Given two curves, one might want to find the minimum distance between two points. It is fairly straightforward to find minimums of the function
$$(x_1-x_2)^2+(y_1-y_2)^2$$
which corresponds to the square of the distance between two points on the curves.
This looks like a variational problem. So I was wondering if the could the Euler-Lagrange equation and similar techniques can be used to solve a problem like this?
If so, how?
I haven't been able to find any such solution looking on google. I've searched for "Euler-Lagrange equation minimum distance between curves" and the like.
Any ideas?
Thanks.