Quaternions: From Classical Mechanics to Computer Graphics, and Beyond

The theory of quaternions was introduced in the mid nineteenth century, and it found many applications in classical mechanics, quantum mechanics, and the theory of relativity. Quaternions were also later used in aerospace applications and flight simulators, particularly when inertial attitude referencing and related control schemes where employed. However, it is only in the recent past that graphics and game programmers discovered the true potential of quaternions and started using it as a powerful tool for describing rotations about an arbitrary axis. From computer graphics, the application domain of quaternions soon expanded into other fields such as visualization, fractals and virtual reality. This paper provides an overview of the various analytical properties of quaternions and their usefulness in the areas of computer/robot vision, computer graphics and animation. Particular emphasis is given to vision algorithms for 3D pose estimation, animation techniques involving viewpoint/object rotations, motion interpolation algorithms, and quaternion fractals. The benefit of using quaternions over other representations such as Euler angles is not just limited to singularity free kinematics relations – Quaternions allow us to derive closed form solutions for algebraic systems involving unknown rotational parameters. Some of the neat mathematical characteristics of quaternions in the complex space together with a set of useful formulae are included for the benefit of the mathematically inclined.