**Figure 1:** example of critical motion : vertical rotation axis

** Horizontal rotation axis.** We have and so, and (see eq:Sx) : eq:sys3 gives and and can then be computed with eq:sys5 and eq:sys6.

if we impose **r=0**, eq:sys2 gives . However, eq:sys1 can't give either or .
if is considered, and can be calculated directly and solved to give **r**, and .

** Vertical rotation axis.** This is the case when , which is similar to the previous one. We have .

if **r=0**, we have and then all parameters except **k** and can be evaluated.
if , leads to a total resolution of calibration

** Rotation axis orthogonal to the image plane.** We have ( and ).

It is the worst critical case in so far as , , **k** and **r** can be calculated, but remains always undetermined, whatever the constraint may be.

As a conclusion, we saw that the problem of affine-to-Euclidean calibration could be easily solved in particular cases (single motion, all parallel axes rotations).

We also showed that using the constraint allowed us to avoid critical cases : there remains then just one real critical motion (rotation axes orthogonal to the image plane).

Mon Dec 7 13:48:06 GMT 1998