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| | The '''Euler angles''' are three angles introduced by [[Leonhard Euler]] to describe the [[Orientation (geometry)|orientation]] of a [[rigid body]]. | | The '''Euler angles''' are three angles introduced by [[Leonhard Euler]] to describe the [[Orientation (geometry)|orientation]] of a [[rigid body]]. |
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| − | (TO DO: fix math templates.)
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| − | == Description ==
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| − | To describe such an orientation in [[dimension|3-dimensional]] [[Euclidean space]] three parameters are required.
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| − | They can be given in several ways, Euler angles being one of them; see [[charts on SO(3)]] for others.
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| − | Euler angles are also used to describe the orientation of a [[frame of reference]] (typically, a [[coordinate system]] or [[Basis (linear algebra)|basis]]) relative to another. They are typically denoted as [[Alpha|{{math|''α''}}]], [[Beta|{{math|''β''}}]], [[Gamma|{{math|''γ''}}]], or [[Phi|{{math|''φ''}}]], [[Theta|{{math|''θ''}}]], [[Psi (letter)|{{math|''ψ''}}]].
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| − | Euler angles represent a sequence of three ''[[elemental rotation]]s'', i.e. rotations about the axes of a [[coordinate system]].
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| − | For instance, a first rotation about {{math|''z''}} by an angle {{math|''α''}}, a second rotation about {{math|''x''}} by an angle {{math|''β''}}, and a last rotation again about {{math|''z''}}, by an angle {{math|''γ''}}. These rotations start from a known standard orientation.
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| − | In physics, this standard initial orientation is typically represented by a motionless (''fixed'', ''global'', or ''world'') [[coordinate system]]; in [[linear algebra]], by a [[standard basis]].
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| − | Any orientation can be achieved by composing three elemental rotations. The elemental rotations can either occur about the axes of the fixed coordinate system ([[#Extrinsic rotations|extrinsic rotations]]) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation ([[#Intrinsic rotations|intrinsic rotations]]).
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| − | The rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a ''local'' coordinate system.
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| | == See also == | | == See also == |
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| | * [[Leonhard Euler]] | | * [[Leonhard Euler]] |
| | * [[Mathematics]] | | * [[Mathematics]] |
| | + | * [[Orientation (geometry)]] |
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| | == External links == | | == External links == |
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| | * [https://en.wikipedia.org/wiki/Euler_angles Euler angles] @ Wikipedia | | * [https://en.wikipedia.org/wiki/Euler_angles Euler angles] @ Wikipedia |
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| | + | [[Category:Geometry]] |
| | [[Category:Mathematics]] | | [[Category:Mathematics]] |
