Quaternions and Rotations

Introduction

Although not covered in the notes on Moodle, this topic may be of interest for those who want to know more about Quaternions. This is an oversimplified explanation of Quaternions and rotations. From Addendum 3, you learned a little about the 4D hypersphere. In this addendum, this knowledge will be further explained in terms of roations with Quaternions.

Rotations using Quaternions

There are a number of special quaternions:

$(1,0,0,0)$ = identity
$(0,1,0,0)$ = pitch by 180^o ( $\pi$ )
$(0,0,1,0)$ = yaw by $\pi$
$(0,0,0,1)$ = roll by $\pi$

Changing the rotaion from $\pi$ to $\frac{\pi}{2}$ , the following quaternions are:

$\left ( \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0,0\right )$ = pitch by 90^o
$\left ( \frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}},0\right )$ = yaw by $\frac{\pi}{2}$
$\left ( \frac{1}{\sqrt{2}},0,0,\frac{1}{\sqrt{2}}\right )$ = roll by $\frac{\pi}{2}$

Equivalents & Inverses of Quaternions

$(a,b,c,d )=(-a,-b,-c,-d)$
$(a,-b,-c,-d)=(-a,b,c,d)$
$(-a,-b,-c,-d)=(-a,b,c,d)^{-1}$

Exercises & Assignments

There are no exercises or assignments associated with this Addendum.

Quaternions and Rotations

Introduction

Rotations using Quaternions

Equivalents & Inverses of Quaternions

Exercises & Assignments

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