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Again, determining whether three points constitute a " left turn " or a " right turn " does not require computing the actual angle between the two line segments, and can actually be achieved with simple arithmetic only.

For three points, and, simply compute the direction of the cross product of the two vectors connecting the point to and the point to, which is given by the expression.

If the result is 0, the points are collinear ; if it is positive, the three points constitute a " left turn " or counter-clockwise orientiation, otherwise a " right turn " or clockwise orientation.