Check If a point on a circle is left or right of a point What is the best way to determine if a point on a circle is to the left or to the right of another point on that same circle?

If you mean in which direction you have to travel the shortest distance from $a$ to $b$ and assuming that the circle is centered at the origin then this is given by the sign of the determinant $\det(a\, b)$ where $a$ and $b$ are columns in a $2\times 2$ matrix.  If this determinant is positive you travel in the counter clockwise direction. If it is negative you travel in the clockwise direction.  If it is zero then both directions result in the same travel distance (either $a=b$ or $a$ and $b$ are antipodes).
