From Classical Mechanics to Computer Graphics and Beyond

Abstract

This paper presents an overview of the analytical advantages of dual-quaternions and their potential in the areas of robotics, graphics, and animation. While quaternions have proven themselves as providing an unambiguous, un-cumbersome, computationally efficient method of representing rotational information, we hope after reading this paper the reader will take a parallel view on dual-quaternions. Despite the fact that the most popular method of describing rigid transforms is with homogeneous transformation matrices they can suffer from several downsides in comparison to dual-quaternions. For example, dual-quaternions offer increased computational efficiency, reduced overhead, and coordinate invariance. We also demonstrate and explain how, dual-quaternions can be used to generate constant smooth interpolation between transforms. Hence, this paper aims to provide a comprehensive step-by-step explanation of dual-quaternions, and it comprising parts (i.e., quaternions and dual-numbers) in a straightforward approach using practical real-world examples and uncomplicated implementation information. While there is a large amount of literature on the theoretical aspects of dual-quaternions there is little on the practical details. So, while giving a clear no-nonsense introduction to the theory, this paper also explains and demonstrates numerous workable aspect using real-world examples with statistical results that illustrate the power and potential of dual-quaternions.