Active and passive transformation: Difference between revisions

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{{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{{mvar|P}} fromis position Ptransformed to point {{mvar|{{′|P'}}}} by rotating clockwise by an [[angle]] {{mvar|θ}} about the [[origin (mathematics)|origin]] of thea fixed coordinate system. In the passive transformation (right), point {{mvar|P}} doesstays not movefixed, while the coordinate system rotates counterclockwise by an angle {{mvar|θ}} about its origin. The coordinates of {{mvar|{{′|P'}}}} inafter the active case (that is,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}}

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]].

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/>

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/>

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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== In abstract vector spaces ==

{{unsourced-section|date=September 2023}}

The distinction between active and passive transformations can be seen mathematically by considering abstract [[vector spaces]].

Fix a finite-dimensional vector space <math>V</math> over a field <math>K</math> (thought of as <math>\mathbb{R}</math> or <math>\mathbb{C}</math>), and a basis <math>\mathcal{B} = \{e_i\}_{1 \leq i \leq n}</math> of <math>V</math>. This basis provides an isomorphism <math>C: K^n \rightarrow V</math> via the component map <math display="inline">(v_i)_{1 \leq i \leq n} = (v_1, \cdots, v_n) \mapsto \sum_i v_i e_i</math>.

An '''active transformation''' is then an [[endomorphism]] on <math>V</math>, that is, a linear map from <math>V</math> to itself. Taking such a transformation <math>\tau \in \text{End}(V)</math>, a vector <math>v \in V</math> transforms as <math>v \mapsto \tau v</math>. The components of <math>\tau</math> with respect to the basis <math>\mathcal{B}</math> are defined via the equation <math display="inline">\tau e_i = \sum_j\tau_{ji}e_j</math>. Then, the components of <math>v</math> transform as <math>v_i \mapsto \tau_{ij}v_j</math>.

A '''passive transformation''' is instead an endomorphism on <math>K^n</math>. This is applied to the components: <math>v_i \mapsto T_{ij}v_j =: v'_i</math>. TheProvided that <math>T</math> is invertible, the new basis <math>\mathcal{B}' = \{e'_i\}</math> is determined by asking that <math>v_ie_i = v'_i e'_i</math>, from which the expression <math>e'_i = (T^{-1})_{ji}e_j</math> can be derived.

Although the spaces <math>\text{End}(V)</math> and <math>\text{End}({K^n})</math> are isomorphic, they are not canonically isomorphic. Nevertheless a choice of basis <math>\mathcal{B}</math> allows construction of an isomorphism.

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Often one restricts to the case where the maps are invertible, so that active transformations are the [[general linear group]] <math>\text{GL}(V)</math> of transformations while passive transformations are the group <math>\text{GL}(n, K)</math>.

The transformations can then be understood as acting on the space of bases for <math>V</math>. An active transformation <math>\tau \in \text{GL}(V)</math> sends the basis <math>\{e_i\} \mapsto \{\tau e_i\}</math>. Meanwhile a passive transformation <math>T \in \text{GL}(n, K)</math> sends the basis <math display="inline">\{e_i\} \mapsto \left\{\sum_{j}(T^{-1})_{ji}e_j\right\}</math>.

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.