# Quaternions vs Axis + angle

I have been trying to find the difference between the two but to no luck minus this:

The primary diff erence between the two representations is that a quaternion’s axis of rotation is scaled by the sine of the half angle of rotation, and instead of storing the angle in the fourth component of the vector, we store the cosine of the half angle.

I have no idea what

sine of the half angle of rotation

or

cosine of the half angle

means, could someone explain?

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Please use blockquotes (>  at the beginning of the line) rather than code formatting for your quotes. Also, where do you have the quotes from? – Harald Hanche-Olsen Feb 24 '12 at 10:46
I'm not conversant in quaternions, but here's an attempt If you rotate it by $\theta$, then they want $\sin\left(\frac{\theta}{2}\right)$. – Manishearth Feb 24 '12 at 10:47
Also, have you looked at Quaternions and spatial rotation on wikipedia? – Harald Hanche-Olsen Feb 24 '12 at 10:49

Let $E = \mathbb{R}^3$ be an euclidien oriented vector space, $\mathbb{H} = \mathbb{R} \oplus E$ be the quaternion algebra. Let $R_{\theta,\vec{u}}$ be a rotation of $E$ with axis $\vec{u}$ a unit vector and angle $\theta$. Then what your quote means is : The quaternion associated to $R_{\theta,\vec{u}}$ is the element $q_{\theta,\vec{u}} := (\cos(\theta/2),\sin(\theta/2)\vec{u}) \in \mathbb{H}$.
More precisely, the map $\mathbb{H} \rightarrow \mathbb{H}, z \mapsto q_{\theta,\vec{u}} z q_{\theta,\vec{u}}^{-1}$ stabilize the subspace $E$ and is equal to $R_{\theta,\vec{u}}$ on that space.