Intrinsic vs extrinsic rotations

Parent: Rotations/SO(3) Group Index
See also:  Active/passive or Alibi/alias rotation transformations

Source:  http://rock-learning.github.io/pytransform3d/transformation_ambiguities.html

We want to rotate first by $R_1$, then by $R_2$.

Extrinsic (global) rotation

In global coordinates, extrinsic rotation: $R_2 \cdot R_1$

Intrinsic (local) rotation

In local coordinates, intrinsic rotation: $R_1 \cdot R_2$
($R_1$ defines new coordinates in which $R_2$ is applied)

Specifying the convention is relevant when dealing with Euler angles!!!

Illustration

Source: bonn-3D-cs

Active translation –> rotation

Passive translation –> rotation

Relation

To convert a global rotation to a local rotation, reverse the order of transformations.

e.g.

• Active global transformation A
  • A: first T from origin, followed by R around origin
  • if this were carried out as an equivalent local transformation, the active local transformation would be first R around local CS, then T from local CS
    
• Active local transformation B
  • B: first T from local CS, followed by R around local CS
  • if this were carried out as an equivalent global transformation, the active global transformation would be first R around origin, then T from origin