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The nine elements rij in a general rotational transformation are not independent quantities. Indeed, a rigid body possesses at most three rotational degrees of freedom, and thus at most three quantities are required to specify its orientation. This can be easily seen by examining the constraints that govern the matrices in SO(3):

(2.25)

(2.26)

Equation (2.25) follows from the fact that the columns of a rotation matrix are unit vectors, and Equation (2.26) follows from the fact that columns of a rotation matrix are mutually orthogonal. Together, these constraints define six independent equations with nine unknowns, which implies that there are three free variables.

In this section we derive three ways in which an arbitrary rotation can be represented using only three independent quantities: the Euler angle representation, the roll-pitch-yaw representation, and the axis-angle representation.