A new Approach to Hyper Dual Split Quaternions with Different Polar Representation

Abstract

Hamilton first introduced quaternions in 1843 as a way to represent rotations in three dimensional space, and since then, they have become the important tool in many fields. One advantage of quaternions over other methods of representing rotations is their ability to avoid the problem of gimbal lock, which can occur when using Euler angles. Quaternions also have a relatively simple algebraic structure and can be efficiently implemented in computer algorithms. In recent years, quaternions have been used in the development of virtual reality systems and computer games, where they are used to represent orientations of objects in three-dimensional space. They have also been applied in robotics, control theory, and signal processing. Overall, quaternions have become the valuable tools in many areas of mathematics and engineering, and their usage continue to expand. Sangwine and Bihan introduced a quaternion polar representation that draws inspiration from the Cayley-Dickson form. In their formulation, they express quaternions using a complex modulus and argument. The Cayley-Dickson construction is a mathematical procedure that extends the concept of complex numbers to higher dimensions, paving the way for the development of quaternions. On the other hand, the complex argument represents the direction or orientation of the quaternion in a manner analogous to the argument of a complex number. This approach provides a concise and insightful way to represent quaternions, offering a geometric interpretation that aligns with the principles of complex analysis.