Further results on the minimum bi-path length for rigid objects with dual steering. Solution to a problem posed by Ulam

The optimal motion of a rigid segment in the plane is determined for starting from some giving position and orientation and terminating at a prespecified position and orientation. Optimality is with respect to the sum of the distances traveled by the endpoints. As such, the solution is relevant for applications in optimal path planning for certain robotic vehicles. A new proof and geometric characterization is given. It adds to a growing repertoire of optimal path planning curves, and solves a problem posed by Ulam in 1960.