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.
Best Answer
I can't see the page in Google Books, but what you apparently have there is the logarithm of a unit quaternion $\mathbf q$, 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 $\mathbf q=(s,\;\;v)$ is defined as
$\ln \mathbf q=\left(\ln|\mathbf q|,\;\;\left(\frac1{\|v\|}\arccos\frac{s}{|q|}\right)v\right)$
where $|\mathbf q|$ is the norm of the quaternion and $\|v\|$ is the norm of the vector part (and note that the vector part of $\ln\mathbf q$ has a scalar multiplier); 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.