Dual Quaternions
Contents...
Dual Quaternions
In this paper, we talk about dual quaternions, what they are, and how we can use them to quickly describe complex motion of Xenaʻs powers. A dual quaternion is an ordered pair $(A, B)$ such that $A, B \in \mathbb{H}$ meaning that both $A$ and $B$ are quaternions. We represent them algebraically as…
\[\hat{A} = (A, B) \equiv A + \epsilon B\]where $\epsilon$ is an infinitesimal number such that $\epsilon^2 = 0$. We are here restricting the root we take of $\epsilon$ as positive. We will now define addition and multiplication over these special types of numbers. We canʻt define division for them since this implies some form of dividing by zero, which isnʻt well formed. Hereʻs what addition looks like when we use plain algebra…
\[\hat{A} + \hat{C} = (A + \epsilon B) + (C + \epsilon D) \\ = A + C + \epsilon(B + D) \\ \equiv (A + C, B + D)\]Now, we will define multiplication for dual quaternions using plain algebra…
\[\hat{A} \cdot \hat{C} = (A + \epsilon B)(C + \epsilon D) \\ = (AC + \epsilon AD + \epsilon BC + \epsilon^2BD) \\ = (AC + \epsilon(AD + BC)) \\ \equiv (AC, AD+BC)\]The last thing we will do is represent them in polar form…
\[\hat{A} = (A, B) = a_0 + a_1i + a_2j + a_3k + \epsilon(b_0 + b_1i + b_2j + b_3k)\\ = ||A|| e^{\hat{n}\varphi_0} + \epsilon||B||e^{\hat{n}\varphi_1} \\ = ||A|| e^{\hat{n}\varphi_0} + ||B||\epsilon e^{\hat{n}\varphi_1}\]This is all as desired. $\blacksquare$