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

Leonidas J. Guibas, Micha Sharir, Shmuel Sifrony

Research output: Contribution to journalArticlepeer-review


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(λs(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 λs(n) is the (almost linear) maximum length of (n, s) Davenport-Schinzel sequences. This follows from an upper bound of O(λs(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.

Original languageEnglish
Pages (from-to)491-521
Number of pages31
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Dec 1989


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