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Let S be a set of n (possibly intersecting) line segments in the plane. We show that the arrangement of S can be stored implicitly into a data structure of size O(n log2n) so that the following query can be answered in time O(n1/2 log2n): Given two query points, determine whether they lie in the same face of the arrangement of S and, if so, return a path between them that lies within the face. This version of the implicit point location problem is motivated by the following motion planning problem: Given a polygonal robot R with m vertices and a planar region bounded by polygonal obstacles with n vertices in total, preprocess them into a data structure so that, given initial and final positions of R, one can quickly determine whether there exists a continuous collision-free translationsal motion of R from the initial to the final position. We show that such a query can be answered in time O((mn)1/2 log2mn) using O(mn log2mn) storage.

Implicit point location in arrangements of line segments, with an application to motion planning