On the Number of Views of Polyhedral Scenes

Abstract

It is known that a scene consisting of k convex polyhedra of total complexity n has at most O(n ⁴ k ²) distinct orthographic views, and that the number of such views is Ω((nk ²+n ²)²) in the most case. The corresponding bounds for perspective views are O(n ⁶ k ³) and Ω((nk ²+n ²)³), respectively. In this papers, we close these gaps by improving the lower bounds. We construct an example of a scene θ(n ⁴ k ²) orthographic views, and another with θ(n ⁶ k ³) 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.