Eliminating Depth Cycles Among Triangles in Three Dimensions

Abstract

The vertical depth relation among n pairwise openly disjoint triangles in 3-space may contain cycles. We show that, for any \(\varepsilon >0\), the triangles can be cut into \(O(n^{3/2+\varepsilon })\) connected semialgebraic 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. 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.