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 ${D^0to K_S^0pi^+pi^-}$ 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 $x=(0.56pm0.19^{+0.03}_{-0.09}{^{+0.06}_{-0.09}})%$ and $y=(0.30pm0.15^{+0.04}_{-0.05}{^{+0.03}_{-0.06})}%$, 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 $arg(q/p)=(-6pm11{pm3}{^{+3}_{-4}})^{circ}$ are found to be consistent with conservation of $CP$ symmetry in mixing and in the interference between mixing and decay, respectively.
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