Editing Euler angles (section)

===Precession, nutation and intrinsic rotation===
[[Image:Praezession.svg|thumb|170px|right|Euler basic motions of the Earth. Intrinsic rotation (green), Precession (blue) and Nutation (red)]]
[[Precession]], [[nutation]], and [[rotation|intrinsic rotation]] are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not all expressed in terms of the external frame, or all in terms of the co-moving rotated body frame, but in a mixture. They constitute a '''mixed axes of rotation''' system{{dash}}precession moves the line of nodes around the external axis ''z'', nutation rotates around the line of nodes ''N'', and intrinsic rotation is around ''Z'', an axis fixed in the body that moves.

Note: If an object undergoes a certain change of orientation this can be described as a combination of precession, nutation, and internal rotation, but how much of each depends on what XYZ coordinate system one has chosen for the object.

As an example, consider a [[Spinning top|top]]. If we define the Z axis to be the symmetry axis of the top, then the top spinning around its own axis of symmetry corresponds to intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top may wobble up and down (if it is not what is called a [[symmetric top]]); the change of inclination angle is nutation. The same example can be seen with the movements of the earth.

Though all three movements can be represented by rotation matrices, only precession can be expressed in general as a matrix in the basis of the space without dependencies on the other angles.

These movements also behave as a gimbal set. Given a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.