We present a general formulation of the basic equations of large-scale magnetosphere-ionosphere coupling in terms of Euler potentials and describe a specific numerical implementation of this formulation in the context of the Rice Convection Model (RCM). When written in terms of Euler potentials, both the Vasyliunas magnetosphere-ionosphere coupling equation and the expression for bounce-averaged adiabatic drift assume particularly elegant forms, while the equation for the conservation of ionospheric current is only slightly more complicated to solve than the corresponding formula for a dipole case. For simplicity, large-scale models of convection in the inner magnetosphere have typically assumed strict symmetry between the northern and southern hemispheres, explicitly assuming the internal planetary magnetic field to be a dipole aligned with Earth's rotation axis and oriented perpendicular to the solar wind flow velocity. These approximations have precluded the realistic treatment of ionospheric longitude and seasonal effects as well as dipole-tilt and IMF-By-penetration effects in the magnetosphere. We present a scheme for constructing an Euler-potential-based computational mesh, in which the Euler potential α is set to zero at the dip equator for a reference altitude of 90 km, and ¿ lines in the northern ionosphere follow lines of constant centered dipole magnetic longitude but are spaced equally in terms of total latitude-integrated magnetic flux. Properties of the Euler-potential-based grid are illustrated using an IGRF model for the Earth's internal field. Our procedure yields an Euler-potential-based grid that covers the entire ionosphere, except for the southern polar cap and cusp.