Orientation:
Ialways let open upward (in the direction of ). From a given value of tan θ , I take 0 < θ < π .
The angles are signed. I have used complex rotation, not because I like being abstract, but because (for example) complex multiplication by corresponds always to "rotation by angle α". More on
Signed Angles:http://www.geogebratube.org/material/show/id/99837
Complex Rotation: http://www.geogebratube.org/material/show/id/115348

Equation (3) usually has ONE real solution, and the parabola is uniquely determined by the constraints. When there are instead THREE real solutions, I choose the value of θ which is closest to the previous value. This can always be done so that the axes rotate smoothly (no leaps).
As a result, if A and C begin on opposite arms of the parabola, they will stay on opposite arms. Put another way, v will always point inside the triangle ΔVAC).