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Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Is Can we have a polynomial spline (e.g natural cubic spline) passing through all these points, such that: (a) successive segments of the spline have are continuous and have equal 1st & 2nd derivative at the meeting point (Eg . If $S_1(x)$ joins $p_1-p_2$ and $S_2(x)$ joins $p_2-p_3$, then $S_1''(x_2)=S_2''(x_2)$.) and (b) the curvature of the spline is bounded above by $\rho$?

Note that Natural polynomial splines obey (a) but it's hard to say anything about (b). I am also unaware of any means to bound the curvature of a spline, and a literature search online didn't turn up much of interest.

Here are 2 other variations of the question above that I am unable to answer: (V1) If the spline needs to be closed, i.e. $p_{n+1}=p_1$, how, if at all,does the answer change? (V2) If we allow any type of interpolation spline at all that obeys (a) and (b), do we have a solution?

FYI, this isn't a homework problem. I ran into this question when trying to write code for an Engineering application.

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Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Is a polynomial spline (e.g natural cubic spline) passing through all these points, such that: (a) successive segments of the spline have the same second are continuous and have equal 1st & 2nd derivative at the meeting point (Eg . If $S_1(x)$ joins $p_1-p_2$ and $S_2(x)$ joins $p_2-p_3$, then $S_1''(x_2)=S_2''(x_2)$.) and (b) the curvature of the spline is bounded above by $\rho$?
Here are 2 other variations of the question above that I am unable to answer: (V1) If the spline needs to be closed, i.e. $p_{n+1}=p_1$, how, if at all,does the answer change? (V2) If we allow any type of interpolation spline at all that obeys (a) and (b), do we have a solution?