A Comparison of Representations of General Spatial Screw Displacement

This paper presents a comparison of eight methods of representing a general spatial rigid-body screw displacement. A distinction is made between point transformations and line transformations. The following three representations involving point transformations are considered: the real 4 × 4 matrix; the complex 4 × 4 matrix; and the real 6 × 6 matrix. These utilise homogeneous coordinates. The remaining five representations considered here are concerned with line transformations and are: the dual orthogonal 3 × 3 matrix; the dual special unitary 2 × 2 matrix; the dual Pauli spin matrices; the dual unit quaternion; and the dual special unitary 3 × 3 matrix. These make use of dual-number techniques. The most commonly used of these representations is the 4 × 4 real matrix and the conclusion reached is that this is certainly the best of the point transformations. The dual unit quaternion is similarly found to be the best of the line transformations.