As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form $$\forall z \in \mathbb{C}, \quad f(z) = a z + b$$ with $a \in \mathbb{C}\backslash \{0\}$ and $b \in \mathbb{C}$. This gives an especially appealing way of describing and parameterizing similarities. By writing $a = r\mathrm{e}^{\mathrm{i} \theta}$ with $r >0$ and $\theta \in [0,2\pi)$, we recover that a similarity is the combination of a rotation (of angle $\theta$), a homothety (of parameter $r$), and a translation (of $b$).

I am curious about possible extension of this result in dimension 3. Of course, there is no three-dimensional space such as the complex numbers. However, it is possible to describe 3D direct similarities in terms of a combination of homotethies (around a certain point, possibly non-zero), rotations (idem), and translations. Since we can represent 3D-rotations with unit-quaternions (see https://www.youtube.com/watch?v=zjMuIxRvygQ for a nice video), I am wondering if there is a nice comparable relation to $f(z) = az + b$ above in the 3D case.

As is well-known, in dimension 2, a linear map $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is a direct similarity if, once we identify $\mathbb{R}^2$ with $\mathbb{C}$, $f$ is of the form $$\forall z \in \mathbb{C}, \quad f(z) = a z + b$$ with $a \in \mathbb{C}\backslash \{0\}$ and $b \in \mathbb{C}$. This gives an especially appealing way of describing and parameterizing similarities. By writing $a = r\mathrm{e}^{\mathrm{i} \theta}$ with $r >0$ and $\theta \in [0,2\pi)$, we recover that a similarity is the combination of a rotation (of angle $\theta$), a homothety (of parameter $r$), and a translation (of $b$).

I am curious about possible extension of this result in dimension 3. Of course, there is no three-dimensional space such as the complex numbers. However, it is possible to describe 3D direct similarities in terms of a combination of homotheties (around a certain point, possibly non-zero), rotations (idem), and translations. Since we can represent 3D-rotations with unit-quaternions (see this nice Youtube video), I am wondering if there is a nice comparable relation to $f(z) = az + b$ above in the 3D case.