A Survey of Representations of Spatial Rotation about a Fixed Point

This paper presents a comparison of five methods of representing a general spatial rigid-body rotation about a fixed point. The following representations are considered: the real orthogonal 3 × 3 matrix; the special unitary 2 × 2 matrix; the Pauli spin matrices; the unit quaternion; and the special unitary 3 × 3 matrix together with spherical harmonics of the first degree. Although the first of these representations is certainly the most commonly used, particularly in engineering and technological applications, it is shown that it is not the best or most efficient representation. The conclusion reached is that the most concise and efficient representation in practice is the unit quaternion, although the special unitary 2 × 2 matrices follow closely behind.