Is it possible to find the equation of the curve, $y=f(x)$, that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),\dots$ , and$(x_n,y_n)$ so that for any $j \in [1,n)$, the length of the curve between $(x_j,y_j)$ and $(x_{j+1},y_{j+1})$ is as small as possible?
That is; the curve is almost a line between $(x_j,y_j)$ and $(x_{j+1},y_{j+1})$. In other words, we need $f(x)$ such that $f'(x_j<x<x_{j+1})=c$, where $c$ is a constant.
What I know is only joining the points, (using Lagrange's polynomial method), but that does not satisfy the condition that we require (to minimise the length of the curve).
I believe it is possible using infinite series as Fourier series or other, but I am not sure.
Any help would be really appreciated. THANKS!