We study the common intersection of arrangements of double-wedges. We consider arrangements where double-wedges may be either bowties (which do not contain a vertical line) or hourglasses (which contain a vertical line), in contrast to earlier studies that focused on arrangements of only bowties. This generalization changes the setting drastically, in particular, with respect to all arguments involving the point-line duality. Namely, a point in the intersection of all double-wedges is equivalent to a line that stabs a set of segments $\mathcal{S}$ (corresponding to the bowties) while it avoids a different set of segments $\mathcal{A}$ (corresponding to the complement of the hourglasses). We show that in this general setting, the intersection of $n$ double-wedges may consist of $Ω(n^2)$ interior-disjoint regions. Further, we discuss Gallai-type results for arrangements of segments and anti-segments, and we provide algorithms for computing the intersection of such arrangements with worst-case optimal running time. Finally, we also prove that we can find a single intersection point in almost optimal running time, assuming that 3SUM admits no truly subquadratic-time algorithm.

翻译：我们研究双楔形体排列的公共交集。与早期仅关注领结形楔形体排列的研究不同，本文考虑双楔形体既可以是领结形（不包含垂直线）也可以是沙漏形（包含垂直线）的排列。这一推广彻底改变了研究背景，尤其是涉及点线对偶性的论证。具体而言，所有双楔形体交集中的一个点等价于一条刺穿线段集$\mathcal{S}$（对应领结形）同时规避另一线段集$\mathcal{A}$（对应沙漏形的补集）的直线。我们证明，在此一般化设定下，$n$个双楔形体的交集可能包含$\Omega(n^2)$个内部不相交的区域。进一步，我们讨论了线段与反线段排列的Gallai型结论，并提出了具有最坏情形最优运行时间的算法来计算此类排列的交集。最后，在假设3SUM问题不存在真正次二次时间算法的前提下，我们证明能够以近乎最优的运行时间找到单个交点。