Recently much attention has been devoted to the problem of translating a set of geometrical objects in a given direction, one at a time, without allowing collisions between the objects. This paper studies the translation problem in three dimensions on a set of c-oriented faces", that is, the faces whose bounding edges have a constant number c of orientations. We solve the problem in O(N log2 NK) time and O(N log N) space, where N is the total number of edges of the faces and K is the number of edge intersections in the projection plane. As an intermediate step, we also solve a problem related to ray-shooting. The algorithm for translating c-oriented faces finds uses in computer graphic systems.

We examine the problem of reporting all intersecting pairs in a set of c-oriented polygons, each having at most k edges (for some constant k). Polygons are called c-oriented if the edges of all polygons have a constant number of orientations. The problem arises in many applications such as VLSI design rule checking and architecture databases. We present an asymptotically optimal algorithm that reports all pairs in O(n log nt) time and O(n) space, where n is the number of polygons and t the number of intersecting pairs. Since the optimal algorithm may report an intersecting pair more than once (but at most a constant number of times), we also give a further algorithm which reports each pair exactly once but is not space-optimal, namely, requires O(n log n) space.