Active and passive transformation: Difference between revisions

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{{For|the concept of "passive transformation" in grammar|active voice|passive voice}}
 
[[File:PassiveActive.JPG|thumb|310px|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.that is, 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''',<ref>[http://mathworld.wolfram.com/AlibiTransformation.html Weisstein, Eric W. "Alibi Transformation." From MathWorld--A Wolfram Web Resource.]</ref> is a [[Transformation (mathematics)|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''',<ref>[http://mathworld.wolfram.com/AliasTransformation.html Weisstein, Eric W. "Alias Transformation." From MathWorld--A Wolfram Web Resource.]</ref> 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'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either.
|isbn=0-19-856245-4 |year=2004 |publisher=Oxford University Press}}
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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.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.<ref name = Davidson/>
 
== Example ==
 
===Active transformation===
As an active transformation, ''R'' rotates the initial vector '''v''', and a a new vector '''v'''' is obtained. For a counterclockwise rotation of '''v''' with respect to the fixed coordinate system:
:<math>\mathbf{v'}=R\mathbf{v}=\begin{pmatrix}
\cos \theta & -\sin \theta\\
On the other hand, when one views ''R'' as a passive transformation, the initial vector '''v''' is left unchanged, while the coordinate system and its 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:
:<math>\mathbf{v}=v^a\mathbf{e}_a=v'^aR\mathbf{e}_a.</math>
From this equation one sees that the new coordinates (''i.e.''that is, coordinates with respect to the new basis) are given by
:<math>v'^a=(R^{-1})_b^a v^b</math>
so that