I've been interested in quaternions for their ability to represent positions and motions in spacetime in application to game development; I have searched for articles regarding representations of quaternions online for a few days, and a bit of it helped, but my knowledge of higher mathematics is limited and I struggled to make sense of many of the equations shown in these articles. I realize the study of quaternions is a deep subject closely linked with trigonometry and linear algebra, but at the moment my only interest is in the polar/exponential representation of quaternions.

During my search for information on this notion, I found an article that illustrated a particular form of expressing quaternions, 'Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form' by Stephen J. Sangwine and Nicolas Le Bihan

In the introduction of this document, it says any quaternion of the form $q = a+b\hat i+c\hat j+d\hat k$ may be expressed as $q = |q|e^{\hat i\phi}e^{\hat k\psi}e^{\hat j\theta}$.

This article, 'Hypercomplex Signals—A Novel Extension of the Analytic Signal to the Multidimensional Case' by Thomas Bülow and Gerald Sommer, states the same idea in part five of the document.

I've read through these articles and others, taking what I can from them with my current insight on trigonometry, but I have yet to grasp how to correctly convert a quaternion from rectangular form $a+b\hat i+c\hat j+d\hat k$ to the portrayed exponential form $|q|e^{\hat i\phi}e^{\hat k\psi}e^{\hat j\theta}$. I feel that seeing some examples of quaternions in both forms would help me understand the conversion.

Thus, given that $q = a+b\hat i+c\hat j+d\hat k = |q|e^{\hat i\phi}e^{\hat k\psi}e^{\hat j\theta}$, what would the following quaternions be in the form $|q|e^{\hat i\phi}e^{\hat k\psi}e^{\hat j\theta}$?

  • $3+4\hat i+2\hat j+3\hat k$
  • $3\hat i+4\hat j$
  • $5\hat i+3\hat j+\hat k$

Since $\phi$, $\psi$ and $\theta$ are just Euler angles you can easily convert back and forth from quaternions to Euler angles following the formulae showed in wikipedia:



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