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 points 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").
[1][2] By transformation, mathematicians usually refer to active transformations, while physicists and engineers could mean either.
[citation needed] 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.
[2] 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.
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.
may consist of a translation and a linear transformation.
In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3 matrix
as a new basis, then the coordinates of the new vector
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.
as a passive transformation, the initial vector
is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation
[4] This gives a new coordinate system XYZ with basis vectors:
with respect to the new coordinate system XYZ are given by:
From this equation one sees that the new coordinates are given by
Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
The distinction between active and passive transformations can be seen mathematically by considering abstract vector spaces.
Fix a finite-dimensional vector space
This basis provides an isomorphism
Often one restricts to the case where the maps are invertible, so that active transformations are the general linear group
The transformations can then be understood as acting on the space of bases for
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.
This observation is made more natural by viewing bases
from the left by composition, while passive transformations, identified with
This turns the space of bases into a left
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.
