On the Number of Views of Polyhedral Scenes

It is known that a scene consisting of k convex polyhedra of total complexity n has at most O(n^4 k²) distinct orthographic views, and that the number of such views is Ω ((nk² + n²)²) in the worst case. The corresponding bounds for perspective views are O(n^6 k³) and Ω((nk² + n²)³), respectively. In this paper, we close these gaps by improving the lower bounds. We construct an example of a scene with Θ(n^4 k²) orthographic views, and another with Θ(n^6 k³) perspective views. Our construction can also be used to improve the known lower bounds for the number of silhouette views and for the number of distinct views from a viewpoint moving along a straight line.