Triangles in space or building (and analyzing) castles in the air

Abstract

We show that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n 7/3 logn) and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement and an alternative less efficient, but still subcubic algorithm for calculating all non-convex cells, analyze some special cases of the problem where improved bounds (and faster algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

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Work on this paper by the first author has been supported by an AT&T Bell Laboratories PhD Scholarship. Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation, the IBM Corporation, and by a research grant from the NCRD — the Israeli National Council for Research and Development. A preliminary version of this paper has appeared inProc. 4th ACM Symp. on Computational Geometry, Urbana, Illinois, 1988.

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Aronov, B., Sharir, M. Triangles in space or building (and analyzing) castles in the air. Combinatorica 10, 137–173 (1990). https://doi.org/10.1007/BF02123007

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AMS subject classification (1980)

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