Topic B2

Consider these important definitions that apply to all following explanations.

The quaternion set is given by

\(\mathbb{H}\triangleq\left\{ h_{1}+\imi h_{2}+\imj h_{3}+\imk h_{4}\,:\,h_{1},h_{2},h_{3},h_{4}\in\mathbb{R}\right\}\)

in which the imaginary units \(\imi\), \(\imj\), and \(\imk\) have the following properties:

\(\hat{\imath}^{2}=\hat{\jmath}^{2}=\hat{k}^{2}=\hat{\imath}\hat{\jmath}\hat{k}=-1\)

The dual quaternion set is given by

\(\mathcal{H}\triangleq\left\{ \quat h+\dual\quat h'\,:\,\quat h,\quat h'\in\mathbb{H},\,\dual^{2}=0,\,\dual\neq0\right\}\)

where \(\dual^2=0\) but \(\dual\neq0\).