Translation
A translation is a proper isometry of Euclidean space performed by the operator
- $$\mathbf T = {\dfrac{\tau_{x\vphantom{y}}}{2} \mathbf e_{235} + \dfrac{\tau_y}{2} \mathbf e_{315} + \dfrac{\tau_z{\vphantom{y}}}{2} \mathbf e_{125} + \large\unicode{x1d7d9}}$$ .
This operator is identical to the translation operator in rigid geometric algebra but with the extra factor of $$\mathbf e_5$$. It translates an object $$\mathbf x$$ by the displacement vector $$\boldsymbol \tau = (\tau_x, \tau_y, \tau_z)$$ when used with the sandwich antiproduct $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}}$$. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\boldsymbol \tau$$, where the magnitude of $$\boldsymbol \tau/2$$ is the tangent of half the angle of rotation.
Exponential Form
A translation by a distance $$\delta$$ perpendicular to a unitized plane $$\mathbf g$$ can be expressed as an exponential of the plane's attitude as
- $$\mathbf T = \exp_\unicode{x27C7}\left(\dfrac{1}{2}\delta \operatorname{att}(\mathbf g)\right) = \dfrac{\delta}{2} \operatorname{att}(\mathbf g) + {\large\unicode{x1d7d9}}$$
Matrix Form
When a translation $$\mathbf T$$ is applied to a round point, it is equivalent to premultiplying the point by the $$5 \times 5$$ matrix
- $$\begin{bmatrix} 1 & 0 & 0 & \tau_x & 0 \\ 0 & 1 & 0 & \tau_y & 0 \\ 0 & 0 & 1 & \tau_z & 0 \\ 0 & 0 & 0 & 1 & 0 \\ \tau_x & \tau_y & \tau_z & \dfrac{\tau^2}{2} & 1 \end{bmatrix}$$ .