Euler angles
The angles , \psi and \theta that determine the position of one Cartesian rectangular coordinate system 0xyz relative to another one 0x'y'z' with the same origin and orientation. The Euler angles are regarded as the angles through which the former must be successively rotated about the axes of the latter so that in the end the two systems coincide (see Fig.).
Figure: e036390a
Let u be the axis coinciding with the line of intersection of the planes 0xy and 0x'y', oriented so that the three lines 0z, 0z' and u form a right-handed triple. Then \psi is the angle between 0x and u, measured in the plane 0xy from 0x in the direction of the shortest rotation of 0x to 0y, \theta is the angle between 0z and 0z' not exceeding \pi, and \phi is in the direction of the shortest rotation of 0x' to 0y'. The coordinates x,y,z, and x',y',z' are connected by the relations
x'=(\cos\psi\cos\phi-\sin\psi\cos\theta\sin\phi)x'+
+(-\cos\psi\sin\phi-\sin\psi\cos\theta\cos\phi)y'+(\sin\psi\sin\theta)z',
y=(\sin\psi\cos\phi+\cos\psi\cos\theta\sin\phi)x'+
+(-\sin\psi\sin\phi+\cos\psi\cos\theta\cos\phi)y'+(-\cos\psi\sin\theta)z',
z=(\sin\theta\sin\phi)x'+(\sin\theta\cos\phi)y'+(\cos\theta)z'.
These angles were introduced by L. Euler (1748).
Comments
For other formulas, as well as applications, see [a1]–[a3].
References
[a1] | L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1965) (Translated from Russian) |
[a2] | G. Gallavotti, "The elements of mechanics" , Springer (1983) |
[a3] | H. Goldstein, "Classical mechanics" , Addison-Wesley (1959) |
Euler angles. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_angles&oldid=34483