  Using hydrodynamical simulations, we explore the use of the mean and
percentiles of the curvature distribution function to recover the equation of
state of the high-$z$ ($2 < z < 4$) intergalactic medium (IGM). We find that
the mean and percentiles of the absolute curvature distribution exhibit tight
correlation with the temperatures measured at respective characteristic
overdensities $\bar{\Delta}_i$'s at each redshift. Hence, they provide
complementary probes of the same underlying temperature-density distribution,
and can in principle be used to simultaneously recover both parameters $T_0$
and $\gamma$ of the IGM effective equation of state. We quantify the associated
errors in the recovered parameters $T_0$ and $\gamma$ from the intrinsic
scatter in the characteristic overdensities and the uncertainties in the
curvature measurement.
