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The unprecedented quality of the observations available from the AtacamaLarge Millimetre/sub-millimetre Array (ALMA) calls for analysis methods makingthe best of them. Reconstructing in space the morphology and kinematics ofradio sources is an underdetermined problem that requires imposing additionalconstraints for its solution. The hypothesis of rotational invariance about awell-defined star axis, which is a good approximation to the description of thegas envelopes of many evolved stars and protostars, is particularly efficientin this role. In the first part of the article, a systematic use of simulatedobservations allows for identifying the main problems and for constructingquantities aimed at solving them. In particular the evaluation of theorientation of the star axis in space and the differentiation between expansionalong the star axis and rotation about it are given special attention. The useof polar rather than Cartesian sky coordinates is shown to better match themorphology and kinematics of actual stars. The radial dependence of the gasdensity and temperature and the possible presence of velocity gradients arebriefly considered. In the second part, the results obtained in the first partare applied to a few stars taken as examples with the aim of evaluating theirusefulness when applied to concrete cases. A third part takes stock of whatprecedes and formulates some guidelines for modelling the radio emission ofaxisymmetric radio sources, limited however to the mathematics and geometry ofthe problem, physics considerations being ignored.

Space reconstruction of the morphology and kinematics of axisymmetric radio sources