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| {{For|the concept of "passive transformation" in grammar|active voice|passive voice}}
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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 are the same as the coordinates of P relative to the rotated coordinate system.]]
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| In [[physics]] and [[engineering]], an '''active transformation''', or '''alibi transformation''', is a [[Transformation (mathematics)|transformation]] which actually changes the physical position of a point, or [[rigid body]], and makes sense even in the absence of a [[coordinate system]] whereas a '''passive transformation''', or '''alias transformation''', is a change in the position of the coordinate system from which the object is observed ([[change of basis]]). By default, by ''transformation'', [[mathematician]]s usually mean active transformations, while [[physicist]]s and [[engineer]]s could mean either.
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| Put differently, a ''passive'' transformation refers to observation of the ''same'' event from two different coordinate systems.<ref name= Davidson>
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| {{cite book |title=Robots and screw theory: applications of kinematics and statics to robotics
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| |author=Joseph K. Davidson, Kenneth Henderson Hunt
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| |chapter=§4.4.1 The active interpretation and the active transformation |page=74 ''ff'' |url=http://books.google.com/books?id=OQq67Tr7D0cC&pg=PA74
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| |isbn=0-19-856245-4 |year=2004 |publisher=Oxford University Press}}
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| </ref>
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| On the other hand, the ''active transformation'' is a new position of all points, relative to the same coordinate system. For instance, the active transformation is useful to describe successive positions of a [[rigid body]]. On the other hand, the passive transformation may be useful in human motion analysis to observe the motion of the [[tibia]] relative to the [[femur]], i.e. 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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| == Example ==
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| As an example, in the vector space ℝ<sup>2</sup>, let {'''e'''<sub>1</sub>,'''e'''<sub>2</sub>} be a [[basis (linear algebra)|basis]], and consider the vector '''v''' = ''v''<sup>1</sup>'''e'''<sub>1</sub> + ''v''<sup>2</sup>'''e'''<sub>2</sub>. A rotation through angle θ is given by the matrix:
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| :<math>R=
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| \begin{pmatrix}
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| \cos \theta & -\sin \theta\\
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| \sin \theta & \cos \theta
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| \end{pmatrix},
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| </math>
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| which can be viewed either as an [[Active and passive transformation#Active transformation|active transformation]] or a [[Active and passive transformation#Passive transformation|passive transformation]], as described below.
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| ===Active transformation===
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| As an active transformation, ''R'' rotates '''v''' . Thus a new vector '''v'''' is obtained. For a counterclockwise rotation of '''v''' with respect to the fixed coordinate system:
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| :<math>\mathbf{v'}=R\mathbf{v}=\begin{pmatrix}
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| \cos \theta & -\sin \theta\\
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| \sin \theta & \cos \theta
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| \end{pmatrix}\begin{pmatrix}
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| v^1 \\
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| v^2
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| \end{pmatrix}.</math>
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| If one views {''R'''''e'''<sub>1</sub>,''R'''''e'''<sub>2</sub>} as a new basis, then the coordinates of the new vector '''v′''' in the new basis are the same as those of '''v''' in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.
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| === Passive transformation === | |
| On the other hand, when one views ''R'' as a passive transformation, the vector '''v''' is left unchanged, while the 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:
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| :<math>\mathbf{v}=v^a\mathbf{e}_a=v'^aR\mathbf{e}_a.</math>
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| From this equation one sees that the new coordinates (''i.e.'', coordinates with respect to the new basis) are given by
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| :<math>v'^a=(R^{-1})_b^a v^b</math>
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| so that
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| :<math>\mathbf{v}=v'^a\mathbf{e}'_a=v^b(R^{-1})_b^a R_a^c \mathbf{e}_c=v^b\mathbf{e}_b.</math>
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| Thus, in order for the vector to remain unchanged by the passive transformation, the coordinates of the vector ''must'' transform according to the inverse of the active transformation operator.<ref name=Amidror>
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| {{cite book |isbn=1-4020-5457-2 |year=2007 |publisher=Springer |title=The theory of the Moiré phenomenon: Aperiodic layers |author=Isaac Amidror
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| |url=http://books.google.com/books?id=Z_QRomE5g3QC&pg=PT361 |chapter=Appendix D: Remark D.12 |page=346 }}
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| </ref> | |
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| ==See also==
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| * [[Change of basis]]
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| * [[Covariance and contravariance of vectors]]
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| ==References==
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| <references/> | |
| * [[Dirk Struik]] (1953) ''Lectures on Analytic and Projective Geometry'', page 84, [[Addison-Wesley]].
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| ==External links==
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| * [http://www.google.com/buzz/114134834346472219368/AWqcUGXVjcs/Consider-an-equilateral-triangle-in-a-plane-whose UI ambiguity]
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| [[Category:Systems theory]] | |
| [[Category:Mathematical terminology]]
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