SOME NEW RESULTS ON MOVING POLYGONS IN THE PLANE.

Let P equals (p//1,p//2,. . . , p//n) and Q equals (q//1,q//2,. . . ,q//m) be two non-intersecting polygons in the plane specified by their cartesian coordinates in order. Given a direction d we can ask whether P can be translated an arbitrary distance in direction d without colliding with Q. An algorithm is presented for answering the above translation query in O(n plus m) time. It is also shown that all the directions of movability (translation) of P with respect to Q can be computed in O(nm) time. For the more general case of a set of M non-intersecting n-gons P equals (P//1,P//2,. . . ,P//M) we say that it exhibits the translation ordering property if for all fixed directions there exists an ordering for translating the polygons by a single common vector without any collisions occurring with those polygons not yet moved. It is shown that for a given collection P, the translation ordering property query can be answered in O(Mn plus M**2log n) time.