  We report a measurement of ${D^{0}}$-${\overline{D}{}^{0}}$ mixing parameters
and a search for indirect $CP$ violation through a time-dependent
  amplitude analysis of ${D^0\to K_S^0\pi^+\pi^-}$ decays. The results are
based on 921~fb$^{-1}$ of data accumulated with the Belle detector at the KEKB
asymmetric-energy $e^+e^-$ collider. Assuming $CP$ conservation, we measure the
mixing parameters $x=(0.56\pm0.19^{+0.03}_{-0.09}{^{+0.06}_{-0.09}})\%$ and
$y=(0.30\pm0.15^{+0.04}_{-0.05}{^{+0.03}_{-0.06})}\%$, where the errors are
statistical, experimental systematic, and systematic due to the amplitude
model, respectively. With $CP$ violation allowed, the parameters
$|q/p|=0.90^{+0.16}_{-0.15}{^{+0.05}_{-0.04}}{^{+0.06}_{-0.05}}$ and
$\arg(q/p)=(-6\pm11{\pm3}{^{+3}_{-4}})^{\circ}$ are found to be consistent with
conservation of $CP$ symmetry in mixing and in the interference between mixing
and decay, respectively.
