I tried to implement points location algorithm using Fortune's algorithm to get Voronoi diagram and another sweepline algorithm to locate many points in $O(n\cdot\log(n))$. If there are multiple concentric points on some step I get a pencil of radius vectors origins from the center of a circle. I need to sort them by angle (or at least to find minmax elements). I use the next formula to compare raduis vectors $\mathbf{v_i} = (x_i, \;y_i)$ and $\mathbf{v_j} = (x_j, \;y_j)$:
$$ atan2(y_i, \;x_i) < atan2(y_j, \;x_j) $$
I sure result can be achieved avoiding trigonometric functions. Can it be expressed without comparisons?
Currently I can sort them by quadrants, if both points are in the same quadrant, then I just look at cross product sign, otherwise I compare quadrant numbers.
PS: can someone create point-location
tag?