Eliminating Depth Cycles Among Triangles in Three Dimensions
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Eliminating Depth Cycles Among Triangles in Three Dimensions Boris Aronov1
· Edward Y. Miller2 · Micha Sharir3
Received: 7 March 2019 / Revised: 7 April 2020 / Accepted: 28 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any ε > 0, the triangles can be cut into O(n 3/2+ε ) connected semialgebraic pieces, whose description complexity depends only on the choice of ε, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. The pieces can be constructed efficiently. This work extends the recent study by two of the authors (Discrete Comput. Geom. 59(3), 725–741 (2018)) on eliminating depth cycles among lines in 3-space. Our approach is again algebraic, and makes use of a recent variant of the polynomial partitioning technique, due to Guth (Math. Proc. Camb. Philos. Soc. 159(3), 459–469 (2015)), which leads to a recursive algorithm for cutting the triangles. In contrast to the case of lines, our analysis here is considerably more involved, due to the two-dimensional nature of the objects being cut, so additional tools, from topology and algebra, need to be brought to bear. Our result makes significant progress towards resolving a decades-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics. In addition, we generalize our bound to well-behaved patches of two-dimensional algebraic surfaces of constant degree. To Ricky, a mathematician, an inspiration, a friend, who brought us all together
Work on this paper by B.A. has been partially supported by NSF Grants CCF-11-17336, CCF-12-18791, and CCF-15-40656, and by BSF grant 2014/170. Work by M.S. has been supported by Grant 2012/229 from the U.S.–Israel Binational Science Foundation, by Grants 892/13 and 260/18 from the Israel Science Foundation, by the Israeli Centers for Research Excellence (I-CORE) program (center no. 4/11), by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University, and by Grant G-1367-407.6/2016 from the German–Israeli Foundation for Scientific Research and Development. An earlier version of this work appeared in SODA’17 [6]. Extended author information available on the last page of the article
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Discrete & Computational Geometry
Keywords Depth order · Depth cycles · Cycle elimination · Painter’s algorithm · Algebraic methods in combinatorial geometry · Polynomial partitioning Mathematics Subject Classification 68U05 · 52C99 · 52C35 We are honored to have been invited to contribute this paper to this special issue of Discrete & Computational Geometry dedicated to the memory of Ricky Pollack. Ricky had been a great friend of ours—in fact a friend of the entire communities of discrete and computational geometry. Together with his long-time associate, Eli Goodman, they have fostered and nurtured both communities, bringing them together in a series of major confe
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