For any motion, the relation eq:ambiguity implies that the form of is :
where commutes with .
When the rotation component of the displacement is performed around an axis parallel to the basis axes of the camera, takes special forms :
If the rotation axis is parallel to the horizontal axis of the camera : eq:Sx Q=(
) and S S_x (
If the rotation axis is parallel to the vertical axis of the camera :
In practice, r is often negligible in comparison with k and we can consider that : eq:Sy S S_y (
Finally, if the rotation axis is orthogonal to the image plane, is the identity and : eq:Sz S S_z (
We can observe that, in these 3 cases, the structure of is independent of any ambiguity in the real Jordan decomposition. It will be shown later that these cases correspond to critical motions for affine-to-Euclidean calibration.