Active and passive transformation: Difference between revisions

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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 [[physics]] and [[engineering]], spatial transformations in the 3-dimensional Euclidian space <math>\R^3</math> are distinguished into '''active''' or '''alibi transformations''', and '''passive''' or '''alias transformstions'''. 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 even 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 default, by ''transformation'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either. Both type of transformations may consist of a combination of a [[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>

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</math>

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 Euclidian space <math>\R^3</math>==

In general a spatial transformation <math>T:\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, ''R'' rotates<math>T</math> transforms the initial vector '''<math>\mathbf{v'''}=(v_x,v_y,v_z)</math> andinto a new vector <math>\mathbf{v}'=(v'_x,v'_y,v'''' is obtained_z)=T\mathbf{v}=T(v_x,v_x,v_x)</math>. For a counterclockwise rotation of '''v''' with respect to the fixed coordinate system:

:<math>\mathbf{v}'=R\mathbf{v}=\begin{pmatrix}

\cos \theta & -\sin \theta\\

\sin \theta & \cos \theta

\end{pmatrix}\begin{pmatrix}

v_1 \\

v_2

\end{pmatrix}.</math>

If one views <math>\{\mathbf{e}'_1_x=RT(1,0,0),\ \mathbf{e}'_2_y=RT(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_1v_x\mathbf{e}'_1_x+v_2v_y\mathbf{e}'_2_y+v_z\mathbf{e}'_z</math> in the new basis are the same as those of <math>\mathbf{v}=v_1v_x\mathbf{e}_1_x+v_2v_y\mathbf{e}_2_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.

=== Passive transformation ===

On the other hand, when one views <math>RT</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 rotatedtransformed in the opposite direction, i.e. with the rotationinverse transformation <math>RT^{-1}=R^*</math>. This Ingives ordera fornew thecoordinate vectorsystem toXYZ remain fixed, the coordinates in terms of the newwith basis must change, according tovectors:

:<math>\mathbf{ve}_X=(v_1,v_2)=v'_1RT^*{-1}(1,0,0)+v'_2R*,\ \mathbf{e}_Y=T^{-1}(0,1,0),\ \mathbf{e}_Z=RT^*{-1}(v'_10,v'_20,1)</math>.

From this equation one sees that the new coordinates (that is, coordinates with respect to the new basis) are given by

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:

:<math>(v'_1,v'_2)=R(v_1,v_2)</math>.

:<math>\mathbf{v}=(v_x,v_y,v_z)=v_Xe_X+v_Ye_Y+v_Ze_Z=T^{-1}(v_X,v_Y,v_Z)</math>.

so that the coordinates of the vector ''must'' transform according to the inverse of the active transformation of the coordinate system.<ref name=Amidror>

From this equation one sees that the new coordinates (that is, coordinates with respect to the new basis) are given by

:<math>(v'_1v_X,v'_2v_Y,v_Z)=RT(v_1v_x,v_y,v_2v_z)</math>.

As a passive transformation <math>T</math> transforms the old coordinates into the new ones.

Note the difference between <math>(v_X,v_Y,v_Z)</math> and <math>(v'_x,v'_y,v'_z)</math>

{{cite book |isbn=1-4020-5457-2 |year=2007 |publisher=Springer |title=The theory of the Moiré phenomenon: Aperiodic layers |first=Isaac|last=Amidror