Active and passive transformation: Difference between revisions

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[[File:PassiveActive.JPG|thumb|310px|In the active transformation (left), a point {{mvar|P}} is transformed to point {{mvar|{{′|P}}}} by rotating clockwise by [[angle]] {{mvar|θ}} about the [[origin (mathematics)|origin]] of a fixed coordinate system. In the passive transformation (right), point {{mvar|P}} stays fixed, while the coordinate system rotates counterclockwise by an angle {{mvar|θ}} about its origin. The coordinates of {{mvar|{{′|P}}}} after the active transformation relative to the original coordinate system are the same as the coordinates of {{mvar|P}} relative to the rotated coordinate system.]]

[[Geometric transformation]]s can be distinguished into two types: '''active''' or '''alibi transformations''' which change the physical position of a set of [[point (geometry)|point]]s relative to a fixed [[frame of reference]] or [[coordinate system]] (''[[alibi]]'' meaning "being somewhere else at the same time"); and '''passive''' or '''alias transformations''' which leave points fixed but change the frame of reference or coordinate system relative to which they are described (''[[pseudonym|alias]]'' meaning "going under a different name").<ref>{{cite book |last1=Crampin |first1=M. |first2=F.A.E. |last2=Pirani |year=1986 |title=Applicable Differential Geometry |publisher=Cambridge University Press |page=22 |isbn=978-0-521-23190-9 |url=https://books.google.com/books?id=iDfk7bjI5qAC&pg=PA22 }}</ref><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> By ''transformation'', [[mathematician]]s usually refer to active transformations, while [[physicist]]s and [[engineer]]s could mean either.{{cn|date=September 2023}}

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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In [[three-dimensional Euclidean space]], any [[rigid transformation|proper rigid transformation]], whether active or passive, can be represented as a [[screw displacement]], the composition of a [[translation (geometry)|translation]] along an axis and a [[rotation (mathematics)|rotation]] about that axis.

The terms ''active transformation'' and ''passive transformation'' were first introduced in 1957 by [[Valentine Bargmann]] for describing [[Lorentz transformations]] in [[special relativity]].<ref>{{cite journal |last=Bargmann |first=Valentine |title=Relativity |journal=Reviews of Modern Physics |volume=29 |number=2 |year=1957 |pages=161–174 |doi=10.1103/RevModPhys.29.161 |bibcode=1957RvMP...29..161B }}</ref>

== Example ==

[[File:AliasAlibi and alibialias rotations.pngsvg|thumb|upright=1.8|Rotation considered as aan passiveactive (''aliasalibi'') or activepassive (''alibialias'') 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]]:

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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 display="block">\mathbf{v} = (v_x,v_y,v_z) = v_Xe_Xv_X\mathbf{e}_X+v_Ye_Yv_Y\mathbf{e}_Y+v_Ze_Zv_Z\mathbf{e}_Z = T^{-1}(v_X,v_Y,v_Z).</math>

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

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The inverse in the passive transformation ensures the ''components'' transform identically under <math>\tau</math> and <math>T</math>. This then gives a sharp distinction between active and passive transformations: active transformations [[left action|act from the left]] on bases, while the passive transformations act from the right, due to the inverse.

This observation is made more natural by viewing bases <math>\mathcal{B}</math> as a choice of isomorphism <math>\Phi_{\mathcal{B}}: VK^n \rightarrow K^nV</math>. The space of bases is equivalently the space of such isomorphisms, denoted <math>\text{Iso}(V, K^n, V)</math>. Active transformations, identified with <math>\text{GL}(V)</math>, act on <math>\text{Iso}(V, K^n, V)</math> from the left by composition, whilethat is if <math>\tau</math> represents an active transformation, we have <math>\Phi_{\mathcal{B'}} = \tau \circ \Phi_{\mathcal{B}}</math>. On the opposite, passive transformations, identified with <math>\text{GL}(n, K)</math> acts on <math>\text{Iso}(V, K^n, V)</math> from the right by pre-composition, that is if <math>T</math> represents a passive transformation, we have <math>\Phi_{\mathcal{B''}} = \Phi_{\mathcal{B}} \circ T</math>.

This turns the space of bases into a ''left'' <math>\text{GL}(V)</math>-[[torsor]] and a ''right'' <math>\text{GL}(n, K)</math>-torsor.