Quaternion

• A way of respresenting the orientation of a 3 dimension body in space, using a 4D object
• A 4-dimensional extension of the complex numbers
  • $⟹\mathbb{R}\subset \mathbb{C}\subset \mathbb{H}$
• The quaternion describes the orientation of the object’s local reference frame relative to the global reference frame

Definition

$$\begin{array}{c}\mathbf{q}=a+bi+cj+dk=a+V\\ \text{where }a,b,c,d\in \mathbb{R}\\ \text{where }i,j,k\text{ are imaginary basis vectors}\\ \text{where }V=bi+cj+dk\end{array}$$ $$\mathbb{H}=\left\{a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}|a,b,c,d\in \mathbb{R}\right\}$$

$i$ is precisely the $i$ used in complex numbers. $j$ and $k$ function the same as $i$, but are along different axes.

$$i^2=j^2=k^2=ijk=-1$$ $$ij=-ji=kjk=-kj=iki=-ik=j$$

Operations

Addition

${\mathbf{q}}_1+{\mathbf{q}}_2$

Multiplication

${\mathbf{q}}_1{\mathbf{q}}_2$

$${\mathbf{q}}_1{\mathbf{q}}_2\ne {\mathbf{q}}_2{\mathbf{q}}_1$$

$k{\mathbf{q}}_2$ $k{\mathbf{q}}_2={\mathbf{q}}_{2}k$

Norm

$\Vert \mathbf{q}\Vert$ $\Vert \mathbf{q}\Vert =\sqrt{a^2+b^2+c^2+d^2}$

Conjugation

${\mathbf{q}}^{\ast }=\bar{\mathbf{q}}$

$$(a+bi+cj+dk{)}^{\ast }=(a-bi-cj-dk)$$

Usage

Let $\mathbf{v}\in \mathbb{H}=0+xi+yj+zk$, correspond to a vector in ${\mathbb{R}}^3$. This is the vector that we want to rotate.

Let $\mathbf{q}\in \mathbb{H}=a+bi+cj+dk$, be the quaternion that we will use to manipulate $\mathbf{v}$

• Left multiplying by $\mathbf{q}$ rotates, and translates negative to the axis of rotation
• Right multiplying by ${\mathbf{q}}^{\ast }$ rotates the same amount, and translates positive to the axis of rotation
• Therefore $\mathbf{qv}{\mathbf{q}}^{\ast }$ results in that rotation, twice, with the translation cancelled out. This is the form of quaternion rotation.

Let $\hat{V}=\frac{V}{|V|}$

The following is a quaternion that rotates about $V$ by $\theta$ degrees

$$\mathbf{q}=\cos \frac{1}{2}\theta +\hat{V}\sin \frac{1}{2}\theta$$