I move an ellipse (light green) along a Bézier curve (black) and let it intersect (red) with a single normal (dark green) on that curve.
The dark blue curve represents the distance of each ellipse, the light blue curve is it's 1st derivative. Which is a polynomial of degree 18. (Thanks to the trial version of Mathematica for figuring that out.)
Which I could tackle with Newton-Horner.
And then I would have still to include rotation and pressure (scaling)...
Or I just take, say, a hundred samples, and take the maximum/minimum as start for the Newton method.