General 3-D Motion
Let us fix a point o on the body and coordinate system xyz.
Three coordinates are needed to specify the coordinates of O. since total six degrees of freedom are required to specify the orientation of a rigid body, the 3 coordinates are needed to specify the orientation of the primed axis.
There are many ways of specifying the orientation of a Cartesian set of axes relative to another set with common origin, the most common being the Euler angles.
Euler Angles:
The Euler angles are defined as the three successive angles of rotation. Many conventions are followed.
The sequence employed here is started by rotating the initial system of axes, xyz, by an angle 
about X-axis. The resulted coordinate system is labeled as the x2, y2 and z2 . Note that z2 axis is the same as z1 axis.
In the second stage, the intermeadiate axes x2,y2,z2 is rotated about y2 to produce another intermeadiate set, x3,y3,z3 . Note that y2 and y3 are same and the rotation is by amount 
. The intersection of 
and 
planes is known as the line of nodes. Finally the 
is rotated about z3 by an angle 
.