Eliminating Depth Cycles among Triangles in Three Dimensions

TL;DR

This paper presents a near-optimal method to eliminate depth cycles among triangles in 3D space by cutting them into a subquadratic number of pieces, enabling a proper partial order in their depth relation.

Contribution

It introduces a novel algebraic and topological approach using polynomial partitioning to cut triangles into nearly minimal pieces, solving a longstanding open problem in computational geometry.

Findings

Cuts triangles into O(n^{3/2+ε}) pieces for any ε>0

Ensures the depth relation among pieces is acyclic

Nearly tight bound in the worst case

Abstract

Given $n$ pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any $\varepsilon>0$, the triangles can be cut into $O(n^{3/2+\varepsilon})$ connected semi-algebraic pieces, whose description complexity depends only on the choice of $\varepsilon$, 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, in this full generality, with a subquadratic bound on the number of pieces. This work extends the recent study by two of the authors (Aronov, Sharir~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, 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.