Euler angles

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The angles , and that determine the position of one Cartesian rectangular coordinate system relative to another one 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 be the axis coinciding with the line of intersection of the planes and , oriented so that the three lines , and form a right-handed triple. Then is the angle between and , measured in the plane from in the direction of the shortest rotation of to , is the angle between and not exceeding , and is in the direction of the shortest rotation of to . The coordinates , and are connected by the relations

These angles were introduced by L. Euler (1748).


For other formulas, as well as applications, see [a1][a3].


[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)
How to Cite This Entry:
Euler angles. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article