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