Thickening Dubins Curves

Using optimal control theory, the Markov-Dubins problem of finding a shortest path between two points in the plane with given initial and final tangent direction and an upper bound on curvature has been thoroughly solved. Minimizers for that problem have a simple form consisting only of straight line segments and circular arcs. Naturally, such curves also occur as strongly critical curves for the ropelength problem, if we neglect critical self-distance. Criticality was shown before by analyzing possible kink tension functions, but a more direct way to show criticality of minimizers would be to prove that they are thickness regular, which means that there is a variation vector field in space around the curve in the direction of which the right-derivative of the Thi_∞-functional is positive. But it turns out that this is not always possible.