VulkanLearn/DualQuaternion.md at master · vcoda/VulkanLearn · GitHub

$\large \mathbf{D}=\mathbf{p}+\varepsilon \mathbf{q}$

$\large \mathbf{D}^{*}=\mathbf{p}^{*}+\varepsilon \mathbf{q}^{*}$

$\large \mathbf{D}^{\diamond}=\mathbf{p}^{*}-\varepsilon \mathbf{q}^{*}$

$\large \mathbf{p}=\mathbf{r}$

$\large \mathbf{q}=\frac{1}{2}\mathbf{tr}$

This stands for rotation followed by a translation:

$\large \mathbf{D}^{\diamond }=\mathbf{DvD}^{\diamond }$

$\large \mathbf{v}^{'}=(\mathbf{p}+\varepsilon \mathbf{q})(1+\varepsilon \mathbf{v})(\mathbf{p}^{*}-\varepsilon \mathbf{q}^{*})$

$\large \mathbf{v}^{'}=\mathbf{pp}^{*}+\varepsilon (-\mathbf{pq}^{*}+(\mathbf{q}+\mathbf{pv})\mathbf{p}^{*})$

$\large \mathbf{v}^{'}=1+\varepsilon (\mathbf{qp}^{*}+\mathbf{pvp}^{*}-\mathbf{pq}^{*})$

$\large \mathbf{v}^{'}=1+\varepsilon (\frac{1}{2}\mathbf{trr}^{*}+\mathbf{rvr}^{*}-\frac{1}{2}\mathbf{rr}^{*}\mathbf{t}^{*})$

$\large \mathbf{v}^{'}=1+\varepsilon (\mathbf{rvr}^{*}+\mathbf{t})$

Similarly, $\large \mathbf{q}=\frac{1}{2}\mathbf{rt}$ stands for a translation followed by a rotation.