Before revision:  and the fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments , i. e., into Jordan arcs, each pair of which intersect at most once^3, where we show a nearly-optimal algorithm to cut n pairwise disjoint non-vertical triangles in \mathbb{R}^3 into pieces that form a depth order} .
After revision:  and the fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments . The fifth application is for eliminating depth cycles among triangles in \mathbb{R^3, where we show a nearly-optimal algorithm to cut n pairwise disjoint non-vertical triangles in \mathbb{R}^3 into pieces that form a depth order} .