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Although the Mars Express (MEX) does not carry a magnetometer, it is in principle possible to derive the interplanetary magnetic field (IMF) orientation from the three dimensional velocity distribution of pick-up ions measured by the Ion Mass Analyser (IMA) on board MEX because pick-up ions' orbits, in velocity phase space, are expected to gyrate around the IMF when the IMF is relatively uniform on a scale larger than the proton gyroradius. During bow shock outbound crossings, MEX often observed cycloid distributions (two dimensional partial ring distributions in velocity phase space) of protons in a narrow channel of the IMA detector (only one azimuth for many polar angles). We show two such examples. Three different methods are used to derive the IMF orientation from the observed cycloid distributions. One method is intuitive (intuitive method), while the others derive the minimum variance direction of the velocity vectors for the observed ring ions. These velocity vectors are selected either manually (manual method) or automatically using simple filters (automatic method). While the intuitive method and the manual method provide similar IMF orientations by which the observed cycloid distribution is well arranged into a partial circle (representing gyration) and constant parallel velocity, the automatic method failed to arrange the data to the degree of the manual method, yielding about a 30° offset in the estimated IMF direction. The uncertainty of the derived IMF orientation is strongly affected by the instrument resolution. The source population for these ring distributions is most likely newly ionized hydrogen atoms, which are picked up by the solar wind. [ABSTRACT FROM AUTHOR]