The four parameters , , , and describing a finite rotation about an arbitrary axis. The Euler parameters are defined by
(1)
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(2)
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(3)
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where is the unit normal vector, and are a quaternion in scalar-vector representation
(4)
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Because Euler's rotation theorem states that an arbitrary rotation may be described by only three parameters, a relationship must exist between these four quantities
(5)
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(6)
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(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by
(7)
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(8)
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(9)
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and
(10)
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The Euler parameters may be given in terms of the Euler angles by
(11)
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(12)
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(13)
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(14)
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(Goldstein 1980, p. 155).
Using the Euler parameters, the rotation formula becomes
(15)
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and the rotation matrix becomes
(16)
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where the elements of the matrix are
(17)
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Here, Einstein summation has been used, is the Kronecker delta, and is the permutation symbol. Written out explicitly, the matrix elements are
(18)
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(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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