Appendix C: Quaternions and special relativity

Abstract

Complex numbers unify algebra and two dimensional geometry representing rotations on the plane as multiplication by a complex number. This appendix introduces quaternions as an extension of this unification to four dimensions. The quaternion gradient is defined, the quaternion analogues of the Cauchy-Riemann equations is obtained, and basic quaternion integral theorems are derived. A pedagogic derivation of the quaternion representation of three-dimensional rotations widely applied in robotics and computer graphics is presented. The appendix closes with a detailed discussion on the application of biquaternions in relativity.