I've got a 2D cubic Bézier curve which, by construction, is a f(x) function : no loops, a single solution for each defined x.
Is there a common mean to convert this curve to a 3rd degree polynomial ? 3 should be enough since there can only be two "bumps".
Thanks!
On
I assume you want a polynomial function $y=f(x)$ that has the same shape as your original parametric cubic Bezier curve.
Well, you already have $y$ as a function of $x$, but the function is unpleasant. For a given value of $x$, there is no closed form formula for the corresponding $y$ -- you have to find $y$ numerically (by intersecting a vertical line with the given Bezier curve).
Let's denote this unpleasant function by $y=g(x)$. So, now the problem is just to approximate $g$ by a polynomial $f$. There are lots of standard ways to do this, depending on how you want to measure the approximation error and what extra conditions you want $f$ to satisfy. You say you want exact matching of end points and end tangents.
So, you choose a degree $m$ for $f$. As you say, $m$ will have to be bigger than 4. Then $f$ will have $m+1$ coefficients, which you can determine by solving a system of $m+1$ linear equations. You get 4 equations from the end-point constraints. Pick another $m-3$ points $(x_i,y_i)$ in the interior of the curve, and, for each $i$, write down the equation that expresses the requirement that $y_i = f(x_i)$. Solve the system of linear equations.
One subtlety is that the extra $m-3$ points have to be chosen fairly carefully. You can't just distribute them uniformly, or else $f$ might wiggle badly. They have to be more dense towards the ends of the curve, and less dense in the middle. The "Chebyshev nodes" are a good choice.
This process is called Hermite interpolation if you use derivative information, and Lagrange interpolation if you don't. Both are covered in Wikipedia articles.
Well, it turns out it's not really possible.
The following Processing code shows my results for a 4th order polynomial. The equations of the parameters are from Maple.
I'll try with adding the derivative of x2 and adding another point in the middle (with a 5rth order polynomial) but I'm not sure it'll appropriate for my use.