Abstract
We show that any locally-fat (or α, β-covered) polyhedron with convex fat faces can be decomposed into O(n) tetrahedra, where n is the number of vertices of the polyhedron. We also show that the restriction that the faces are fat is necessary: there are locally-fat polyhedra with non-fat faces that require Ω(n 2) pieces in any convex decomposition. Furthermore, we show that if we want the polyhedra in the decomposition to be fat themselves, then the worst-case number of tetrahedra cannot be bounded as a function of n. Finally, we obtain several results on the problem where we want to only cover the boundary of the polyhedron, and not its entire interior.
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This research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.
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de Berg, M., Gray, C. (2008). Decompositions and Boundary Coverings of Non-convex Fat Polyhedra. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_15
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DOI: https://doi.org/10.1007/978-3-540-87744-8_15
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