Active and passive transformation: Difference between revisions

Revision as of 01:56, 9 April 2020

Revision as of 15:20, 1 July 2021

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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 [[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>

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[[File:Alias and alibi rotations.png|thumb|upright=1.8|Rotation considered as a passive (''alias'') or active (''alibi'') transformation]]

[[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]]:

:<math>R=

\begin{pmatrix}

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\end{pmatrix},

</math>

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 <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 (linear algebra)|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.

=== 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>

<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:

:<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>.

From this equation one sees that the new coordinates are given by

:<math>(v_X,v_Y,v_Z) = T(v_x,v_y,v_z)</math>.

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