Active and passive transformation

For the concept of "passive transformation" in grammar, see active voice and passive voice.

In the active transformation (left), point P moves relative to the coordinate frame to location P', while the coordinate frame remains unchanged (in this case, the position of P has rotated clockwise by angle θ), while in a passive transformation (right), point P is observed in two different coordinate frames (pictured rotated relative to one another by angle θ) The coordinates of P' in the active case are the same as the coordinates of P in the rotated frame in the passive case if the rotation of P is clockwise and the rotation of the axes is counterclockwise.

In the physical sciences, an active transformation is one which actually changes the physical position of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is a change in the coordinate description of the physical system (change of basis). The distinction between active and passive transformations is important. By default, by transformation, mathematicians usually mean active transformations, while physicists could mean either.

Put differently, a passive transformation refers to observation of the same event from two different coordinate frames. On the other hand, the active transformation is a new mapping of all points from the same coordinate frame. If we describe successive displacements of a rigid body, the active transformation is useful. If we describe the displacements of individual arms of a robot, each with its own coordinate frame, the passive interpretation is useful to depict all the arm displacements from a common perspective.

In short, the active transform changes object's position while the observer moves in passive transform. When the screen scrolls up under "up" key press, the transform is active, and passive otherwise.

Example

As an example, in the vector space ℝ², let {e₁,e₂} be a basis, and consider the vector v = v¹e₁ + v²e₂. A rotation through angle θ is given by the matrix:

$R= \begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{pmatrix},$

which can be viewed either as an active transformation or a passive transformation, as described below.

Active transformation

As an active transformation, R rotates v . Thus a new vector v' is obtained. For a counterclockwise rotation of v with respect to the fixed coordinate system:

$\mathbf{v'}=R\mathbf{v}=\begin{pmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{pmatrix}\begin{pmatrix} v^1 \\ v^2 \end{pmatrix}.$

If one views {Re₁,Re₂} as a new basis, then the coordinates of the new vector v′ in the new basis are the same as those of v in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.

Passive transformation

On the other hand, when one views R as a passive transformation, the vector v is left unchanged, while the basis vectors are rotated. In order for the vector to remain fixed, the coordinates in terms of the new basis must change. For a counterclockwise rotation of frames:

$\mathbf{v}=v^a\mathbf{e}_a=v'^aR\mathbf{e}_a.$

From this equation one sees that the new coordinates (i.e., coordinates with respect to the new basis) are given by

$v'^a=(R^{-1})_b^a v^b$

so that

$\mathbf{v}=v'^a\mathbf{e}'_a=v^b(R^{-1})_b^a R_a^c \mathbf{e}_c=v^b\mathbf{e}_b.$

Thus, in order for the vector to remain unchanged by the passive transformation, the coordinates of the vector must transform according to the inverse of the active transformation operator.

See also

• Change of basis
• Covariance and contravariance of vectors

References

• Dirk Struik (1953) Lectures on Analytic and Projective Geometry, page 84, Addison-Wesley.