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Rotation Formula

RotationFormula

A rotation formula is a formula which transforms a given coordinate system by rotating it through a counterclockwise angle Phi about an axis n^^. Referring to the above figure (Goldstein 1980), the equation for the "fixed" vector in the transformed coordinate system (i.e., the above figure corresponds to an alias transformation), is

r^'=ON^->+NV^->+VQ^->
(1)
=n^^(n^^·r)+[r-n^^(n^^·r)]cosPhi+(rxn^^)sinPhi
(2)
=rcosPhi+n^^(n^^·r)(1-cosPhi)+(rxn^^)sinPhi
(3)

(Goldstein 1980; Varshalovich et al. 1988, p. 24). The angle Phi and unit normal n^^ may also be expressed as Euler angles. In terms of the Euler parameters,

r^'=r(e_0^2-e_1^2-e_2^2-e_3^2)+2e(e·r)+2(rxe)e_0.
(4)

The 3×3 rotation matrix can be calculated in the Wolfram Language as follows: 

  With[{n = {nx, ny, nz}},
    Cos[phi] IdentityMatrix[3] + (1 - Cos[p]) Outer[Times, n, n]
      + Sin[p] {{0, n[[3]], -n[[2]]}, {-n[[3]], 0, n[[1]]}, {n[[2]], -n[[1]], 0}}
  ]

See also

Alias Transformation, Alibi Transformation, Euler Angles, Euler Parameters, Rodrigues' Rotation Formula

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References

Gibbs, J. W. and Wilson, E. B. Vector Analysis: A Text-Book for the Use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs. New York: Dover, p. 338, 1960.Goldstein, H. "Finite Rotations." §4-7 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 164-166, 1980.Grubin, C. "Derivation of the Quaternion Scheme via the Euler Axis and Angle." J. Spacecraft 7, 1251-1263, 1970.Hamel, G. Theoretische Mechanik: Eine Einheitliche Einführung in die Gesamte Mechanik. Berlin: New York: Springer-Verlag, p. 103, 1949.Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotations in Terms of Rotation Axis and Rotation Angle." §1.4.2 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 23-24, 1988.

Referenced on Wolfram|Alpha

Rotation Formula

Cite this as:

Weisstein, Eric W. "Rotation Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RotationFormula.html

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