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| {{About|the object in geometric algebra|the vector operator|curl (mathematics)}}
| | Records Manager Lucien Lezcano from Esterhazy, has several interests that include illusion, ganhando dinheiro na internet and button collecting. Recommends that you visit Kronborg Castle.<br><br>Here is my page ... [http://comoganhardinheironainternet.comoganhardinheiro101.com/ ganhe dinheiro] |
| {{Unreferenced|date=March 2007}}
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| A '''rotor''' is an object in [[geometric algebra]] that [[Rotation (mathematics)|rotate]]s any [[Blade (geometry)|blade]] or general [[multivector]] about the [[origin (mathematics)|origin]]. They are normally motivated by considering an [[Parity (mathematics)|even number]] of [[Reflection (mathematics)|reflection]]s, which generate rotations (see also the [[Cartan–Dieudonné theorem]]).
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| ==Generation using reflections==
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| ===General formulation===
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| {{multiple image|
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| |image1=Rotation of a vector as double reflection along vectors (large angle).svg
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| |caption1=''α'' > ''θ''/2
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| |width1=140
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| |image2=Rotation of a vector as double reflection along vectors (small angle).svg
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| |caption2=''α'' < ''θ''/2
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| |width2=130
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| |footer=Rotation of a vector ''a'' through angle ''θ'', as a double reflection ''along'' two unit vectors ''n'' and ''m'', separated by angle ''θ''/2 (not just ''θ''). Each prime on ''a'' indicates a reflection. The plane of the diagram is the plane of rotation.}}
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| [[Geometric algebra#Reflection along a vector|Reflections along a vector]] in geometric algebra may be represented as (minus) sandwiching a multivector ''M'' between a [[Null vector|non-null]] vector ''v'' perpendicular to the [[hyperplane]] of reflection and that vector's [[multiplicative inverse|inverse]] ''v''<sup>−1</sup>:
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| :<math>-vMv^{-1}</math>
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| and are of even grade. Under a rotation generated by the rotor ''R'', a general multivector ''M'' will transform double-sidedly as
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| :<math>RMR^{-1}.</math>
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| ===Restricted alternative formulation===
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| For a [[Euclidean space]], it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a ''unit'' (i.e. normalized) multivector:
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| :<math>-vMv, \quad v^2=1 ,</math>
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| forming rotors that are automatically normalised:
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| :<math>R\tilde{R}=\tilde{R}R=1 .</math>
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| The derived rotor action is then expressed as a sandwich product with the reverse:
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| :<math>RM\tilde{R}</math>
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| For a reflection for which the associated vector squares to a negative scalar, as may be the case with a [[pseudo-Euclidean space]], such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortcoming.
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| ===Rotations of multivectors and spinors===
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| However, though as multivectors rotors also transform double-sidedly, rotors can be combined and form a [[Group (mathematics)|group]], and so multiple rotors compose single-sidedly. The alternative formulation above is not self-normalizing and motivates the definition of [[spinor]] in geometric algebra as an object that transforms single-sidedly – i.e. spinors may be regarded as non-normalised rotors in which the reverse rather than the inverse is used in the sandwich product.
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| ==Homogeneous representation algebras==
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| In homogeneous representation algebras such as [[conformal geometric algebra]], a rotor in the representation space corresponds to a [[rotation]] about an arbitrary [[Point (geometry)|point]], a [[Translation (geometry)|translation]] or possibly another transformation in the base space.
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| ==See also==
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| *[[Double rotation]]
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| *[[Lie group]]
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| *[[Euler's formula]]
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| *[[Generator (mathematics)]]
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| {{geometry-stub}}
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| [[Category:Geometric algebra]]
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Records Manager Lucien Lezcano from Esterhazy, has several interests that include illusion, ganhando dinheiro na internet and button collecting. Recommends that you visit Kronborg Castle.
Here is my page ... ganhe dinheiro