Active and passive transformation: Difference between revisions

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[[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 (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]], spatial transformations in the 3-dimensional Euclidian space <math>\R^3</math> are distinguished into '''active''' or '''alibi transformations''', and '''passive''' or '''alias transformstions'''. An '''active 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 (alibi, elsewhere) of a point, or [[rigid body]], which can be defined even in the absence of a [[coordinate system]]; whereas a '''passive 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 (alias, other name) (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. Both typetypes of transformationstransformation maycan consistbe ofrepresented by a combination of a [[Translation (geometry)|translation]] and a [[linear transformation]].

Put differently, a ''passive'' transformation refers to description of the ''same'' object in two different coordinate systems.<ref name= Davidson>

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|isbn=0-19-856245-4 |year=2004 |publisher=Oxford University Press}}

</ref>

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.<ref name = Davidson/>

== Example ==