Constructing the Exact Voronoi Diagram of Arbitrary Lines in Three-Dimensional Space - with Fast Poi

We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires $O(n^{3+\varepsilon})$ time and space, where $n$ is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in $O(\log^2 n)$ expected time. The implementation employs the CGAL packages for constructing arrangements and lower envelopes on parametric surfaces together with advanced algebraic tools. Supplementary material and in particular the prototypical code of our implementation can be found in the website: http://acg.cs.tau.ac.il/projects/internal-projects/3d-lines-vor/project-page.

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