A modeldependent amplitude analysis of $B^{\pm} \rightarrow D K^{\pm}$ with $D \rightarrow K_S^0 \pi^+ \pi^$ decays is performed using protonproton collision data, corresponding to an integrated luminosity of $1$ fb$^{1}$, recorded by LHCb at a centreofmass energy of $7$ TeV in $2011$. Values of the $CP$ violation observables $x_{\pm}$ and $y_{\pm}$, which are sensitive to the CKM angle $\gamma$, are measured to be \begin{align*} x_ &= +0.027 \pm 0.044 ^{+0.010}_{0.008} \pm 0.001, \\ y_ &= +0.013 \pm 0.048 ^{+0.009}_{0.007} \pm 0.003, \\ x_+ &= 0.084 \pm 0.045 \pm 0.009 \pm 0.005, \\ y_+ &= 0.032 \pm 0.048 ^{+0.010}_{0.009} \pm 0.008, \end{align*} where the first uncertainty is statistical, the second systematic and the third arises from the uncertainty of the $D \rightarrow K_S^0 \pi^+ \pi^$ amplitude model. The value of $\gamma$ is determined to be $(84^{+49}_{42})^\circ$, including all sources of uncertainty. Neutral $D$ meson mixing is found to have negligible effect.
Invariant mass distributions for (a) $ B ^\pm \rightarrow D (\rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^ ) K ^\pm $ {\it long}, (b) $ B ^\pm \rightarrow D (\rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^ ) \pi ^\pm $ {\it long}, (c) $ B ^\pm \rightarrow D (\rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^ ) K ^\pm $ {\it downstream} and (d) $ B ^\pm \rightarrow D (\rightarrow K ^0_{\rm\scriptscriptstyle S} \pi ^+ \pi ^ ) \pi ^\pm $ {\it downstream} candidates. The fit results, including signal and background components, are superimposed. The lower plots are normalised residual distributions. 
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Dalitz plot and its projections, with fit result superimposed, for $B^ \rightarrow D \pi^$ candidates; $m_{\pm}^2 \equiv m_{ K ^0_{\rm\scriptscriptstyle S} \pi ^\pm }^2$ and $m_{0}^2 \equiv m_{\pi ^+ \pi ^ }^2$. The lower parts of the figures are normalised residual distributions. 
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Dalitz plot and its projections, with fit result superimposed, for $B^+ \rightarrow D \pi^+$ candidates; $m_{\pm}^2 \equiv m_{ K ^0_{\rm\scriptscriptstyle S} \pi ^\pm }^2$ and $m_{0}^2 \equiv m_{\pi ^+ \pi ^ }^2$. The lower parts of the figures are normalised residual distributions. 
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Dalitz plot and its projections, with fit result superimposed, for $B^ \rightarrow D K^$ candidates; $m_{\pm}^2 \equiv m_{ K ^0_{\rm\scriptscriptstyle S} \pi ^\pm }^2$ and $m_{0}^2 \equiv m_{\pi ^+ \pi ^ }^2$. The lower parts of the figures are normalised residual distributions. 
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Dalitz plot and its projections, with fit result superimposed, for $B^+ \rightarrow D K^+$ candidates; $m_{\pm}^2 \equiv m_{ K ^0_{\rm\scriptscriptstyle S} \pi ^\pm }^2$ and $m_{0}^2 \equiv m_{\pi ^+ \pi ^ }^2$. The lower parts of the figures are normalised residual distributions. 
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Likelihood contours at 39.35%, 86.47%, 98.89% and 99.97% confidence level for $(x_{+},y_{+})$ (blue) and $(x_{},y_{})$ (red). 
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Projections of the pvalue regions onto the $(\gamma, \delta_{ B })$ and $(\gamma, r_{ B })$ planes with all sources of uncertainty taken into account. 
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Animated gif made out of all figures. 
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Signal and background yields for components contributing to the $ C P$ asymmetry fit, in the region $\pm 50 {\mathrm{ Me V /}c^2} $ around the known $ B ^\pm$ meson mass. 
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Absolute values of systematic uncertainties. The $ C P$ asymmetry fit bias is considered as a onesided uncertainty and is included in the quadrature sum on that side only. 
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Model related systematic uncertainties for each alternative model. The relative signs indicate full correlation or anticorrelation. 
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Supplementary material full pdf 
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This ZIP file contains supplementary material for the publication LHCbPAPER2014017. The files are: Supplementary.pdf : An overview of the extra figures (colour versions of Figure 7 of the paper) *.pdf, *.png, *.eps, *.C : The figures in various formats 
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Created on 16 August 2019.Citation count from INSPIRE on 20 August 2019.