Perhaps there is a result along these lines?
Given any set of distint distinct points in the plane, there exists a simple (nonintersecting) path through them in a specified order, with the path composed of smoothly joined arcs of circles of the same radius $r$, where $r$ is some function of the minimum point separation.
This is likely useless for any application, but it might make a nice theorem, especially if the largest $r$ could be achieved or at least approached.
