative on the other (and zero for points on the line itself).

The vector (y1 - y2, x2 - x1) is perpendicular to the line, and always pointing right (or always pointing left, if you plane orientation is different from mine).

You can then compute the dot product of that vector and (x3 - x1, y3 - y1) to determine if the point lies on the same side of the line as the perpendicular vector (dot product > 0) or not.

Using the equation of the line ab, get the x-coordinate on the line at the same y-coordinate as the point to be sorted.

• If point's x > line's x, the point is to the right of the line.
• If point's x < line's x, the point is to the left of the line.
• If point's x == line's x, the point is on the line.