#### 4.3.2. Quaternion Transforms

We will now study a subclass of the quaternion set, namely those of unit length, called unit quaternions. The most important fact about unit quaternions is that they can represent any three-dimensional rotation, and that this representation is extremely compact and simple.

Now we will describe what makes unit quaternions so useful for rotations and orientations. First, put the four coordinates of a point or vector $p={({p}_{x}{p}_{y}{p}_{z}{p}_{w})}^{T}$ into the components of a quaternion $\widehat{p}$ , and assume that we have a unit quaternion $\widehat{q}=(sin\varphi {u}_{q},cos\varphi )$ . One can prove that

(4.43)

$$\begin{array}{c}\hfill \widehat{q}\widehat{p}{\widehat{q}}^{-1}\end{array}$$rotates $\widehat{p}$ (and thus the point p) around the axis ...

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