Given two second order (quadratic) Bézier curves which both start at the origin $[0, 0]$ and end at two arbitrary points, and both of whose control points lie at the positive portion of the X-axis ($y = 0, x > 0$), how to determine which of the two curves lies higher in the Y dimension at the limit point immediately right of the origin?
Normally, I would use the derivative at the starting point, but that doesn't work in this case because it is horizontal for both. My first idea was to instead use the second derivative (equal to $2(P_0-2P_1+P_2)$ at all points), but I wasn't able to come up with a working formula.
I am not interested in an approximate solution (I know how to do that).
Alright, I tried some more in the morning and I think I solved it. If $c_{x1}, c_{x2}$ are the X coordinates of the control points, and $e_{y1}, e_{y2}$ are the Y coordinates of the endpoints, the answer is the sign of the expression:
$c_{x1}\cdot\sqrt{e_{y2}} - c_{x2}\cdot\sqrt{e_{y1}}$
Interestingly, it seems that the answer doesn't depend on the endpoints' X coordinates.