TY - GEN

T1 - Eliminating depth cycles among triangles in three dimensions

AU - Aronov, Boris

AU - Miller, Edward Y.

AU - Sharir, Micha

N1 - Publisher Copyright:
Copyright © by SIAM.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

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U2 - 10.1137/1.9781611974782.164

DO - 10.1137/1.9781611974782.164

M3 - Conference contribution

AN - SCOPUS:85016228317

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 2476

EP - 2494

BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

A2 - Klein, Philip N.

PB - Association for Computing Machinery

T2 - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

Y2 - 16 January 2017 through 19 January 2017

ER -