Eliminating depth cycles among triangles in three dimensions

Boris Aronov, Edward Y. Miller, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Given n non-vertical pairwise disjoint triangles in 3-space, their vertical depth (above/below) relation may contain cycles. We show that, for any " > 0, the triangles can be cut into O(n3=2+ϵ) pieces, where each piece is a connected semi-algebraic set whose description complexity depends only on the choice of ", such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. We are not aware of any previous study of this problem with a subquadratic bound on the number of pieces. This work extends the recent study by two of the authors 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, which leads to a recursive procedure 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 essentially settles a 35-year-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

Original languageEnglish (US)
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
EditorsPhilip N. Klein
PublisherAssociation for Computing Machinery
Number of pages19
ISBN (Electronic)9781611974782
StatePublished - 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: Jan 16 2017Jan 19 2017

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

ASJC Scopus subject areas

  • Software
  • General Mathematics


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