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 <italic>n</italic> triangles in three-dimensional space is <italic>O</italic>(<italic>n</italic><supscrpt>2+ε</supscrpt>+<italic>K</italic>), where <italic>K</italic> is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to be <italic>O</italic>(<italic>n</italic><supscrpt>2+ε</supscrpt>). 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 <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt>log<italic>n</italic>+<italic>V</italic>log<italic>n</italic>) time, where <italic>V</italic> 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.

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