I have a piecewise linear path in two dimensions (with finitely many pieces). I would like to approximate it by a curve whose radius of curvature is bounded away from 0 (i.e. I specify the bound a priori). Any suggestions for an algorithm?

I don't have any specific metric in mind -- this is in order to produce a nice visualization, not a theorem -- but, for the sake of argument, let's say I want an approximation that minimizes the Skorohod J1 distance from the piecewise linear path.