Before revision:  the third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in \mathbb{R}^{d} in O(\log^2 n) time ; 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:  the third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in \mathbb{R}^{d} in O(\log^2 n) time , with O(n^{d+\epsilon 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} .