On the Complexity of Visibility Problems with Moving Viewpoints

Summary

We investigate visibility problems with moving viewpoints in n-dimensional space. We show that these problems are NP-hard if the underlying bodies are balls, H-polytopes, or ν-polytopes. This is contrasted by polynomial time solvability results for fixed dimension. We relate the computational complexity to existing algebraic-geometric aspects of the visibility problems, to the theory of packing and covering, and to the view obstruction problem from diophantine approximation.

1. Introduction

Computer graphics and visualization deal with preparing data in order to show (“visualize“) these data on a (two-dimensional) computer screen. In computer graphics, the original data typically stem from the three-dimensional Euclidean space R3, whereas in scientific visualization the data might originate from spaces of much higher dimension (e.g., in information visualization or high-dimensional sphere models in statistical mechanics) [Swayne et al. 1998].

In these scenarios, visibility computations play a central role [O'Rourke 1997]. In the simplest case, we are given a fixed viewpoint υ ∈ ℝ ⁿ, and the scene consists of a set B of bodies. Now the task is to compute a suitable two-dimensional projection of the scene (“to render the scene“) that reflects which part of the scene is visible from the viewpoint υ;. Ina more dynamic setting, the viewpoint can be moved interactively (see [Bern et al. 1994; Lenhof and Smid 1995], for example). However, in general, after each movement of the viewpoint a new rendering process is necessary. In order to speed up this process, commercial Tenderers apply caching techniques [Wernecke 1994].