Composition of rotations
- Consider two rotation \((R_1,R_2)\) associated to their unit quaternions \((q_1,q_2)\).
- The product \(q_1 \, q_2\) represents the composition \(R_1 \circ R_2\).
Demonstration
- We show that \(\mathcal{R}_{q_1\,q_2}(v)=\mathcal{R}_{q_1}\circ\mathcal{R}_{q_2}(v)\).
- \(\mathcal{R}_{q_1\,q_2}(v)=(q_1 q_2)\;v\;(q_1 q_2)^{\star}\)
- \(\mathcal{R}_{q_1\,q_2}(v)=(q_1\,q_2)\;v\;(q_2^{\star}\,q_1^{\star})\), as \((q_1 q_2)^{\star}=q_2 ^ {\star} \,q_1^{\star}\)
- \(\mathcal{R}_{q_1\,q_2}(v)=q_1\,(q_2\;v\;q_2^{\star})\,q_1^{\star}\)
- \(\mathcal{R}_{q_1\,q_2}(v)=q_1\,\mathcal{R}_{q_2}(v)\, q_1^{\star} = \mathcal{R}_{q_1}\circ\mathcal{R}_{q_2}(v)\)