Shortest 3-dimensional paths with a prescribed curvature bound

We present the solution of the three-dimensional case of a problem regarding the structure of minimum-length paths with a prescribed curvature bound and prescribed initial and terminal positions and directions. In particular, we disprove a conjecture, according to which every minimizer is a concatenation of circles and straight lines. We show that there are many minimizers-the “helicoidal arcs”-that are not of this form. These arcs are smooth and are characterized by the fact that their torsion satisfies a second-order ordinary differential equation. The solution is obtained by applying optimal control theory. An essential feature of the problem is that it requires the use of optimal control on manifolds. The natural state space of the problem is the product of three-dimensional Euclidean space and a two-dimensional sphere. Although the problem is obviously embeddable in 6-dimensional Euclidean space, the maximum principle for the embedded problem yields no information, whereas a careful application of the maximum principle on manifolds yields a very strong result, namely, that every minimizer is either a helicoidal arc or of the form C, S, CS, SC, CSC, CCC, where C, S stand for “circle” and “segment”, respectively