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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.]]
[[Geometric transformation]]s incan anybe [[geometricdistinguished space]]into (suchtwo as the [[three-dimensional Euclidean space]]) are distinguished intotypes: '''active''' or '''alibi transformations''', and '''passive''' or '''alias transformations'''. An active transformation is a transformation which changeschange the physical position (''alibi'' means "elsewhere") of a set of [[point (geometry)|point]]s, orrelative [[rigidto body]],a which can be defined independently of anyfixed [[frame of reference]] or [[coordinate system]];<ref>Weisstein, Ericand W.'''passive''' [http://mathworld.wolfram.com/AlibiTransformation.htmlor "Alibi Transformation."]'''alias transformations''Mathworld''.</ref> whereaswhich aleave passivepoints transformationfixed is abut change in the frame of reference or coordinate system relative to which thethey object isare described (''alias'' means "other name") (change of coordinate map, or [[change of basis]]).<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/AlibiTransformation.html "Alibi Transformation"], [http://mathworld.wolfram.com/AliasTransformation.html "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}}
On the other hand, an ''active transformation'' is a transformation of an object with respect to a fixed 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/> ▼Put differently, a ''passive'' transformation refers to description of the ''same'' object relative to two different coordinate systems.<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>
▲On the other hand, an ''active transformation'' is a transformation of an object with respect to a fixed 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/>
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.
== Example ==
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