Active and passive transformation

In the active transformation (left), a point moves from position P to P' by rotating clockwise by an angle θ about the origin of the coordinate system. In the passive transformation (right), point P does not move, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P' in the active case (i.e. relative to the original coordinate system) are the same as the coordinates of P relative to the rotated coordinate system.

In physics and engineering, an active transformation, or alibi transformation, is a transformation which actually changes the physical position of a point, or rigid body, which can be defined even in the absence of a coordinate system; whereas a passive transformation, or alias transformation, is merely a change in the coordinate system in which the object is described (change of coordinate map, or change of basis). By default, by transformation, mathematicians usually refer to active transformations, while physicists and engineers could mean either.

Put differently, a passive transformation refers to description of the same object in two different coordinate systems.^[1] On the other hand, an active transformation is a transformation of one or more objects with respect to the same coordinate system. For instance, active transformations are useful to describe successive positions of a rigid body. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the tibia relative to the femur, i.e. its motion relative to a (local) coordinate system which moves together with the femur, rather than a (global) coordinate system which is fixed to the floor.^[1]

Example

As an example, in the vector space ℝ², let {e₁,e₂} be a basis, and consider the vector v = v¹e₁ + v²e₂. A rotation of the vector 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 (where the matrix is inverse), 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 original basis are the same as those of v in the new 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 coordinate systems:

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

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

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

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.^[2]

See also

• Change of basis
• Covariance and contravariance of vectors
• Rotation of axes
• Translation of axes

References

1. Joseph K. Davidson, Kenneth Henderson Hunt (2004). "§4.4.1 The active interpretation and the active transformation". Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press. p. 74 ff. ISBN 0-19-856245-4.
2. Amidror, Isaac (2007). "Appendix D: Remark D.12". The theory of the Moiré phenomenon: Aperiodic layers. Springer. p. 346. ISBN 1-4020-5457-2.

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

External links

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