We prove that the number of tangencies between the members of two families, each of which consists of $n$ pairwise disjoint curves, can be as large as $\Omega(n^{4/3})$. If the families are doubly-grounded, this is sharp. We also show that if the curves are required to be $x$-monotone, then the maximum number of tangencies is $\Theta(n\log n)$, which improves a result by Pach, Suk, and Treml.

翻译：我们证明,两个家庭的成员之间的时间差,每个家庭由双向脱节曲线组成,其数额可能与Omega(n ⁇ 4/3})美元一样大。如果这些家庭有双倍的缘故,情况就非常明显。我们还表明,如果曲线需要x美元,那么最多的时间差是$Theta(n\log n)美元,这改善了Pach、Suk和Treml的结果。