  In this paper we calculate the fully differential cross sections for
inclusive heavy quark production in deep-inelastic scattering. We construct
proper projection operators to give all possible azimuthal angle distributions
of the heavy quark for unpolarized and longitudinally polarized scatterings.
These projection operators are expressed in terms of momenta of incoming
hadron, virtual photon and detected heavy quark. The azimuthal angle
distributions are calculated to next-to-leading order of $\alpha_s$, i.e.,
$O(\alpha_s^2)$, in a unified way. Analytic expressions of the hard
coefficients are given. Numerical results on future electron-ion colliders are
also given. It is found that at least three azimuthal angle asymmetries can be
more than $1\%$ in typical kinematical regions of these colliders.
