Vertical decompositions for triangles in 3-space

We prove that, for any constant ε > 0, the complexity of the vertical decomposition of a set of n triangles in three-dimensional space is O(n^2+ε + K), where K is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to be O(n^2+ε). These bounds are almost tight in the worst case. We also give a deterministic output-sensitive algorithm for computing the vertical decomposition that runs in O(n² log n + V log n) time, where V is the complexity of the decomposition. The algorithm is reasonably simple (in particular, it tries to perform as much of the computation in two-dimensional spaces as possible) and thus is a good candidate for efficient implementations.