On the Number of Critical Free Contacts of a Convex Polygonal Object Moving in Two-Dimensional Polygonal Space

Given a convex polygonal robot B with k edges capable of translation and rotation, and a set of polygonal obstacles {A1, . . . , Am} each with ni sides and n = ∑ m i=1 ni, Leven and Sharir [1] prove that the number of critical placements of B at which it makes 3 simultaneous contacts with the obstacles but does not intersect their interior is O(knλ6(kn)), where λs(u) is the maximum length of a Davenport-Schinzel sequence of u symbols of order s. The main idea of their proof technique is to transform this 3-dimensional (since both translation and rotation are allowed) problem into a set of 2-dimensional problems involving the complexity of lower envelopes of certain functions. They then use the results on the complexity of lower envelopes to establish this bound. In this talk I will mainly focus on this transformation and the associated case analysis for the proof of the above bound.