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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 [[analyticGeometric geometrytransformation]],s spatialin transformationsany in[[geometric space]] (such as the [[3three-dimensional Euclidean space]] <math>\R^3</math>) are distinguished into '''active''' or '''alibi transformations''', and '''passive''' or '''alias transformations'''. 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,'' means "elsewhere") of a set of [[point (geometry)|point]]s, or [[rigid body]], which can be defined inindependently theof absenceany [[frame of areference]] or [[coordinate system]];<ref>Weisstein, whereasEric aW. '''passive transformation'''<ref>[http://mathworld.wolfram.com/AliasTransformationAlibiTransformation.html Weisstein, Eric W. "AliasAlibi Transformation."] From MathWorld--A Wolfram Web Resource''Mathworld''.]</ref> iswhereas merelya passive transformation is a change in the frame of reference or coordinate system inrelative to which the object is described (''alias,'' means "other name") (change of coordinate map, or [[change of basis]]).<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/AliasTransformation.html "Alias Transformation."] ''Mathworld''.</ref> By ''transformation'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either. BothIn typesthree ofdimensional Euclidean space, any [[rigid transformation|proper rigid transformation]], whether active or passive, can be represented byas a combination[[screw displacement]], the composition of a [[Translationtranslation (geometry)|translation]] along an axis and a [[linearrotation transformation(mathematics)|rotation]] about that axis.
Put differently, a ''passive'' transformation refers to description of the ''same'' object inrelative 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 onean or more objectsobject with respect to thea samefixed 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 ==
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