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Consider the following polynomial in Bezier form:

$$p(u)=\sum_{i=0}^n b_i B_i^n \Big(\frac{u-a}{b-a}\Big)$$

I am supposed to derive Bezier points $b_i$ for $i=0,1,...,n$ in general form, assuming that I know the values of the function, first derivatives and second derivatives at $a$ and $b$. I have tried using the recursive definition for the derivatives of Bezier polynomials but can't get anywhere.

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You have six items of data (value, and first and second derivatives at each end). Therefore, you can calculate values for at most 6 of the control points $b_i$. So, what you're trying to do is possible only if $n \le 5$.

So, assuming $n = 5$, you just write down equations that express the various derivatives in terms of $b_0, \ldots, b_5$. So, for example \begin{align} p(a) &= b_0 \\ p(b) &= b_5 \\ p'(a) &= \frac{5}{b-a}(b_1 - b_0) \end{align} and so on. The left-hand sides are all known quantities, so you can solve these equations to get $b_0, \ldots, b_5$.

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