Euler angles
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
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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=15870