Translating polygons with applications to hidden surface removal

Abstract

Let S be a set of polygons in the plane with a total number of n vertices. A translation ordering for S (in direction d) is an ordering of the polygons such that, if the polygons are moved one by one to infinity in direction d according to this ordering, no collisions occur between the polygons. We show that, after O(n log n) preprocessing using 0(n) space, it is possible to determine, for any given d, in 0(log n) time whether such an ordering exists and, if so, to compute an ordering in O(n) time. Translation orderings correspond to valid orderings for hidden surface removal schemes where objects that are closer to the viewpoint are displayed later than objects that are farther away. Thus our technique can be used to generate displaying orderings for polyhedral terrains. One of the main advantages of our approach is that it can easily be adapted to handle perspective views within the same time and space bounds.