A beginners guide to dual-quaternions: What they are, how they work, and how to use them for 3D character hierarchies

In this paper, we give a beginners guide to the practicality of using dual-quaternions to represent the rotations and translations in character-based hierarchies. Quaternions have proven themselves in many fields of science and computing as providing an unambiguous, un-cumbersome, computationally efficient method of representing rotational information. We hope after reading this paper the reader will take a similar view on dual-quaternions. We explain how dual number theory can extend quaternions to dual-quaternions and how we can use them to represent rigid transforms (i.e., translations and rotations). Through a set of examples, we demonstrate exactly how dual-quaternions relate rotations and translations and compare them with traditional Euler's angles in combination with Matrix concatenation. We give a clear-cut, step-by-step introduction to dual-quaternions, which is followed by a no-nonsense how-to approach on employing them in code. The reader, I believe, after reading this paper should be able to see how dual-quaternions can offer a straightforward solution of representing rigid transforms (e.g., in complex character hierarchies). We show how dual-quaternions propose a novel alternative to pure Euler-Matrix methods and how a hybrid system in combination with matrices results in a faster more reliable solution. We focus on demonstrating the enormous rewards of using dual-quaternions for rigid transforms and in particular their application in complex 3D character hierarchies.