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In [[analytic geometry]], spatial transformations in the [[3-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, elsewhere) of a point, or [[rigid body]], which can be defined 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 ''transformation'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either. Both types of transformation can be represented 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>{{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 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/>
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[[File:Alias and alibi transformations 1 en.png|thumb|upright=1.8|Translation and rotation as passive (''alias'') or active (''alibi'') transformations]]
As an example, let the vector <math>\mathbf{v}=(v_1,v_2) \in \R^2</math>, be a vector in the plane. A rotation of the vector through an angle ''θ'' in counterclockwise direction is given by the [[rotation matrix]]:
\begin{pmatrix}
\cos \theta & -\sin \theta\\
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which can be viewed either as an ''active transformation'' or a ''passive transformation'' (where the above [[matrix (mathematics)|matrix]] will be [[inverse matrix|inverted]]), as described below.
==Spatial transformations in the Euclidean space '''R'''<sup>3</sup>==
In general a spatial transformation <math>T\colon\R^3\to \R^3</math> may consist of a translation and a linear transformation. In the following, the translation will be omitted, and the linear transformation will be represented by a 3×3 matrix <math>T</math>.
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=== Passive transformation ===
On the other hand, when one views <math>T</math> as a passive transformation, the initial vector <math>\mathbf{v}=(v_x,v_y,v_z)</math> is left unchanged, while the coordinate system and its basis vectors are transformed in the opposite direction, that is, with the inverse transformation <math>T^{-1}</math>.<ref name=Amidror>{{cite book |isbn=978-1-4020-5457-0 |year=2007 | publisher=Springer |title=The theory of the Moiré phenomenon: Aperiodic layers |first=Isaac|last=Amidror | chapter-url=https://books.google.com/books?id=Z_QRomE5g3QC&pg=PT361 |chapter=Appendix D: Remark D.12 |page=346 }}</ref> This gives a new coordinate system ''XYZ'' with basis vectors:
▲:<math>\mathbf{e}_X = T^{-1}(1,0,0),\ \mathbf{e}_Y = T^{-1}(0,1,0),\ \mathbf{e}_Z = T^{-1}(0,0,1)</math>
The new coordinates <math>(v_X,v_Y,v_Z)</math> of <math>\mathbf{v}</math> with respect to the new coordinate system ''XYZ'' are given by:
From this equation one sees that the new coordinates are given by
As a passive transformation <math>T</math> transforms the old coordinates into the new ones.
Note the equivalence between the two kinds of transformations: the coordinates of the new point in the active transformation and the new coordinates of the point in the passive transformation are the same, namely
==See also==
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