On the general motion-planning problem with two degrees of freedom

We show that, under reasonable assumptions, any collision-avoiding motion-planning problem for a moving system with two degrees of freedom can be solved in time O(λ_s(n) log²n), where n is the number of collision constraints imposed on the system, s is a fixed parameter depending, e.g., on the maximum algebraic degree of these constraints, and λ_s(n) is the (almost linear) maximum length of (n, s) Davenport-Schinzel sequences. This follows from an upper bound of O(λ_s(n)) that we establish for the combinatorial complexity of a single connected component of the space of all free placements of the moving system. Although our study is motivated by motion planning, it is actually a study of topological, combinatorial, and algorithmic issues involving a single face in an arrangement of curves. Our results thus extend beyond the area of motion planning, and have applications in many other areas.