{\varepsilon}q$ with quaternions $p$ and $q$, the *primal* and *dual part* of $h$, respectively. The conjugate dual quaternion is ${\overline{h}} = {\overline{p}} + {\varepsilon}{\overline{q}}$ and quaternions in ${\mathbb{H}}$ are conjugated by multiplying the coefficients of ${\mathbf{i}}$, ${\mathbf{j}}$, and ${\mathbf{k}}$ with $-1$. The dual quaternions with non-zero primal part are invertible. The inverse of $h$ is $h^{-1} = {N(h)}^{-1}{\overline{h}}$. Here, ${N(h)} \coloneqq h{\overline{h}} = p{\overline{p}} + {\varepsilon}(p{\overline{q}} + q{\overline{p}})$ denotes the *dual quaternion norm.* It is a dual number and, provided $a \neq 0$, the inverse of number $a + {\varepsilon}b \in {\mathbb{D}}$ is $a^{-1} - {\varepsilon}ba^{-2}$.

Denote by ${\mathbb{S}}$ the multiplicative subgroup of dual quaternions with real, nonzero norm. It acts on ${\mathbb{R}}^3 = \langle {\mathbf{i}}, {\mathbf{j}}, {\mathbf{k}}\rangle$ according to $$\label{eq:1}
  z \mapsto \frac{pz{\overline{p}}+p{\overline{q}}-q{\overline{p}}}{{N(p)}}.$$ This equation defines a homomorphism from ${\mathbb{S}}$ to ${\mathrm{SE}(3)}$. It is surjective and the kernel is the real multiplicative group ${\mathbb{R}}^\star$. Hence, there exists an isomorphism between ${\mathbb{S}}/{\mathbb{R}}^\star$ and ${\mathrm{SE}(3)}$. This is actually Study’s well-known kinematic mapping (or its inverse). Factorizing by ${\mathbb{R}}^\star$ turns ${{\mathbb{D}}{\mathbb{H}}}$ into real projective space ${\mathbb{P}}^7$ and ${\mathbb{S}}$ becomes the Study quadric ${S}\subset {\mathbb{P}}^7$ minus the exceptional three-space of classes of dual quaternions with vanishing primal part. More details can be found 