Dual quaternion
Dual quaternions are an 8-dimensional number system (i.e. isomorphic to \(\displaystyle \mathbb{R}^8\)) which can be used to jointly represent rotations and translations in 3D space. They can be used in place of the standard \(\displaystyle 4 \times 4\) homogeneous transformation matrices.
Background
A dual quaternion can be written as \(\displaystyle \mathbf{q} = \mathbf{q}_r + \mathbf{q}_d \epsilon\).
Here, \(\displaystyle \epsilon^2=0\).
- Scalar Multiplication
\(\displaystyle s\mathbf{q} = s\mathbf{q}_r + s \mathbf{q}_d \epsilon\)
- Addition
\(\displaystyle \mathbf{q}_1 + \mathbf{q}_2 = \mathbf{q}_{r1} +\mathbf{q}_{r2} + (\mathbf{q}_{d1} + \mathbf{q}_{d2}) \epsilon\)
- Multiplication
\(\displaystyle \mathbf{q}_1 \mathbf{q}_2 = \mathbf{q}_{r1} \mathbf{q}_{r2} + (\mathbf{q}_{r1}\mathbf{q}_{d2} + \mathbf{q}_{d1} \mathbf{q}_{r2})\epsilon\).
- Conjugate
\(\displaystyle \mathbf{q}^* = \mathbf{q}_{r}^* + \mathbf{q}_{d}^*\epsilon\)
- Magnitude
\(\displaystyle \Vert \mathbf{q} \Vert = \mathbf{q}\mathbf{q}^*\)
Rotations and Translations
A translation is represented as:
\(\displaystyle \mathbf{q}_t = [1,0,0,0][0, \frac{t_x}{2}, \frac{t_y}{2}, \frac{t_z}{2}] = 1 + \frac{\epsilon}{2}\mathbf{t}\)
A rotation is represented as:
\(\displaystyle \mathbf{q}_r = [\cos(\frac{\theta}{2}), \sin(\frac{\theta}{2})n_x, \sin(\frac{\theta}{2})n_y, \sin(\frac{\theta}{2})n_z][0,0,0,0] = \cos(\frac{\theta}{2}) + \sin(\frac{\theta}{2}) \mathbf{n}\)
These can be combined as \(\displaystyle \mathbf{q} = \mathbf{q}_t * \mathbf{q}_r = \mathbf{q}_r + \frac{\epsilon}{2}\mathbf{t}\mathbf{q}_r\).
Applying the transformation to a point \(\displaystyle \mathbf{v} \in \mathbb{R}^3\) is:
\(\displaystyle \mathbf{p}' = \mathbf{q}(1 + \epsilon\mathbf{v})\mathbf{q}^*\)