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Hanle-effect observations of forbidden coronal line transitions and recently also longitudinal Zeeman-effect measurements of coronal lines show quantitative signatures of the weak coronal magnetic field. The interpretation of these observations is, however, complicated by the fact that they are the result of line-of-sight integrations through the optically thin corona. We study by means of simulated observations the possibility of applying tomo-graphic techniques in order to reconstruct the 3D magnetic field configuration in the solar corona from these observations. The reconstruction problem relates to a family of similar problems termed vector tomography. It is shown that Zeeman data and Hanle data alone obtained from vantage points in the ecliptic plane alone are sensitive only to certain magnetic field structures. For a full reconstruction it is necessary to combine the longitudinal Zeeman and Hanle effect data.