Fig 3. The concept of alignment requires a coupling of positions and orientations.

The pair of position and orientation (y ₀,n ₀) is better aligned with (y ₁,n ₁) than with (y ₂,n ₂), even though spatial and angular distances are equal. Formally we can say that the sub-Riemannian distance on ℝ³ ⋊ S ² [27] is smaller between (y ₀,n ₀) and (y ₁,n ₁).