Camera calibration with one-dimensional objects is based on an algebraic constraint on the image of the absolute conic. We give an alternative derivation to this constraint, allowing a geometrical interpretation. From this, we derive the degenerate cases, or critical motions, where the calibration algorithm fails. We also show that constraints on the intrinsic parameters lead to simplified closed-form solutions and a reduced set of critical motions. A simulation and a real data experiment is performed to evaluate the accuracy of the calibration result for motions close to being critical.