I'm having trouble recasting the following question in a form amenable to the calculus of variations.

Question: Given a set of $n$ points $P=\{(x_1,y_1),..(x_n,y_n)\}$ what is the curve passing through these points such that:

(C1) its length is minimal

(C2) the maximum curvature it takes on, as it interpolates the above points, is as small as possible?

(Note: this is related to my earlier MO question here , I wasn't sure if this was good enough for MO.)

My partial answer: Suppose we express the curve in the form $y=f(x)$; suppose further that $L(y)$ and $\kappa(y)$ denote length and curvature as a function of $y$. Then, since we need to reduce both length and curvature, we need to minimize $L(y) + \lambda \kappa(y)$ over the interval $[x_1,x_n]$. This reduces to minimizing $F(y) = \int_{x1}^{x_n} \sqrt{1+y'^2} dx + \lambda \frac{ y''} { (1+y'^2)^{3/2}}$.


(1) Is $F(y)$ the correct functional? It seems not.

(2) How do I alter $F(y)$ to ensure that it passes through all points in $P$?

  • 1
    $\begingroup$ Your $F$ should be a function of the curve you choose. Hence $F$ is a function of $y$. Your problem should read minimize $F(y) = \displaystyle \int_{x_1}^{x_n} \sqrt{1+y'^2} dx + \lambda \sup_{x \in [x_1,x_n]} \left |\frac{y''}{(1+y'^2)^{3/2}} \right|$ $\endgroup$ – user17762 Feb 22 '11 at 3:04
  • $\begingroup$ @ Sivaram: thanks, corrected the $F(y)$ part $\endgroup$ – Ganesh Feb 22 '11 at 3:08
  • $\begingroup$ @Ganesh: To ensure that $y$ passes through your points, simply impose constraints: $y(x_k) = y_k$ for all $k$. Also shouldn't the curvature only concern the interpolation points? Something like $\lambda \sum_k (y''(x_k))^2$ in your objective? $\endgroup$ – Dominique Oct 28 '11 at 17:56

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