  Let $\Gamma$ be a finite set of Jordan curves in the plane. For any curve
$\gamma \in \Gamma$, we denote the bounded region enclosed by $\gamma$ as
$\tilde{\gamma}$. We say that $\Gamma$ is a non-piercing family if for any two
curves $\alpha , \beta \in \Gamma$, $\tilde{\alpha} \setminus \tilde{\beta}$ is
a connected region. A non-piercing family of curves generalizes a family of
$2$-intersecting curves in which each pair of curves intersect in at most two
points. Snoeyink and Hershberger (``Sweeping Arrangements of Curves'', SoCG
'89) proved that if we are given a family $\mathcal{C}$ of $2$-intersecting
curves and a fixed curve $C\in\mathcal{C}$, then the arrangement can be
\emph{swept} by $C$, i.e., $C$ can be continuously shrunk to any point $p \in
\tilde{C}$ in such a way that the we have a family of $2$-intersecting curves
throughout the process. In this paper, we generalize the result of Snoeyink and
Hershberger to the setting of non-piercing curves. We show that given an
arrangement of non-piercing curves $\Gamma$, and a fixed curve $\gamma\in
\Gamma$, the arrangement can be swept by $\gamma$ so that the arrangement
remains non-piercing throughout the process. We also give a shorter and simpler
proof of the result of Snoeyink and Hershberger and cite applications of their
result, where our result leads to a generalization.
