We investigate the motions that lead to ambiguous Euclidean scene reconstructions under several common calibration constraints, giving a complete description of such critical motions for: (i) internally calibrated orthographic and perspective cameras; (ii) in two images, for cameras with unknown focal lengths, either different or equal. One aim of the work was to evaluate the potential of modern algebraic geometry tools for rigorously proving properties of vision algorithms, so we use ideal theoretic calculations as well as classical algebra and geometry. We also present numerical experiments showing the effects of near-critical configurations for the varying and fixed focal length methods.