  We obtain, by extensive direct numerical simulations, trajectories of heavy
inertial particles in two-dimensional, statistically steady, homogeneous, and
isotropic turbulent flows, with friction. We show that the probability
distribution function $\mathcal{P}(\kappa)$, of the trajectory curvature
$\kappa$, is such that, as $\kappa \to \infty$, $\mathcal{P}(\kappa) \sim
\kappa^{-h_{\rm r}}$, with $h_{\rm r} = 2.07 \pm 0.09$. The exponent $h_{\rm
r}$ is universal, insofar as it is independent of the Stokes number ($\rm{St}$)
and the energy-injection wave number. We show that this exponent lies within
error bars of their counterparts for trajectories of Lagrangian tracers. We
demonstrate that the complexity of heavy-particle trajectories can be
characterized by the number $N_{\rm I}(t,{\rm St})$ of inflection points (up
until time $t$) in the trajectory and $n_{\rm I} ({\rm St}) \equiv
\lim_{t\to\infty} \frac{N_{\rm I} (t,{\rm St})}{t} \sim {\rm St}^{-\Delta}$,
where the exponent $\Delta = 0.33 \pm0.02$ is also universal.
