0

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.

DLV
  • 1,740
  • 4
  • 17
  • 28
  • Why not use Lagrange multipliers? One objective function, two constraints. – A.S. Nov 18 '15 at 01:37
  • @A.S. Since $y_1$ is a function of $x_1$ and $y_2$ is a function of $x_2$, I suppose that we could do it without Lagrange multipliers (which for sure would work). – Claude Leibovici Nov 18 '15 at 05:28
  • 1
    @Claude You need Lagrange multipliers if curves are defined implicitly (which is what I think when I see "curves". That, or explicit parameterization which indeed doesn't require Lagrange multipliers). – A.S. Nov 18 '15 at 05:35
  • @A.S. You are correct, indeed. I supposed, as an idiot, that the functions were explicit. Sorry for that !! Cheers. – Claude Leibovici Nov 18 '15 at 05:37

0 Answers0