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I'm looking for an interpolating polynomial curve $p:\mathbb{R} \longrightarrow \mathbb{R}^2$ of order $m$ such that it passes for the points $x_0,x_1,x_2,\dots,x_{n+1}$ with n+1< m. I have to find the times $t_1,t_2,\dots,t_n$ such that $p(t_1)=x_1,\dots,p(t_n)=x_n$ in order to minimize $$ \int_{0}^{T} \|\dot p\|^2 dt $$ where $p(0)=x_0$ and $p(T)=x_{n+1}$ and $0\le t_1 \le t_2 \le \dots \le t_n \le T$.

Can anyone giveme some information or literature about this kind of problem. Does this kind of problem have a name? If the times are fixed the problem becomes a constrained QP. In my case, it appears as a non linear problem. I'm concerned because it seems to be an useful problem in computer graphics and other areas where we want to find a smooth curve passing through points, but I have not found any information about this.

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