Active/passive or Alibi/alias transformations

Parent: Rotations / SO(3) group index
See also: Intrinsic vs extrinsic rotations

Sources:  Wiki , bonn-3D-cs

Alibi / Active	Alias / Passive
	Point $\mathcal{x}$ doesn’t change, but the coord vector representation changes to the ‘prime’ system
CS $S(x,, y)$ is fixed	CS $S(x,, y)$ is rotated
Point $\mathcal{x}$ rotates within fixed CS	Point $\mathcal{x}$ remains stationary but is represented within a new CS
$\mathcal{x}_1 \rightarrow \mathcal{x}_2$	$S_1(x^\prime,y^\prime) \rightarrow S_2(x^{\prime\prime}, y^{\prime\prime})$
$\left( \begin{array}\cos\theta & -\sin\theta & 0\ \sin\theta & \cos\theta & 0\0 & 0 & 1\end{array} \right)$Counterclockwise rotation by theta	$\left( \begin{array}\cos\theta & \sin\theta & 0\ -\sin\theta & \cos\theta & 0\0 & 0 & 1\end{array} \right)$
mathematics	physics, robotics

Source:  http://rock-learning.github.io/pytransform3d/transformation_ambiguities.html To transform between one another: use the inverse 

Translations

Source: bonn-3D-cs

Active

Equation

$$\mathbf{x}_2 = {}_2 \mathbf{T}^1, \mathbf{x}_1$$

Passive

Equation

$${}^2\mathbf{x} = {}^2 \mathbf{T}_1, {}^1\mathbf{x}$$

Relation between active and passive transformations

The passive transformation is the inverse of the active transformation!

Active $$ \mathbf{x}_2 = \mathbf{T} \cdot \mathbf{x}_1 $$

Passive $$ {}^2\mathbf{x} = \mathbf{T}^{-1} \cdot {}^1\mathbf{x} $$

Extended notation $$ \begin{align} {}_2 \mathbf{T}^1 &= \left( {}^2 \mathbf{T}_1 \right)^{-1}\
{}_2 \mathbf{T}^1 &= {}^1 \mathbf{T}_2 \end{align}$$