Active and passive transformation: Difference between revisions

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Typo/general fixes, replaced: Euclidian → Euclidean (3)
(Inserted {{short description|Distinction between meanings of Euclidian space transformations}})
m (Typo/general fixes, replaced: Euclidian → Euclidean (3))
{{short description|Distinction between meanings of EuclidianEuclidean space transformations}}
{{For|the concept of "passive transformation" in grammar|active voice|passive voice}}
 
[[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 [[analytic geometry]], spatial transformations in the 3-dimensional EuclidianEuclidean 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>
which can be viewed either as an ''active transformation'' or a ''passive transformation'' (where the above matrix will be inverted), as described below.
 
==Spatial transformations in the EuclidianEuclidean space <math>\R^3</math>==
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>.
 
===Active transformation===
As an active transformation, <math>T</math> transforms the initial vector <math>\mathbf{v}=(v_x,v_y,v_z)</math> into a new vector <math>\mathbf{v}'=(v'_x,v'_y,v'_z)=T\mathbf{v}=T(v_x,v_y,v_z)</math>.
 
If one views <math>\{\mathbf{e}'_x=T(1,0,0),\ \mathbf{e}'_y=T(0,1,0),\ \mathbf{e}'_z=T(0,0,1)\}</math> as a new basis, then the coordinates of the new vector <math>\mathbf{v}'=v_x\mathbf{e}'_x+v_y\mathbf{e}'_y+v_z\mathbf{e}'_z</math> in the new basis are the same as those of <math>\mathbf{v}=v_x\mathbf{e}_x+v_y\mathbf{e}_y+v_z\mathbf{e}_z</math> 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.