### Abstract

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(n^{3=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 language | English (US) |
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Title of host publication | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |

Editors | Philip N. Klein |

Publisher | Association for Computing Machinery |

Pages | 2476-2494 |

Number of pages | 19 |

ISBN (Electronic) | 9781611974782 |

State | Published - Jan 1 2017 |

Event | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain Duration: Jan 16 2017 → Jan 19 2017 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |
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Country | Spain |

City | Barcelona |

Period | 1/16/17 → 1/19/17 |

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### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017*(pp. 2476-2494). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms). Association for Computing Machinery.