Abstract: The group representation of the line transformation matrices of finite and instantaneous motions of a rigid body in 3D space is investigated. The exponential mapping from the instantaneous motion to the finite motion is developed in an explicit manner using the differential equation of line transformation. This is followed by a vigorous proof that the entire set of finite (instantaneous) motion matrices of the line transformation is a faithful representation of SE(3) (se(3)), which, in turn, is isomorphic with the adjoint representation of SE(3) (se(3)) itself. The merit of this work lies in that it allows different representations of finite and instantaneous motions of a rigid body to be unified under the framework of group representation theory.

Key words: description of rigid body motions, group representation theory, line transformation