Lower bounds on moving a ladder in two and three dimensions

It is shown that Ω(n2) distinct moves may be necessary to move a line segment (a “ladder”) in the plane from an initial to a final position in the presence of polygonal obstacles of a total ofn vertices, and that Ω(n4) moves may be necessary for the same problem in three dimensions. These two results establish lower bounds on algorithms that solve the motion-planning problems by listing the moves of the ladder. The best upper bounds known areO(n2 logn) in two dimensions, andO(n5 logn) in three dimensions.