# the logarithm of quaternion

I'm reading 3D math primer for graphics and game development by Fletcher Dunn and Ian Parberry. On page 170, the logarithm of quaternion is defined as

\begin{align} \log \mathbf q &= \log \left( \begin{bmatrix} \cos \alpha & \mathbf n \sin \alpha \end{bmatrix} \right) \\\\ &\equiv \begin{bmatrix} 0 & \alpha \mathbf n \end{bmatrix} \end{align}

I don't see how $\log \left( \begin{bmatrix} \cos \alpha & \mathbf n \sin \alpha \end{bmatrix} \right)$ is equal to $\begin{bmatrix} 0 & \alpha \mathbf n \end{bmatrix}$. Can anyone help me out?

Thanks.

• I don't see how you can have a log of a quat, since quat addition is commutative but multiplication not. The standard rule for logarithms, log(ab)=log(a)+log(b) would not apply except over a subset where log(ab)=log(ba). Sorry, Lie algebra is above my paygrade. There are other unreal objects that have perfectly valid logarithms, like elements of a finite field with a single generator of the multiplicative group. I have in mind the Hebrew alphabet and the practice of gematria, as it maps onto the integers 1 to 22, and arithmetic is done modulo 23. – richard1941 Jul 21 '17 at 17:54
• @richard1941 The log map is still defined even if the group operator is not commutative: see e.g. en.wikipedia.org/wiki/Logarithm_of_a_matrix#Properties – Gus May 10 at 18:39

I can't see the page in Google books, but what you apparently have there is the logarithm of a unit quaternion q, which has scalar part $\cos(\theta)$ and vector part $\sin(\theta)\vec{n}$ where $\vec{n}$ is a unit vector.

Since the logarithm of an arbitrary quaternion $q=(s\;\;v)$ is defined as

$\ln q=(\ln\,|q|\;\;\frac{v}{\|v\|}\arccos\frac{s}{|q|})$

where $|q|$ is the norm of the quaternion and $\|v\|$ is the norm of the vector part, applying that formula to a unit quaternion yields a scalar part of 0 (the logarithm of the norm of a unit quaternion is zero), and you should now be able to derive the formula for the vector part.

• Hi JM, it's possible my browser's not rendering that correctly, or maybe there's a typo in the markup, should there be a comma between ln |q| and v/|v|? Tweaking that line to $\ln q=(\ln|q|,\frac{v}{\|v\|}\arccos\frac{s}{|q|})$ makes it appear the way I think it should. (Currently it's q=(\ln\\,|q|\;\;...), which looks like parts might be double-escaped.) – John P Nov 16 '18 at 9:33
• @John, I defined the "arbitrary quaternion" to be $q=(s\;\;v)$ in the preceding sentence, so that convention is followed in the logarithm. If you prefer, you can separate the scalar and vector parts with a comma in your own notation. – J. M. is a poor mathematician Dec 31 '18 at 10:32
• Thanks, that makes sense. There's nothing wrong with minimal notation, I just didn't recognize the spacing as a part of the formatting. – John P Jan 6 at 19:32

Just recall $\exp(\alpha i) = \cos \alpha + i \sin \alpha$ for complex numbers, the quaternion (remember a quaternion is just 3 complex numbers which all have the same real part) version is by direct analogue and take logarithm of both sides.