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 $F(y) = \int_{x1}^{x_n} \sqrt{1+y'^2} dx + \lambda \frac{ y''} { (1+y'^2)^{3/2}}$.
Doubts:
(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 $P$?