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Wikipedia / Active and passive transformation

Geometric transformations can be distinguished into two types: active or alibi transformations which change the physical position of a set of pointss relative to a fixed frame of reference or coordinate system (alibi meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (alias meaning "going under a different name").

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, that is, 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.

In three-dimensional Euclidean space, any proper rigid transformation, whether active or passive, can be represented as a screw displacement, the composition of a translation along an axis and a rotation about that axis.

The terms active transformation and passive transformation were first introduced in 1957 by Valentine Bargmann for describing Lorentz transformations in special relativity.

Example

, be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix which can be viewed either as an active transformation or a passive transformation (where the above matrix will be inverted), as described below.

Spatial transformations in the Euclidean space R3

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Active transformation

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as a new basis 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

. This gives a new coordinate system XYZ

with respect to the new coordinate system XYZ

transforms the old coordinates into the new ones.

In abstract vector spaces

The distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces.

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An active transformation is then an endomorphism.

A passive transformation can be derived.

allows construction of an isomorphism.

As left- and right-actions

Often one restricts to the case where the maps are invertible, so that active transformations are the general linear group.

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The inverse in the passive transformation ensures the components. This then gives a sharp distinction between active and passive transformations: active transformations act from the left on bases, while the passive transformations act from the right, due to the inverse.

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This turns the space of bases into a left-torsor and a right-torsor.

From a physical perspective, active transformations can be characterized as transformations of physical space, while passive transformations are characterized as redundancies in the description of physical space. This plays an important role in mathematical gauge theory, where gauge transformations are described mathematically by transition maps which act from the right on fibers.

See also

References