  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.
