Consider a robot R that is either a line segment or the Minkowski sum of a line segment and a 3-ball, and a set S of polyhedral obstacles with a total of n vertices in R 3. We design near-optimal exact algorithms for planning the motion of R among S when R is allowed to translate and rotate. Specifically, we can preprocess S in time O(n 4+ε) for any ε> 0 into a data structure that given two placements α and β of R, can decide in time O(log n) whether a collision-free rigid motion of R between α and β exists and if so, output such a motion in time asymptotically proportional to its complexity. Furthermore, we can find in time O(n 4+ε) for any ε> 0 the largest placement of a similar (translated, rotated and scaled) copy of R that does not intersect S. A number of additional stronger results are provided. Our line segment motion planning algorithm improves the result of Ke and O’Rourke by two orders of magnitude and almost matches their lower bound, thus settling a classical motion planning problem first considered by Schwartz and Sharir in 1984. This implies a number of natural directions for future work concerning rigid motion planning in three dimensions.

Given k finite point sets A1,..., Ak in R2, we are inter-ested in finding one translation for each point set such that the union of the translated point sets is in convex position. We show that if k is part of the input, then it is NP-hard to determine if such translations exist, even when each point set has at most three points. The original motivation of this problem comes from the question of whether a given triangulation T of a point set is the empty shape triangulation with respect to some (strictly convex) shape S. In other words, we want to find a shape S such that the triangles of T are precisely those triangles about which we can circum-scribe a homothetic copy of S that does not contain any other vertices of T. This is the Delaunay criterion with respect to S; for the usual Delaunay triangulation, S is the circle. 1