Active and passive transformation
In analytic geometry, spatial transformations in the 3-dimensional Euclidean space are distinguished into active or alibi transformations, and passive or alias transformations. An active transformation is a transformation which actually changes the physical position (alibi, elsewhere) of a point, or rigid body, which can be defined in the absence of a coordinate system; whereas a passive transformation is merely a change in the coordinate system in which the object is described (alias, other name) (change of coordinate map, or change of basis). By transformation, mathematicians usually refer to active transformations, while physicists and engineers could mean either. Both types of transformation can be represented by a combination of a translation and a linear transformation.
Put differently, a passive transformation refers to description of the same object in two different coordinate systems. 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, 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.
As an example, let the vector , 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 
In general a spatial transformation 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 an active transformation, transforms the initial vector into a new vector .
If one views as a new basis, then the coordinates of the new vector in the new basis are the same as those of 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.
On the other hand, when one views 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 .  This gives a new coordinate system XYZ with basis vectors:
The new coordinates of with respect to the new coordinate system XYZ are given by:
From this equation one sees that the new coordinates are given by
As a passive transformation transforms the old coordinates into the new ones.
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
- Weisstein, Eric W. "Alibi Transformation." From MathWorld--A Wolfram Web Resource.
- Weisstein, Eric W. "Alias Transformation." From MathWorld--A Wolfram Web Resource.
- 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.
- Amidror, Isaac (2007). "Appendix D: Remark D.12". The theory of the Moiré phenomenon: Aperiodic layers. Springer. p. 346. ISBN 978-1-4020-5457-0.