Active and passive transformation: Difference between revisions

Revision as of 18:57, 28 September 2023

Revision as of 19:06, 28 September 2023

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[[File:PassiveActive.JPG|thumb|310px|In the active transformation (left), a point {{mvar|P}} is transformed to point {{mvar|{{′|P}}}} by rotating clockwise by [[angle]] {{mvar|θ}} about the [[origin (mathematics)|origin]] of a fixed coordinate system. In the passive transformation (right), point {{mvar|P}} stays fixed, while the coordinate system rotates counterclockwise by an angle {{mvar|θ}} about its origin. The coordinates of {{mvar|{{′|P}}}} after the active transformation relative to the original coordinate system are the same as the coordinates of {{mvar|P}} relative to the rotated coordinate system.]]

[[Geometric transformation]]s can be distinguished into two types: '''active''' or '''alibi transformations''' which change the physical position (''alibi'' meansmeaning "elsewherebeing somewhere else at the same time") of a set of [[point (geometry)|point]]s relative to a fixed [[frame of reference]] or [[coordinate system]]; 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'' meansmeaning "othergoing under a different name").<ref>Weisstein,{{cite Ericbook W|last1=Crampin |first1=M. [http://mathworld|first2=F.wolframA.com/AlibiTransformationE.html "Alibi|last2=Pirani Transformation"],|year=1986 [http|title=Applicable Differential Geometry |publisher=Cambridge University Press |page=22 |url=https://mathworldbooks.wolframgoogle.com/AliasTransformation.htmlbooks?id=iDfk7bjI5qAC&pg=PA22 "Alias Transformation"]. ''Mathworld''.}}</ref><ref name= Davidson>{{cite book | title=Robots and screw theory: applications of kinematics and statics to robotics | author=Joseph K. Davidson, Kenneth Henderson Hunt | chapter=§4.4.1 The active interpretation and the active transformation | page=74 ''ff'' | chapter-url=https://books.google.com/books?id=OQq67Tr7D0cC&pg=PA74 | isbn=0-19-856245-4 |year=2004 | publisher=Oxford University Press}}</ref> By ''transformation'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either.{{cn|date=September 2023}}

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/>

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

The terms ''active transformation'' and ''passive transformation'' were first introduced in 18571957 by [[Valentine Bargmann]] for describing [[Lorentz transformations]] in [[special relativity]].<ref>{{cite journal |last=Bargmann |first=Valentine |title=Relativity |journal=Reviews of Modern Physics |volume=29 |number=2 |year=1957 |pages=161–174 |doi=10.1103/RevModPhys.29.161 }}</ref>

== Example ==