  In this paper, we consider a transformation of $k$ disjoint paths in a graph.
For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$
connecting the same set of terminal pairs, we aim to determine whether
$\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one
path with another path so that the intermediates are also $k$ disjoint paths.
The problem is called Disjoint Paths Reconfiguration. We first show that
Disjoint Paths Reconfiguration is PSPACE-complete even when $k=2$. On the other
hand, we prove that, when the graph is embedded on a plane and all paths in
$\mathcal{P}$ and $\mathcal{Q}$ connect the boundaries of two faces, Disjoint
Paths Reconfiguration can be solved in polynomial time. The algorithm is based
on a topological characterization for rerouting curves on a plane using the
algebraic intersection number. We also consider a transformation of disjoint
$s$-$t$ paths as a variant. We show that the disjoint $s$-$t$ paths
reconfiguration problem in planar graphs can be determined in polynomial time,
while the problem is PSPACE-complete in general.
