Geometric applications of Davenport-Schinzel sequences

We present efficient algorithms for the following geometric problems: (i) Preprocessing of a 2-D polyhedral terrain so as to support fast ray shooting queries from a fixed point. (ii) Determining whether two disjoint interlocking simple polygons can be separated from one another by a sequence of translations. (iii) Determining whether a given convex polygon can be translated and rotated so as to fit into another given polygonal region. (iv) Motion planning for a convex polygon in the plane amidst polygonal barriers. All our algorithms make use of Davenport Schinzel sequences and on some generalizations of them; these sequences are a powerful combinatorial tool applicable in contexts which involve the calculation of the pointwise maximum or minimum of a collection of functions.