[3] H. Plantinga and C. R. Dyer, Real-time Hidden-line Elimination for a Rotating Polyhedral Scene Using the Aspect Representation, Proc. of Graphics Interface 90, 9{16. 12 Figure 2: Worst Case Number of Changes for Visibility Graph 4 Concluding Remarks

First we consider the case when each of the polygons has piecewise linear motions. Suppose each vertex has m motion pieces within each of which it moves in a xed direction at a constant speed. It is simple to show that both Algorithms I and II can work with a simple modication. Complexities of the algorithms are the same except that quantity n is to be replaced by nm. If the motion of each vertex f (t) is polynomial of higher degree, then two trajectories in (r; ; t) space may intersect more than twice. Let s be the maximum number of intersections between any two curves. Noting that construction of the arrangement is the dominant factor in Algorithm II, its running time is O(n 2 s+2 (n)). Thus, it does not change asymptotic computation time very much. 11 For problem (a), we start following along the curve segment corresponding to the trajec-tory of a certain vertex, say V i. Since the vertical coordinate indicates time, the sequence of t intervals during which V i is visible in the planar projection represents the time intervals during which V i is visible from the observer. The complexity of the zone of a curve in an arrangement is O(n s+2 (n)), where a zone of a curve is the collection of faces crossed by the curve [2]. For problem (b), we consider a vertical line = 0. All curves intersected by this vertical line provides information about objects visible from direction 0 at various times. This can be done by tracing the boundaries of faces intersected by the vertical line. Problem (c) can be resolved in a similar manner by considering a horizontal line. Thus, all queries of (a-c) can be answered in near linear time. If problem (a) were to be solved for all vertices in the polygon, then it would take close to quadratic time. This compares with our earlier results of O(n 2 log n). In fact, this bound can be shown to be worst case near-optimal. Figure 2 contains an example in which O(n 2) intervals actually occur. A total of n line segments are divided into three groups, A, B, and C each of size n 3. Line segments of Group A and C are arranged such that each vertex of group A and each vertex of group C are mutually visible. Suppose that the observer is …