Summary: Recognizing an object that consists of different rigid parts which can move with respect to each other, may be performed by characterizing each rigid part separately and treating all these descriptions as a whole. But then the question pops up whether or not such an object can be characterized in a simpler way. The aim of this note is to develop a mathematical framwork to compute the relevant transformation group induced on the image plane by the relative motions of the parts, together with the overall Euclidean motions (or other relevant transformations) of the object as a whole. Once this transformation group is identified, the Lie group theory machinery for computing invariants can be put to work. As an example the existence of invariants under overall Euclidean motions of a planar object consisting of two rigid components which can translate, rotate or do both, in this plane with respect to each other, is completely analysed for images obtained by perspective and by pseudo-orthographic projection. In particular, it turns out that for perspective projection no other invariants exist than those obtained by considering each part separately as a rigid object, whereas in the pseudo-orthographic case simpler invariants (using partial information from each component) do exist. Examples of such invariants are given.