Minimizing the trace length of a rod endpoint in the presence of polygonal obstacles is NP-hard

We continue, and in a sense complete, our study the motion of a rod (line segment) in the plane in the presence of polygonal obstacles, under an optimality criterion based on minimizing the trace length of a fixed but arbitrary point (called the focus) on the rod. In an earlier paper, we showed that this problem is NP-hard when the focus is in the relative interior of the rod. Our proof did not cover the case where the focus lies at one of the rod endpoints. Indeed, considerable evidence suggested that this special case might admit a polynomial time solution. In this paper we settle this open problem by proving, by means of a non-trivial adaptation of our earlier construction, that this remaining case is also NP-hard.