  We show that the two-dimensional structure of a rigidly rotating
self-gravitating body is accessible with relatively good precision by assuming
a purely spheroidal stratification. With this hypothesis, the two-dimensional
problem becomes one-dimensional, and consists in solving two coupled
fixed-point equations in terms of equatorial mass density and eccentricity of
isopycnics. We propose a simple algorithm of resolution based on the
self-consistent field method. Compared to the full unconstrained-surface
two-dimensional problem, the precision in the normalized enthalpy field is
better than $10^{-3}$ in absolute, and the computing time is drastically
reduced. In addition, this one-dimensional approach is fully appropriate to
fast rotators, works for any density profile (including any barotropic equation
of state), and can account for mass density jumps in the system, including the
existence of an ambient pressure. Several tests are given.
