  Despite the fact that the physics of the cosmic microwave background
anisotropies is most naturally expressed in Fourier space, pixelised maps are
almost always used in the analysis and simulation of microwave data. A
complementary approach is investigated here, in which maps are used only in the
visualisation of the data, and the temperature anisotropies and polarization
are only ever expressed in terms of their spherical multipoles. This approach
has a number of advantages: there is no information loss (assuming a
band-limited observation); deconvolution of asymmetric beam profiles and the
temporal response of the instrument are naturally included; correlated noise
can easily be taken into account, removing the need for additional destriping;
polarization is also analysed in the same framework; and reliable estimates of
the spherical multipoles of the sky and their errors are obtained directly for
subsequent component separation and power spectrum estimation. The formalism
required to analyse experiments which survey the full sky by scanning on
circles is derived here, with particular emphasis on the Planck mission. A
number of analytical results are obtained in the limit of simple scanning
strategies. Although there are non-trivial computational obstacles to be
overcome before the techniques described here can be implemented at high
resolution, if these can be overcome the method should allow for a more robust
return from the next generation of full-sky microwave background experiments.
